| Step |
Hyp |
Ref |
Expression |
| 1 |
|
bdayfinbndlem.1 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0s ) |
| 2 |
|
bdayfinbndlem.2 |
⊢ ( 𝜑 → ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ) |
| 3 |
|
bdayn0p1 |
⊢ ( 𝑁 ∈ ℕ0s → ( bday ‘ ( 𝑁 +s 1s ) ) = suc ( bday ‘ 𝑁 ) ) |
| 4 |
1 3
|
syl |
⊢ ( 𝜑 → ( bday ‘ ( 𝑁 +s 1s ) ) = suc ( bday ‘ 𝑁 ) ) |
| 5 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( bday ‘ ( 𝑁 +s 1s ) ) = suc ( bday ‘ 𝑁 ) ) |
| 6 |
5
|
sseq2d |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑁 +s 1s ) ) ↔ ( bday ‘ 𝑤 ) ⊆ suc ( bday ‘ 𝑁 ) ) ) |
| 7 |
|
bdayelon |
⊢ ( bday ‘ 𝑤 ) ∈ On |
| 8 |
|
bdayelon |
⊢ ( bday ‘ 𝑁 ) ∈ On |
| 9 |
8
|
onsuci |
⊢ suc ( bday ‘ 𝑁 ) ∈ On |
| 10 |
|
onsseleq |
⊢ ( ( ( bday ‘ 𝑤 ) ∈ On ∧ suc ( bday ‘ 𝑁 ) ∈ On ) → ( ( bday ‘ 𝑤 ) ⊆ suc ( bday ‘ 𝑁 ) ↔ ( ( bday ‘ 𝑤 ) ∈ suc ( bday ‘ 𝑁 ) ∨ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) ) ) |
| 11 |
7 9 10
|
mp2an |
⊢ ( ( bday ‘ 𝑤 ) ⊆ suc ( bday ‘ 𝑁 ) ↔ ( ( bday ‘ 𝑤 ) ∈ suc ( bday ‘ 𝑁 ) ∨ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) ) |
| 12 |
|
onsssuc |
⊢ ( ( ( bday ‘ 𝑤 ) ∈ On ∧ ( bday ‘ 𝑁 ) ∈ On ) → ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑤 ) ∈ suc ( bday ‘ 𝑁 ) ) ) |
| 13 |
7 8 12
|
mp2an |
⊢ ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑤 ) ∈ suc ( bday ‘ 𝑁 ) ) |
| 14 |
13
|
orbi1i |
⊢ ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∨ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) ↔ ( ( bday ‘ 𝑤 ) ∈ suc ( bday ‘ 𝑁 ) ∨ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) ) |
| 15 |
14
|
bicomi |
⊢ ( ( ( bday ‘ 𝑤 ) ∈ suc ( bday ‘ 𝑁 ) ∨ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) ↔ ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∨ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) ) |
| 16 |
11 15
|
bitri |
⊢ ( ( bday ‘ 𝑤 ) ⊆ suc ( bday ‘ 𝑁 ) ↔ ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∨ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) ) |
| 17 |
|
fveq2 |
⊢ ( 𝑧 = 𝑤 → ( bday ‘ 𝑧 ) = ( bday ‘ 𝑤 ) ) |
| 18 |
17
|
sseq1d |
⊢ ( 𝑧 = 𝑤 → ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ) ) |
| 19 |
|
breq2 |
⊢ ( 𝑧 = 𝑤 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑤 ) ) |
| 20 |
18 19
|
anbi12d |
⊢ ( 𝑧 = 𝑤 → ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) ↔ ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) ) ) |
| 21 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑤 → ( 𝑧 = 𝑁 ↔ 𝑤 = 𝑁 ) ) |
| 22 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑤 → ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) ) |
| 23 |
22
|
3anbi1d |
⊢ ( 𝑧 = 𝑤 → ( ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) |
| 24 |
23
|
rexbidv |
⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) |
| 25 |
24
|
2rexbidv |
⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) |
| 26 |
21 25
|
orbi12d |
⊢ ( 𝑧 = 𝑤 → ( ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ↔ ( 𝑤 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ) |
| 27 |
20 26
|
imbi12d |
⊢ ( 𝑧 = 𝑤 → ( ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ↔ ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ) ) |
| 28 |
27
|
rspccva |
⊢ ( ( ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ) |
| 29 |
2 28
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ) |
| 30 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → 𝑁 ∈ ℕ0s ) |
| 31 |
|
0n0s |
⊢ 0s ∈ ℕ0s |
| 32 |
31
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → 0s ∈ ℕ0s ) |
| 33 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → 𝑤 = 𝑁 ) |
| 34 |
1
|
n0snod |
⊢ ( 𝜑 → 𝑁 ∈ No ) |
| 35 |
34
|
addsridd |
⊢ ( 𝜑 → ( 𝑁 +s 0s ) = 𝑁 ) |
| 36 |
35
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → ( 𝑁 +s 0s ) = 𝑁 ) |
| 37 |
33 36
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → 𝑤 = ( 𝑁 +s 0s ) ) |
| 38 |
|
0slt1s |
⊢ 0s <s 1s |
| 39 |
38
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → 0s <s 1s ) |
| 40 |
34
|
sltp1d |
⊢ ( 𝜑 → 𝑁 <s ( 𝑁 +s 1s ) ) |
| 41 |
35 40
|
eqbrtrd |
⊢ ( 𝜑 → ( 𝑁 +s 0s ) <s ( 𝑁 +s 1s ) ) |
| 42 |
41
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → ( 𝑁 +s 0s ) <s ( 𝑁 +s 1s ) ) |
| 43 |
|
oveq1 |
⊢ ( 𝑎 = 𝑁 → ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑁 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) |
| 44 |
43
|
eqeq2d |
⊢ ( 𝑎 = 𝑁 → ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑁 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) ) |
| 45 |
|
oveq1 |
⊢ ( 𝑎 = 𝑁 → ( 𝑎 +s 𝑞 ) = ( 𝑁 +s 𝑞 ) ) |
| 46 |
45
|
breq1d |
⊢ ( 𝑎 = 𝑁 → ( ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑁 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 47 |
44 46
|
3anbi13d |
⊢ ( 𝑎 = 𝑁 → ( ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑁 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑁 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 48 |
|
oveq1 |
⊢ ( 𝑏 = 0s → ( 𝑏 /su ( 2s ↑s 𝑞 ) ) = ( 0s /su ( 2s ↑s 𝑞 ) ) ) |
| 49 |
48
|
oveq2d |
⊢ ( 𝑏 = 0s → ( 𝑁 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑁 +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ) |
| 50 |
49
|
eqeq2d |
⊢ ( 𝑏 = 0s → ( 𝑤 = ( 𝑁 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑁 +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ) ) |
| 51 |
|
breq1 |
⊢ ( 𝑏 = 0s → ( 𝑏 <s ( 2s ↑s 𝑞 ) ↔ 0s <s ( 2s ↑s 𝑞 ) ) ) |
| 52 |
50 51
|
3anbi12d |
⊢ ( 𝑏 = 0s → ( ( 𝑤 = ( 𝑁 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑁 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑁 +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ∧ 0s <s ( 2s ↑s 𝑞 ) ∧ ( 𝑁 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 53 |
|
oveq2 |
⊢ ( 𝑞 = 0s → ( 2s ↑s 𝑞 ) = ( 2s ↑s 0s ) ) |
| 54 |
|
2sno |
⊢ 2s ∈ No |
| 55 |
|
exps0 |
⊢ ( 2s ∈ No → ( 2s ↑s 0s ) = 1s ) |
| 56 |
54 55
|
ax-mp |
⊢ ( 2s ↑s 0s ) = 1s |
| 57 |
53 56
|
eqtrdi |
⊢ ( 𝑞 = 0s → ( 2s ↑s 𝑞 ) = 1s ) |
| 58 |
57
|
oveq2d |
⊢ ( 𝑞 = 0s → ( 0s /su ( 2s ↑s 𝑞 ) ) = ( 0s /su 1s ) ) |
| 59 |
|
0sno |
⊢ 0s ∈ No |
| 60 |
|
divs1 |
⊢ ( 0s ∈ No → ( 0s /su 1s ) = 0s ) |
| 61 |
59 60
|
ax-mp |
⊢ ( 0s /su 1s ) = 0s |
| 62 |
58 61
|
eqtrdi |
⊢ ( 𝑞 = 0s → ( 0s /su ( 2s ↑s 𝑞 ) ) = 0s ) |
| 63 |
62
|
oveq2d |
⊢ ( 𝑞 = 0s → ( 𝑁 +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑁 +s 0s ) ) |
| 64 |
63
|
eqeq2d |
⊢ ( 𝑞 = 0s → ( 𝑤 = ( 𝑁 +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑁 +s 0s ) ) ) |
| 65 |
57
|
breq2d |
⊢ ( 𝑞 = 0s → ( 0s <s ( 2s ↑s 𝑞 ) ↔ 0s <s 1s ) ) |
| 66 |
|
oveq2 |
⊢ ( 𝑞 = 0s → ( 𝑁 +s 𝑞 ) = ( 𝑁 +s 0s ) ) |
| 67 |
66
|
breq1d |
⊢ ( 𝑞 = 0s → ( ( 𝑁 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑁 +s 0s ) <s ( 𝑁 +s 1s ) ) ) |
| 68 |
64 65 67
|
3anbi123d |
⊢ ( 𝑞 = 0s → ( ( 𝑤 = ( 𝑁 +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ∧ 0s <s ( 2s ↑s 𝑞 ) ∧ ( 𝑁 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑁 +s 0s ) ∧ 0s <s 1s ∧ ( 𝑁 +s 0s ) <s ( 𝑁 +s 1s ) ) ) ) |
| 69 |
47 52 68
|
rspc3ev |
⊢ ( ( ( 𝑁 ∈ ℕ0s ∧ 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s ) ∧ ( 𝑤 = ( 𝑁 +s 0s ) ∧ 0s <s 1s ∧ ( 𝑁 +s 0s ) <s ( 𝑁 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 70 |
30 32 32 37 39 42 69
|
syl33anc |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 71 |
70
|
expr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( 𝑤 = 𝑁 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 72 |
|
idd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) → 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) ) |
| 73 |
|
idd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( 𝑦 <s ( 2s ↑s 𝑝 ) → 𝑦 <s ( 2s ↑s 𝑝 ) ) ) |
| 74 |
|
n0addscl |
⊢ ( ( 𝑥 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) → ( 𝑥 +s 𝑝 ) ∈ ℕ0s ) |
| 75 |
74
|
n0snod |
⊢ ( ( 𝑥 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) → ( 𝑥 +s 𝑝 ) ∈ No ) |
| 76 |
75
|
3adant2 |
⊢ ( ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) → ( 𝑥 +s 𝑝 ) ∈ No ) |
| 77 |
76
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( 𝑥 +s 𝑝 ) ∈ No ) |
| 78 |
77
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ( 𝑥 +s 𝑝 ) ∈ No ) |
| 79 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → 𝑁 ∈ No ) |
| 80 |
79
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → 𝑁 ∈ No ) |
| 81 |
|
peano2no |
⊢ ( 𝑁 ∈ No → ( 𝑁 +s 1s ) ∈ No ) |
| 82 |
80 81
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ( 𝑁 +s 1s ) ∈ No ) |
| 83 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ( 𝑥 +s 𝑝 ) <s 𝑁 ) |
| 84 |
79
|
sltp1d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → 𝑁 <s ( 𝑁 +s 1s ) ) |
| 85 |
84
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → 𝑁 <s ( 𝑁 +s 1s ) ) |
| 86 |
78 80 82 83 85
|
slttrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) |
| 87 |
86
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( ( 𝑥 +s 𝑝 ) <s 𝑁 → ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) ) |
| 88 |
72 73 87
|
3anim123d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 89 |
|
oveq1 |
⊢ ( 𝑎 = 𝑥 → ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑥 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) |
| 90 |
89
|
eqeq2d |
⊢ ( 𝑎 = 𝑥 → ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑥 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) ) |
| 91 |
|
oveq1 |
⊢ ( 𝑎 = 𝑥 → ( 𝑎 +s 𝑞 ) = ( 𝑥 +s 𝑞 ) ) |
| 92 |
91
|
breq1d |
⊢ ( 𝑎 = 𝑥 → ( ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑥 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 93 |
90 92
|
3anbi13d |
⊢ ( 𝑎 = 𝑥 → ( ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑥 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑥 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 94 |
|
oveq1 |
⊢ ( 𝑏 = 𝑦 → ( 𝑏 /su ( 2s ↑s 𝑞 ) ) = ( 𝑦 /su ( 2s ↑s 𝑞 ) ) ) |
| 95 |
94
|
oveq2d |
⊢ ( 𝑏 = 𝑦 → ( 𝑥 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑞 ) ) ) ) |
| 96 |
95
|
eqeq2d |
⊢ ( 𝑏 = 𝑦 → ( 𝑤 = ( 𝑥 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑞 ) ) ) ) ) |
| 97 |
|
breq1 |
⊢ ( 𝑏 = 𝑦 → ( 𝑏 <s ( 2s ↑s 𝑞 ) ↔ 𝑦 <s ( 2s ↑s 𝑞 ) ) ) |
| 98 |
96 97
|
3anbi12d |
⊢ ( 𝑏 = 𝑦 → ( ( 𝑤 = ( 𝑥 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑥 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑥 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 99 |
|
oveq2 |
⊢ ( 𝑞 = 𝑝 → ( 2s ↑s 𝑞 ) = ( 2s ↑s 𝑝 ) ) |
| 100 |
99
|
oveq2d |
⊢ ( 𝑞 = 𝑝 → ( 𝑦 /su ( 2s ↑s 𝑞 ) ) = ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) |
| 101 |
100
|
oveq2d |
⊢ ( 𝑞 = 𝑝 → ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) |
| 102 |
101
|
eqeq2d |
⊢ ( 𝑞 = 𝑝 → ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) ) |
| 103 |
99
|
breq2d |
⊢ ( 𝑞 = 𝑝 → ( 𝑦 <s ( 2s ↑s 𝑞 ) ↔ 𝑦 <s ( 2s ↑s 𝑝 ) ) ) |
| 104 |
|
oveq2 |
⊢ ( 𝑞 = 𝑝 → ( 𝑥 +s 𝑞 ) = ( 𝑥 +s 𝑝 ) ) |
| 105 |
104
|
breq1d |
⊢ ( 𝑞 = 𝑝 → ( ( 𝑥 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) ) |
| 106 |
102 103 105
|
3anbi123d |
⊢ ( 𝑞 = 𝑝 → ( ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑥 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 107 |
93 98 106
|
rspc3ev |
⊢ ( ( ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ∧ ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 108 |
107
|
ex |
⊢ ( ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) → ( ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 109 |
108
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 110 |
88 109
|
syld |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 111 |
110
|
rexlimdvvva |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 112 |
111
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 113 |
71 112
|
jaod |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( 𝑤 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 114 |
29 113
|
syld |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 115 |
114
|
impr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 116 |
115
|
olcd |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) ) ) → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 117 |
116
|
expr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) |
| 118 |
117
|
expd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) → ( 0s ≤s 𝑤 → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) ) |
| 119 |
5
|
eqeq2d |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ↔ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) ) |
| 120 |
|
df-ne |
⊢ ( 𝑤 ≠ ( 𝑁 +s 1s ) ↔ ¬ 𝑤 = ( 𝑁 +s 1s ) ) |
| 121 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → 𝑤 ∈ No ) |
| 122 |
|
sleloe |
⊢ ( ( 0s ∈ No ∧ 𝑤 ∈ No ) → ( 0s ≤s 𝑤 ↔ ( 0s <s 𝑤 ∨ 0s = 𝑤 ) ) ) |
| 123 |
59 121 122
|
sylancr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( 0s ≤s 𝑤 ↔ ( 0s <s 𝑤 ∨ 0s = 𝑤 ) ) ) |
| 124 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → 𝑤 ∈ No ) |
| 125 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → 0s <s 𝑤 ) |
| 126 |
124 125
|
0elleft |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → 0s ∈ ( L ‘ 𝑤 ) ) |
| 127 |
126
|
ne0d |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( L ‘ 𝑤 ) ≠ ∅ ) |
| 128 |
|
simprrl |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 129 |
128
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 130 |
|
peano2n0s |
⊢ ( 𝑁 ∈ ℕ0s → ( 𝑁 +s 1s ) ∈ ℕ0s ) |
| 131 |
1 130
|
syl |
⊢ ( 𝜑 → ( 𝑁 +s 1s ) ∈ ℕ0s ) |
| 132 |
|
n0sbday |
⊢ ( ( 𝑁 +s 1s ) ∈ ℕ0s → ( bday ‘ ( 𝑁 +s 1s ) ) ∈ ω ) |
| 133 |
131 132
|
syl |
⊢ ( 𝜑 → ( bday ‘ ( 𝑁 +s 1s ) ) ∈ ω ) |
| 134 |
133
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ∈ ω ) |
| 135 |
134
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( bday ‘ ( 𝑁 +s 1s ) ) ∈ ω ) |
| 136 |
129 135
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( bday ‘ 𝑤 ) ∈ ω ) |
| 137 |
|
oldfi |
⊢ ( ( bday ‘ 𝑤 ) ∈ ω → ( O ‘ ( bday ‘ 𝑤 ) ) ∈ Fin ) |
| 138 |
136 137
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( O ‘ ( bday ‘ 𝑤 ) ) ∈ Fin ) |
| 139 |
|
leftssold |
⊢ ( L ‘ 𝑤 ) ⊆ ( O ‘ ( bday ‘ 𝑤 ) ) |
| 140 |
|
ssfi |
⊢ ( ( ( O ‘ ( bday ‘ 𝑤 ) ) ∈ Fin ∧ ( L ‘ 𝑤 ) ⊆ ( O ‘ ( bday ‘ 𝑤 ) ) ) → ( L ‘ 𝑤 ) ∈ Fin ) |
| 141 |
138 139 140
|
sylancl |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( L ‘ 𝑤 ) ∈ Fin ) |
| 142 |
|
leftssno |
⊢ ( L ‘ 𝑤 ) ⊆ No |
| 143 |
|
sltso |
⊢ <s Or No |
| 144 |
|
soss |
⊢ ( ( L ‘ 𝑤 ) ⊆ No → ( <s Or No → <s Or ( L ‘ 𝑤 ) ) ) |
| 145 |
142 143 144
|
mp2 |
⊢ <s Or ( L ‘ 𝑤 ) |
| 146 |
|
fimax2g |
⊢ ( ( <s Or ( L ‘ 𝑤 ) ∧ ( L ‘ 𝑤 ) ∈ Fin ∧ ( L ‘ 𝑤 ) ≠ ∅ ) → ∃ 𝑐 ∈ ( L ‘ 𝑤 ) ∀ 𝑒 ∈ ( L ‘ 𝑤 ) ¬ 𝑐 <s 𝑒 ) |
| 147 |
145 146
|
mp3an1 |
⊢ ( ( ( L ‘ 𝑤 ) ∈ Fin ∧ ( L ‘ 𝑤 ) ≠ ∅ ) → ∃ 𝑐 ∈ ( L ‘ 𝑤 ) ∀ 𝑒 ∈ ( L ‘ 𝑤 ) ¬ 𝑐 <s 𝑒 ) |
| 148 |
142
|
sseli |
⊢ ( 𝑒 ∈ ( L ‘ 𝑤 ) → 𝑒 ∈ No ) |
| 149 |
142
|
sseli |
⊢ ( 𝑐 ∈ ( L ‘ 𝑤 ) → 𝑐 ∈ No ) |
| 150 |
|
slenlt |
⊢ ( ( 𝑒 ∈ No ∧ 𝑐 ∈ No ) → ( 𝑒 ≤s 𝑐 ↔ ¬ 𝑐 <s 𝑒 ) ) |
| 151 |
148 149 150
|
syl2an |
⊢ ( ( 𝑒 ∈ ( L ‘ 𝑤 ) ∧ 𝑐 ∈ ( L ‘ 𝑤 ) ) → ( 𝑒 ≤s 𝑐 ↔ ¬ 𝑐 <s 𝑒 ) ) |
| 152 |
151
|
ancoms |
⊢ ( ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ 𝑒 ∈ ( L ‘ 𝑤 ) ) → ( 𝑒 ≤s 𝑐 ↔ ¬ 𝑐 <s 𝑒 ) ) |
| 153 |
152
|
ralbidva |
⊢ ( 𝑐 ∈ ( L ‘ 𝑤 ) → ( ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ↔ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) ¬ 𝑐 <s 𝑒 ) ) |
| 154 |
153
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ 𝑐 ∈ ( L ‘ 𝑤 ) ) → ( ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ↔ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) ¬ 𝑐 <s 𝑒 ) ) |
| 155 |
|
rightssold |
⊢ ( R ‘ 𝑤 ) ⊆ ( O ‘ ( bday ‘ 𝑤 ) ) |
| 156 |
|
ssfi |
⊢ ( ( ( O ‘ ( bday ‘ 𝑤 ) ) ∈ Fin ∧ ( R ‘ 𝑤 ) ⊆ ( O ‘ ( bday ‘ 𝑤 ) ) ) → ( R ‘ 𝑤 ) ∈ Fin ) |
| 157 |
138 155 156
|
sylancl |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( R ‘ 𝑤 ) ∈ Fin ) |
| 158 |
157
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ( R ‘ 𝑤 ) ∈ Fin ) |
| 159 |
124
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → 𝑤 ∈ No ) |
| 160 |
|
simprrr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → 𝑤 ≠ ( 𝑁 +s 1s ) ) |
| 161 |
160
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → 𝑤 ≠ ( 𝑁 +s 1s ) ) |
| 162 |
161
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → 𝑤 ≠ ( 𝑁 +s 1s ) ) |
| 163 |
162
|
neneqd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ¬ 𝑤 = ( 𝑁 +s 1s ) ) |
| 164 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑤 ∈ Ons ) → 𝑤 ∈ Ons ) |
| 165 |
131
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑤 ∈ Ons ) → ( 𝑁 +s 1s ) ∈ ℕ0s ) |
| 166 |
|
n0ons |
⊢ ( ( 𝑁 +s 1s ) ∈ ℕ0s → ( 𝑁 +s 1s ) ∈ Ons ) |
| 167 |
165 166
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑤 ∈ Ons ) → ( 𝑁 +s 1s ) ∈ Ons ) |
| 168 |
129
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 169 |
168
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑤 ∈ Ons ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 170 |
|
bday11on |
⊢ ( ( 𝑤 ∈ Ons ∧ ( 𝑁 +s 1s ) ∈ Ons ∧ ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) → 𝑤 = ( 𝑁 +s 1s ) ) |
| 171 |
164 167 169 170
|
syl3anc |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑤 ∈ Ons ) → 𝑤 = ( 𝑁 +s 1s ) ) |
| 172 |
163 171
|
mtand |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ¬ 𝑤 ∈ Ons ) |
| 173 |
|
elons |
⊢ ( 𝑤 ∈ Ons ↔ ( 𝑤 ∈ No ∧ ( R ‘ 𝑤 ) = ∅ ) ) |
| 174 |
173
|
notbii |
⊢ ( ¬ 𝑤 ∈ Ons ↔ ¬ ( 𝑤 ∈ No ∧ ( R ‘ 𝑤 ) = ∅ ) ) |
| 175 |
|
imnan |
⊢ ( ( 𝑤 ∈ No → ¬ ( R ‘ 𝑤 ) = ∅ ) ↔ ¬ ( 𝑤 ∈ No ∧ ( R ‘ 𝑤 ) = ∅ ) ) |
| 176 |
174 175
|
bitr4i |
⊢ ( ¬ 𝑤 ∈ Ons ↔ ( 𝑤 ∈ No → ¬ ( R ‘ 𝑤 ) = ∅ ) ) |
| 177 |
172 176
|
sylib |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ( 𝑤 ∈ No → ¬ ( R ‘ 𝑤 ) = ∅ ) ) |
| 178 |
159 177
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ¬ ( R ‘ 𝑤 ) = ∅ ) |
| 179 |
178
|
neqned |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ( R ‘ 𝑤 ) ≠ ∅ ) |
| 180 |
|
rightssno |
⊢ ( R ‘ 𝑤 ) ⊆ No |
| 181 |
|
soss |
⊢ ( ( R ‘ 𝑤 ) ⊆ No → ( <s Or No → <s Or ( R ‘ 𝑤 ) ) ) |
| 182 |
180 143 181
|
mp2 |
⊢ <s Or ( R ‘ 𝑤 ) |
| 183 |
|
fimin2g |
⊢ ( ( <s Or ( R ‘ 𝑤 ) ∧ ( R ‘ 𝑤 ) ∈ Fin ∧ ( R ‘ 𝑤 ) ≠ ∅ ) → ∃ 𝑑 ∈ ( R ‘ 𝑤 ) ∀ 𝑓 ∈ ( R ‘ 𝑤 ) ¬ 𝑓 <s 𝑑 ) |
| 184 |
182 183
|
mp3an1 |
⊢ ( ( ( R ‘ 𝑤 ) ∈ Fin ∧ ( R ‘ 𝑤 ) ≠ ∅ ) → ∃ 𝑑 ∈ ( R ‘ 𝑤 ) ∀ 𝑓 ∈ ( R ‘ 𝑤 ) ¬ 𝑓 <s 𝑑 ) |
| 185 |
180
|
sseli |
⊢ ( 𝑑 ∈ ( R ‘ 𝑤 ) → 𝑑 ∈ No ) |
| 186 |
180
|
sseli |
⊢ ( 𝑓 ∈ ( R ‘ 𝑤 ) → 𝑓 ∈ No ) |
| 187 |
|
slenlt |
⊢ ( ( 𝑑 ∈ No ∧ 𝑓 ∈ No ) → ( 𝑑 ≤s 𝑓 ↔ ¬ 𝑓 <s 𝑑 ) ) |
| 188 |
185 186 187
|
syl2an |
⊢ ( ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ 𝑓 ∈ ( R ‘ 𝑤 ) ) → ( 𝑑 ≤s 𝑓 ↔ ¬ 𝑓 <s 𝑑 ) ) |
| 189 |
188
|
ralbidva |
⊢ ( 𝑑 ∈ ( R ‘ 𝑤 ) → ( ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ↔ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) ¬ 𝑓 <s 𝑑 ) ) |
| 190 |
189
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑑 ∈ ( R ‘ 𝑤 ) ) → ( ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ↔ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) ¬ 𝑓 <s 𝑑 ) ) |
| 191 |
|
simp2l |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑐 ∈ ( L ‘ 𝑤 ) ) |
| 192 |
|
simp2r |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) |
| 193 |
|
simp3l |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑑 ∈ ( R ‘ 𝑤 ) ) |
| 194 |
|
simp3r |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) |
| 195 |
191 192 193 194
|
cutminmax |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) |
| 196 |
|
simpl2l |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑐 ∈ ( L ‘ 𝑤 ) ) |
| 197 |
139 196
|
sselid |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑐 ∈ ( O ‘ ( bday ‘ 𝑤 ) ) ) |
| 198 |
142 196
|
sselid |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑐 ∈ No ) |
| 199 |
|
oldbday |
⊢ ( ( ( bday ‘ 𝑤 ) ∈ On ∧ 𝑐 ∈ No ) → ( 𝑐 ∈ ( O ‘ ( bday ‘ 𝑤 ) ) ↔ ( bday ‘ 𝑐 ) ∈ ( bday ‘ 𝑤 ) ) ) |
| 200 |
7 198 199
|
sylancr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( 𝑐 ∈ ( O ‘ ( bday ‘ 𝑤 ) ) ↔ ( bday ‘ 𝑐 ) ∈ ( bday ‘ 𝑤 ) ) ) |
| 201 |
197 200
|
mpbid |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑐 ) ∈ ( bday ‘ 𝑤 ) ) |
| 202 |
129
|
3ad2ant1 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 203 |
202
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 204 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → 𝑁 ∈ ℕ0s ) |
| 205 |
204
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → 𝑁 ∈ ℕ0s ) |
| 206 |
205
|
3ad2ant1 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑁 ∈ ℕ0s ) |
| 207 |
206
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑁 ∈ ℕ0s ) |
| 208 |
207 3
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) = suc ( bday ‘ 𝑁 ) ) |
| 209 |
203 208
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) |
| 210 |
201 209
|
eleqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑐 ) ∈ suc ( bday ‘ 𝑁 ) ) |
| 211 |
|
bdayelon |
⊢ ( bday ‘ 𝑐 ) ∈ On |
| 212 |
|
onsssuc |
⊢ ( ( ( bday ‘ 𝑐 ) ∈ On ∧ ( bday ‘ 𝑁 ) ∈ On ) → ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑐 ) ∈ suc ( bday ‘ 𝑁 ) ) ) |
| 213 |
211 8 212
|
mp2an |
⊢ ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑐 ) ∈ suc ( bday ‘ 𝑁 ) ) |
| 214 |
210 213
|
sylibr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ) |
| 215 |
|
breq1 |
⊢ ( 𝑒 = 0s → ( 𝑒 ≤s 𝑐 ↔ 0s ≤s 𝑐 ) ) |
| 216 |
|
simpl2r |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) |
| 217 |
|
simpl1 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ) |
| 218 |
217 126
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 0s ∈ ( L ‘ 𝑤 ) ) |
| 219 |
215 216 218
|
rspcdva |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 0s ≤s 𝑐 ) |
| 220 |
|
fveq2 |
⊢ ( 𝑧 = 𝑐 → ( bday ‘ 𝑧 ) = ( bday ‘ 𝑐 ) ) |
| 221 |
220
|
sseq1d |
⊢ ( 𝑧 = 𝑐 → ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ) ) |
| 222 |
|
breq2 |
⊢ ( 𝑧 = 𝑐 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑐 ) ) |
| 223 |
221 222
|
anbi12d |
⊢ ( 𝑧 = 𝑐 → ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) ↔ ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑐 ) ) ) |
| 224 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑐 → ( 𝑧 = 𝑁 ↔ 𝑐 = 𝑁 ) ) |
| 225 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑐 → ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) ) |
| 226 |
225
|
3anbi1d |
⊢ ( 𝑧 = 𝑐 → ( ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) |
| 227 |
226
|
rexbidv |
⊢ ( 𝑧 = 𝑐 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) |
| 228 |
227
|
2rexbidv |
⊢ ( 𝑧 = 𝑐 → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) |
| 229 |
|
oveq1 |
⊢ ( 𝑥 = 𝑔 → ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) |
| 230 |
229
|
eqeq2d |
⊢ ( 𝑥 = 𝑔 → ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) ) |
| 231 |
|
oveq1 |
⊢ ( 𝑥 = 𝑔 → ( 𝑥 +s 𝑝 ) = ( 𝑔 +s 𝑝 ) ) |
| 232 |
231
|
breq1d |
⊢ ( 𝑥 = 𝑔 → ( ( 𝑥 +s 𝑝 ) <s 𝑁 ↔ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ) |
| 233 |
230 232
|
3anbi13d |
⊢ ( 𝑥 = 𝑔 → ( ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ) ) |
| 234 |
233
|
rexbidv |
⊢ ( 𝑥 = 𝑔 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ) ) |
| 235 |
|
oveq1 |
⊢ ( 𝑦 = ℎ → ( 𝑦 /su ( 2s ↑s 𝑝 ) ) = ( ℎ /su ( 2s ↑s 𝑝 ) ) ) |
| 236 |
235
|
oveq2d |
⊢ ( 𝑦 = ℎ → ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) ) |
| 237 |
236
|
eqeq2d |
⊢ ( 𝑦 = ℎ → ( 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) ) ) |
| 238 |
|
breq1 |
⊢ ( 𝑦 = ℎ → ( 𝑦 <s ( 2s ↑s 𝑝 ) ↔ ℎ <s ( 2s ↑s 𝑝 ) ) ) |
| 239 |
237 238
|
3anbi12d |
⊢ ( 𝑦 = ℎ → ( ( 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ) ) |
| 240 |
239
|
rexbidv |
⊢ ( 𝑦 = ℎ → ( ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ) ) |
| 241 |
|
oveq2 |
⊢ ( 𝑝 = 𝑖 → ( 2s ↑s 𝑝 ) = ( 2s ↑s 𝑖 ) ) |
| 242 |
241
|
oveq2d |
⊢ ( 𝑝 = 𝑖 → ( ℎ /su ( 2s ↑s 𝑝 ) ) = ( ℎ /su ( 2s ↑s 𝑖 ) ) ) |
| 243 |
242
|
oveq2d |
⊢ ( 𝑝 = 𝑖 → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 244 |
243
|
eqeq2d |
⊢ ( 𝑝 = 𝑖 → ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) ) |
| 245 |
241
|
breq2d |
⊢ ( 𝑝 = 𝑖 → ( ℎ <s ( 2s ↑s 𝑝 ) ↔ ℎ <s ( 2s ↑s 𝑖 ) ) ) |
| 246 |
|
oveq2 |
⊢ ( 𝑝 = 𝑖 → ( 𝑔 +s 𝑝 ) = ( 𝑔 +s 𝑖 ) ) |
| 247 |
246
|
breq1d |
⊢ ( 𝑝 = 𝑖 → ( ( 𝑔 +s 𝑝 ) <s 𝑁 ↔ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) |
| 248 |
244 245 247
|
3anbi123d |
⊢ ( 𝑝 = 𝑖 → ( ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) |
| 249 |
248
|
cbvrexvw |
⊢ ( ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) |
| 250 |
240 249
|
bitrdi |
⊢ ( 𝑦 = ℎ → ( ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) |
| 251 |
234 250
|
cbvrex2vw |
⊢ ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) |
| 252 |
228 251
|
bitrdi |
⊢ ( 𝑧 = 𝑐 → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) |
| 253 |
224 252
|
orbi12d |
⊢ ( 𝑧 = 𝑐 → ( ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ↔ ( 𝑐 = 𝑁 ∨ ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ) |
| 254 |
223 253
|
imbi12d |
⊢ ( 𝑧 = 𝑐 → ( ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ↔ ( ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑐 ) → ( 𝑐 = 𝑁 ∨ ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ) ) |
| 255 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ) |
| 256 |
255
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ) |
| 257 |
256
|
3ad2ant1 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ) |
| 258 |
257
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ) |
| 259 |
254 258 198
|
rspcdva |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑐 ) → ( 𝑐 = 𝑁 ∨ ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ) |
| 260 |
|
simp1ll |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝜑 ) |
| 261 |
260
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝜑 ) |
| 262 |
|
n0ons |
⊢ ( 𝑁 ∈ ℕ0s → 𝑁 ∈ Ons ) |
| 263 |
1 262
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ Ons ) |
| 264 |
261 263
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑁 ∈ Ons ) |
| 265 |
|
simpl3l |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑑 ∈ ( R ‘ 𝑤 ) ) |
| 266 |
155 265
|
sselid |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑑 ∈ ( O ‘ ( bday ‘ 𝑤 ) ) ) |
| 267 |
|
oldbdayim |
⊢ ( 𝑑 ∈ ( O ‘ ( bday ‘ 𝑤 ) ) → ( bday ‘ 𝑑 ) ∈ ( bday ‘ 𝑤 ) ) |
| 268 |
266 267
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑑 ) ∈ ( bday ‘ 𝑤 ) ) |
| 269 |
268 209
|
eleqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑑 ) ∈ suc ( bday ‘ 𝑁 ) ) |
| 270 |
|
bdayelon |
⊢ ( bday ‘ 𝑑 ) ∈ On |
| 271 |
|
onsssuc |
⊢ ( ( ( bday ‘ 𝑑 ) ∈ On ∧ ( bday ‘ 𝑁 ) ∈ On ) → ( ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑑 ) ∈ suc ( bday ‘ 𝑁 ) ) ) |
| 272 |
270 8 271
|
mp2an |
⊢ ( ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑑 ) ∈ suc ( bday ‘ 𝑁 ) ) |
| 273 |
269 272
|
sylibr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ) |
| 274 |
180 265
|
sselid |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑑 ∈ No ) |
| 275 |
|
madebday |
⊢ ( ( ( bday ‘ 𝑁 ) ∈ On ∧ 𝑑 ∈ No ) → ( 𝑑 ∈ ( M ‘ ( bday ‘ 𝑁 ) ) ↔ ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ) ) |
| 276 |
8 274 275
|
sylancr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( 𝑑 ∈ ( M ‘ ( bday ‘ 𝑁 ) ) ↔ ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ) ) |
| 277 |
273 276
|
mpbird |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑑 ∈ ( M ‘ ( bday ‘ 𝑁 ) ) ) |
| 278 |
|
onsbnd |
⊢ ( ( 𝑁 ∈ Ons ∧ 𝑑 ∈ ( M ‘ ( bday ‘ 𝑁 ) ) ) → 𝑑 ≤s 𝑁 ) |
| 279 |
264 277 278
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑑 ≤s 𝑁 ) |
| 280 |
207
|
n0snod |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑁 ∈ No ) |
| 281 |
274 280
|
slenltd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( 𝑑 ≤s 𝑁 ↔ ¬ 𝑁 <s 𝑑 ) ) |
| 282 |
279 281
|
mpbid |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ¬ 𝑁 <s 𝑑 ) |
| 283 |
|
lltropt |
⊢ ( L ‘ 𝑤 ) <<s ( R ‘ 𝑤 ) |
| 284 |
283
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ( L ‘ 𝑤 ) <<s ( R ‘ 𝑤 ) ) |
| 285 |
284 191 193
|
ssltsepcd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑐 <s 𝑑 ) |
| 286 |
285
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑐 <s 𝑑 ) |
| 287 |
|
breq1 |
⊢ ( 𝑐 = 𝑁 → ( 𝑐 <s 𝑑 ↔ 𝑁 <s 𝑑 ) ) |
| 288 |
287
|
bicomd |
⊢ ( 𝑐 = 𝑁 → ( 𝑁 <s 𝑑 ↔ 𝑐 <s 𝑑 ) ) |
| 289 |
286 288
|
syl5ibrcom |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( 𝑐 = 𝑁 → 𝑁 <s 𝑑 ) ) |
| 290 |
282 289
|
mtod |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ¬ 𝑐 = 𝑁 ) |
| 291 |
|
orel1 |
⊢ ( ¬ 𝑐 = 𝑁 → ( ( 𝑐 = 𝑁 ∨ ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) → ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) |
| 292 |
290 291
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( ( 𝑐 = 𝑁 ∨ ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) → ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) |
| 293 |
259 292
|
syld |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑐 ) → ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) |
| 294 |
|
simp3l1 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑔 ∈ ℕ0s ) |
| 295 |
294
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑔 ∈ ℕ0s ) |
| 296 |
295
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑔 ∈ No ) |
| 297 |
|
simp3l3 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑖 ∈ ℕ0s ) |
| 298 |
297
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑖 ∈ ℕ0s ) |
| 299 |
|
n0addscl |
⊢ ( ( 𝑔 ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) → ( 𝑔 +s 𝑖 ) ∈ ℕ0s ) |
| 300 |
295 298 299
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 𝑖 ) ∈ ℕ0s ) |
| 301 |
300
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 𝑖 ) ∈ No ) |
| 302 |
260
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝜑 ) |
| 303 |
302
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝜑 ) |
| 304 |
303 34
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑁 ∈ No ) |
| 305 |
|
n0sge0 |
⊢ ( 𝑖 ∈ ℕ0s → 0s ≤s 𝑖 ) |
| 306 |
298 305
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 0s ≤s 𝑖 ) |
| 307 |
298
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑖 ∈ No ) |
| 308 |
296 307
|
addsge01d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 0s ≤s 𝑖 ↔ 𝑔 ≤s ( 𝑔 +s 𝑖 ) ) ) |
| 309 |
306 308
|
mpbid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑔 ≤s ( 𝑔 +s 𝑖 ) ) |
| 310 |
|
simp3r3 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( 𝑔 +s 𝑖 ) <s 𝑁 ) |
| 311 |
310
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 𝑖 ) <s 𝑁 ) |
| 312 |
296 301 304 309 311
|
slelttrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑔 <s 𝑁 ) |
| 313 |
303 1
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑁 ∈ ℕ0s ) |
| 314 |
|
n0sltp1le |
⊢ ( ( 𝑔 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ) → ( 𝑔 <s 𝑁 ↔ ( 𝑔 +s 1s ) ≤s 𝑁 ) ) |
| 315 |
295 313 314
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 <s 𝑁 ↔ ( 𝑔 +s 1s ) ≤s 𝑁 ) ) |
| 316 |
312 315
|
mpbid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 1s ) ≤s 𝑁 ) |
| 317 |
|
sltirr |
⊢ ( 𝑁 ∈ No → ¬ 𝑁 <s 𝑁 ) |
| 318 |
304 317
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ¬ 𝑁 <s 𝑁 ) |
| 319 |
|
1n0s |
⊢ 1s ∈ ℕ0s |
| 320 |
|
n0addscl |
⊢ ( ( 𝑔 ∈ ℕ0s ∧ 1s ∈ ℕ0s ) → ( 𝑔 +s 1s ) ∈ ℕ0s ) |
| 321 |
295 319 320
|
sylancl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 1s ) ∈ ℕ0s ) |
| 322 |
321
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 1s ) ∈ No ) |
| 323 |
322 304
|
sltnled |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) <s 𝑁 ↔ ¬ 𝑁 ≤s ( 𝑔 +s 1s ) ) ) |
| 324 |
302
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝜑 ) |
| 325 |
131
|
n0snod |
⊢ ( 𝜑 → ( 𝑁 +s 1s ) ∈ No ) |
| 326 |
324 325
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) ∈ No ) |
| 327 |
324 34
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑁 ∈ No ) |
| 328 |
54
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 2s ∈ No ) |
| 329 |
327 328
|
subscld |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 -s 2s ) ∈ No ) |
| 330 |
|
1sno |
⊢ 1s ∈ No |
| 331 |
330
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 1s ∈ No ) |
| 332 |
329 331 331
|
addsassd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ( 𝑁 -s 2s ) +s 1s ) +s 1s ) = ( ( 𝑁 -s 2s ) +s ( 1s +s 1s ) ) ) |
| 333 |
|
1p1e2s |
⊢ ( 1s +s 1s ) = 2s |
| 334 |
333
|
oveq2i |
⊢ ( ( 𝑁 -s 2s ) +s ( 1s +s 1s ) ) = ( ( 𝑁 -s 2s ) +s 2s ) |
| 335 |
|
npcans |
⊢ ( ( 𝑁 ∈ No ∧ 2s ∈ No ) → ( ( 𝑁 -s 2s ) +s 2s ) = 𝑁 ) |
| 336 |
327 54 335
|
sylancl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 2s ) = 𝑁 ) |
| 337 |
334 336
|
eqtrid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s ( 1s +s 1s ) ) = 𝑁 ) |
| 338 |
332 337
|
eqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ( 𝑁 -s 2s ) +s 1s ) +s 1s ) = 𝑁 ) |
| 339 |
338 327
|
eqeltrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ( 𝑁 -s 2s ) +s 1s ) +s 1s ) ∈ No ) |
| 340 |
329 331
|
addscld |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 1s ) ∈ No ) |
| 341 |
202
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 342 |
341
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 343 |
|
simpl2 |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) |
| 344 |
142 191
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑐 ∈ No ) |
| 345 |
344
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑐 ∈ No ) |
| 346 |
345
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑐 ∈ No ) |
| 347 |
294
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑔 ∈ ℕ0s ) |
| 348 |
|
peano2n0s |
⊢ ( 𝑔 ∈ ℕ0s → ( 𝑔 +s 1s ) ∈ ℕ0s ) |
| 349 |
347 348
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s ) |
| 350 |
349
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑔 +s 1s ) ∈ No ) |
| 351 |
|
subscl |
⊢ ( ( 𝑁 ∈ No ∧ 1s ∈ No ) → ( 𝑁 -s 1s ) ∈ No ) |
| 352 |
34 330 351
|
sylancl |
⊢ ( 𝜑 → ( 𝑁 -s 1s ) ∈ No ) |
| 353 |
324 352
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 -s 1s ) ∈ No ) |
| 354 |
|
simp3r1 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 355 |
354
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 356 |
|
simp3r2 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ℎ <s ( 2s ↑s 𝑖 ) ) |
| 357 |
356
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ℎ <s ( 2s ↑s 𝑖 ) ) |
| 358 |
297
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑖 ∈ ℕ0s ) |
| 359 |
|
expscl |
⊢ ( ( 2s ∈ No ∧ 𝑖 ∈ ℕ0s ) → ( 2s ↑s 𝑖 ) ∈ No ) |
| 360 |
54 358 359
|
sylancr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 2s ↑s 𝑖 ) ∈ No ) |
| 361 |
360
|
mulslidd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 1s ·s ( 2s ↑s 𝑖 ) ) = ( 2s ↑s 𝑖 ) ) |
| 362 |
357 361
|
breqtrrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ℎ <s ( 1s ·s ( 2s ↑s 𝑖 ) ) ) |
| 363 |
|
simp3l2 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ℎ ∈ ℕ0s ) |
| 364 |
363
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ℎ ∈ ℕ0s ) |
| 365 |
364
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ℎ ∈ No ) |
| 366 |
365 331 358
|
pw2sltdivmul2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s 1s ↔ ℎ <s ( 1s ·s ( 2s ↑s 𝑖 ) ) ) ) |
| 367 |
362 366
|
mpbird |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s 1s ) |
| 368 |
365 358
|
pw2divscld |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) ∈ No ) |
| 369 |
347
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑔 ∈ No ) |
| 370 |
368 331 369
|
sltadd2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s 1s ↔ ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) ) ) |
| 371 |
367 370
|
mpbid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) ) |
| 372 |
355 371
|
eqbrtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑐 <s ( 𝑔 +s 1s ) ) |
| 373 |
|
n0sltp1le |
⊢ ( ( ( 𝑔 +s 1s ) ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ) → ( ( 𝑔 +s 1s ) <s 𝑁 ↔ ( ( 𝑔 +s 1s ) +s 1s ) ≤s 𝑁 ) ) |
| 374 |
321 313 373
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) <s 𝑁 ↔ ( ( 𝑔 +s 1s ) +s 1s ) ≤s 𝑁 ) ) |
| 375 |
374
|
biimpd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) <s 𝑁 → ( ( 𝑔 +s 1s ) +s 1s ) ≤s 𝑁 ) ) |
| 376 |
375
|
impr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑔 +s 1s ) +s 1s ) ≤s 𝑁 ) |
| 377 |
|
npcans |
⊢ ( ( 𝑁 ∈ No ∧ 1s ∈ No ) → ( ( 𝑁 -s 1s ) +s 1s ) = 𝑁 ) |
| 378 |
327 330 377
|
sylancl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 1s ) +s 1s ) = 𝑁 ) |
| 379 |
376 378
|
breqtrrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑔 +s 1s ) +s 1s ) ≤s ( ( 𝑁 -s 1s ) +s 1s ) ) |
| 380 |
350 353 331
|
sleadd1d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑔 +s 1s ) ≤s ( 𝑁 -s 1s ) ↔ ( ( 𝑔 +s 1s ) +s 1s ) ≤s ( ( 𝑁 -s 1s ) +s 1s ) ) ) |
| 381 |
379 380
|
mpbird |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑔 +s 1s ) ≤s ( 𝑁 -s 1s ) ) |
| 382 |
346 350 353 372 381
|
sltletrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑐 <s ( 𝑁 -s 1s ) ) |
| 383 |
333
|
oveq2i |
⊢ ( 𝑁 -s ( 1s +s 1s ) ) = ( 𝑁 -s 2s ) |
| 384 |
383
|
oveq1i |
⊢ ( ( 𝑁 -s ( 1s +s 1s ) ) +s 1s ) = ( ( 𝑁 -s 2s ) +s 1s ) |
| 385 |
327 331 331
|
subsubs4d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 1s ) -s 1s ) = ( 𝑁 -s ( 1s +s 1s ) ) ) |
| 386 |
385
|
oveq1d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ( 𝑁 -s 1s ) -s 1s ) +s 1s ) = ( ( 𝑁 -s ( 1s +s 1s ) ) +s 1s ) ) |
| 387 |
|
npcans |
⊢ ( ( ( 𝑁 -s 1s ) ∈ No ∧ 1s ∈ No ) → ( ( ( 𝑁 -s 1s ) -s 1s ) +s 1s ) = ( 𝑁 -s 1s ) ) |
| 388 |
353 330 387
|
sylancl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ( 𝑁 -s 1s ) -s 1s ) +s 1s ) = ( 𝑁 -s 1s ) ) |
| 389 |
386 388
|
eqtr3d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s ( 1s +s 1s ) ) +s 1s ) = ( 𝑁 -s 1s ) ) |
| 390 |
384 389
|
eqtr3id |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 1s ) = ( 𝑁 -s 1s ) ) |
| 391 |
382 390
|
breqtrrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑐 <s ( ( 𝑁 -s 2s ) +s 1s ) ) |
| 392 |
346 340 391
|
ssltsn |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → { 𝑐 } <<s { ( ( 𝑁 -s 2s ) +s 1s ) } ) |
| 393 |
180 193
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑑 ∈ No ) |
| 394 |
393
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑑 ∈ No ) |
| 395 |
394
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑑 ∈ No ) |
| 396 |
|
simprl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑑 = 𝑁 ) |
| 397 |
396
|
oveq1d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑑 -s 1s ) = ( 𝑁 -s 1s ) ) |
| 398 |
397
|
eqcomd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 -s 1s ) = ( 𝑑 -s 1s ) ) |
| 399 |
395
|
sltm1d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑑 -s 1s ) <s 𝑑 ) |
| 400 |
398 399
|
eqbrtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 -s 1s ) <s 𝑑 ) |
| 401 |
390 400
|
eqbrtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 1s ) <s 𝑑 ) |
| 402 |
340 395 401
|
ssltsn |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → { ( ( 𝑁 -s 2s ) +s 1s ) } <<s { 𝑑 } ) |
| 403 |
343 340 392 402
|
ssltbday |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( ( 𝑁 -s 2s ) +s 1s ) ) ) |
| 404 |
342 403
|
eqsstrrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ ( ( 𝑁 -s 2s ) +s 1s ) ) ) |
| 405 |
131 166
|
syl |
⊢ ( 𝜑 → ( 𝑁 +s 1s ) ∈ Ons ) |
| 406 |
324 405
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) ∈ Ons ) |
| 407 |
327 331 328
|
addsubsd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 +s 1s ) -s 2s ) = ( ( 𝑁 -s 2s ) +s 1s ) ) |
| 408 |
|
n0sge0 |
⊢ ( 𝑔 ∈ ℕ0s → 0s ≤s 𝑔 ) |
| 409 |
347 408
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 0s ≤s 𝑔 ) |
| 410 |
331 369
|
addsge01d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 0s ≤s 𝑔 ↔ 1s ≤s ( 1s +s 𝑔 ) ) ) |
| 411 |
409 410
|
mpbid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 1s ≤s ( 1s +s 𝑔 ) ) |
| 412 |
369 331
|
addscomd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑔 +s 1s ) = ( 1s +s 𝑔 ) ) |
| 413 |
411 412
|
breqtrrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 1s ≤s ( 𝑔 +s 1s ) ) |
| 414 |
|
simprr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑔 +s 1s ) <s 𝑁 ) |
| 415 |
331 350 327 413 414
|
slelttrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 1s <s 𝑁 ) |
| 416 |
331 327 415
|
sltled |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 1s ≤s 𝑁 ) |
| 417 |
331 327 331
|
sleadd1d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 1s ≤s 𝑁 ↔ ( 1s +s 1s ) ≤s ( 𝑁 +s 1s ) ) ) |
| 418 |
416 417
|
mpbid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 1s +s 1s ) ≤s ( 𝑁 +s 1s ) ) |
| 419 |
333 418
|
eqbrtrrid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 2s ≤s ( 𝑁 +s 1s ) ) |
| 420 |
|
2nns |
⊢ 2s ∈ ℕs |
| 421 |
|
nnn0s |
⊢ ( 2s ∈ ℕs → 2s ∈ ℕ0s ) |
| 422 |
420 421
|
ax-mp |
⊢ 2s ∈ ℕ0s |
| 423 |
422
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 2s ∈ ℕ0s ) |
| 424 |
302 131
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( 𝑁 +s 1s ) ∈ ℕ0s ) |
| 425 |
424
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) ∈ ℕ0s ) |
| 426 |
|
n0subs |
⊢ ( ( 2s ∈ ℕ0s ∧ ( 𝑁 +s 1s ) ∈ ℕ0s ) → ( 2s ≤s ( 𝑁 +s 1s ) ↔ ( ( 𝑁 +s 1s ) -s 2s ) ∈ ℕ0s ) ) |
| 427 |
423 425 426
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 2s ≤s ( 𝑁 +s 1s ) ↔ ( ( 𝑁 +s 1s ) -s 2s ) ∈ ℕ0s ) ) |
| 428 |
419 427
|
mpbid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 +s 1s ) -s 2s ) ∈ ℕ0s ) |
| 429 |
407 428
|
eqeltrrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 1s ) ∈ ℕ0s ) |
| 430 |
|
n0ons |
⊢ ( ( ( 𝑁 -s 2s ) +s 1s ) ∈ ℕ0s → ( ( 𝑁 -s 2s ) +s 1s ) ∈ Ons ) |
| 431 |
429 430
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 1s ) ∈ Ons ) |
| 432 |
406 431
|
onsled |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 +s 1s ) ≤s ( ( 𝑁 -s 2s ) +s 1s ) ↔ ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ ( ( 𝑁 -s 2s ) +s 1s ) ) ) ) |
| 433 |
404 432
|
mpbird |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) ≤s ( ( 𝑁 -s 2s ) +s 1s ) ) |
| 434 |
340
|
sltp1d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 1s ) <s ( ( ( 𝑁 -s 2s ) +s 1s ) +s 1s ) ) |
| 435 |
326 340 339 433 434
|
slelttrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) <s ( ( ( 𝑁 -s 2s ) +s 1s ) +s 1s ) ) |
| 436 |
326 339 435
|
sltled |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) ≤s ( ( ( 𝑁 -s 2s ) +s 1s ) +s 1s ) ) |
| 437 |
436 338
|
breqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) ≤s 𝑁 ) |
| 438 |
324 1
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑁 ∈ ℕ0s ) |
| 439 |
|
n0sltp1le |
⊢ ( ( 𝑁 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ) → ( 𝑁 <s 𝑁 ↔ ( 𝑁 +s 1s ) ≤s 𝑁 ) ) |
| 440 |
438 438 439
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 <s 𝑁 ↔ ( 𝑁 +s 1s ) ≤s 𝑁 ) ) |
| 441 |
437 440
|
mpbird |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑁 <s 𝑁 ) |
| 442 |
441
|
expr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) <s 𝑁 → 𝑁 <s 𝑁 ) ) |
| 443 |
323 442
|
sylbird |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ¬ 𝑁 ≤s ( 𝑔 +s 1s ) → 𝑁 <s 𝑁 ) ) |
| 444 |
318 443
|
mt3d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑁 ≤s ( 𝑔 +s 1s ) ) |
| 445 |
444
|
a1d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) ≤s 𝑁 → 𝑁 ≤s ( 𝑔 +s 1s ) ) ) |
| 446 |
316 445
|
jcai |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) ≤s 𝑁 ∧ 𝑁 ≤s ( 𝑔 +s 1s ) ) ) |
| 447 |
322 304
|
sletri3d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) = 𝑁 ↔ ( ( 𝑔 +s 1s ) ≤s 𝑁 ∧ 𝑁 ≤s ( 𝑔 +s 1s ) ) ) ) |
| 448 |
446 447
|
mpbird |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 1s ) = 𝑁 ) |
| 449 |
310
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ( 𝑔 +s 𝑖 ) <s 𝑁 ) |
| 450 |
|
simprr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ( 𝑔 +s 1s ) = 𝑁 ) |
| 451 |
449 450
|
breqtrrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ( 𝑔 +s 𝑖 ) <s ( 𝑔 +s 1s ) ) |
| 452 |
297
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 𝑖 ∈ ℕ0s ) |
| 453 |
452
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 𝑖 ∈ No ) |
| 454 |
330
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 1s ∈ No ) |
| 455 |
294
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 𝑔 ∈ ℕ0s ) |
| 456 |
455
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 𝑔 ∈ No ) |
| 457 |
453 454 456
|
sltadd2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ( 𝑖 <s 1s ↔ ( 𝑔 +s 𝑖 ) <s ( 𝑔 +s 1s ) ) ) |
| 458 |
451 457
|
mpbird |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 𝑖 <s 1s ) |
| 459 |
|
n0slt1e0 |
⊢ ( 𝑖 ∈ ℕ0s → ( 𝑖 <s 1s ↔ 𝑖 = 0s ) ) |
| 460 |
452 459
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ( 𝑖 <s 1s ↔ 𝑖 = 0s ) ) |
| 461 |
458 460
|
mpbid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 𝑖 = 0s ) |
| 462 |
354
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 463 |
356
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ℎ <s ( 2s ↑s 𝑖 ) ) |
| 464 |
|
oveq2 |
⊢ ( 𝑖 = 0s → ( 2s ↑s 𝑖 ) = ( 2s ↑s 0s ) ) |
| 465 |
464
|
adantl |
⊢ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) → ( 2s ↑s 𝑖 ) = ( 2s ↑s 0s ) ) |
| 466 |
465
|
adantl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( 2s ↑s 𝑖 ) = ( 2s ↑s 0s ) ) |
| 467 |
466 56
|
eqtrdi |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( 2s ↑s 𝑖 ) = 1s ) |
| 468 |
463 467
|
breqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ℎ <s 1s ) |
| 469 |
363
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ℎ ∈ ℕ0s ) |
| 470 |
|
n0slt1e0 |
⊢ ( ℎ ∈ ℕ0s → ( ℎ <s 1s ↔ ℎ = 0s ) ) |
| 471 |
469 470
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( ℎ <s 1s ↔ ℎ = 0s ) ) |
| 472 |
468 471
|
mpbid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ℎ = 0s ) |
| 473 |
472 467
|
oveq12d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) = ( 0s /su 1s ) ) |
| 474 |
473 61
|
eqtrdi |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) = 0s ) |
| 475 |
474
|
oveq2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) = ( 𝑔 +s 0s ) ) |
| 476 |
294
|
n0snod |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑔 ∈ No ) |
| 477 |
476
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → 𝑔 ∈ No ) |
| 478 |
477
|
addsridd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( 𝑔 +s 0s ) = 𝑔 ) |
| 479 |
462 475 478
|
3eqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → 𝑐 = 𝑔 ) |
| 480 |
56
|
oveq2i |
⊢ ( 𝑔 /su ( 2s ↑s 0s ) ) = ( 𝑔 /su 1s ) |
| 481 |
476
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑔 ∈ No ) |
| 482 |
|
divs1 |
⊢ ( 𝑔 ∈ No → ( 𝑔 /su 1s ) = 𝑔 ) |
| 483 |
481 482
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 /su 1s ) = 𝑔 ) |
| 484 |
|
simprr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑐 = 𝑔 ) |
| 485 |
483 484
|
eqtr4d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 /su 1s ) = 𝑐 ) |
| 486 |
480 485
|
eqtrid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 /su ( 2s ↑s 0s ) ) = 𝑐 ) |
| 487 |
486
|
sneqd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → { ( 𝑔 /su ( 2s ↑s 0s ) ) } = { 𝑐 } ) |
| 488 |
56
|
oveq2i |
⊢ ( ( 𝑔 +s 1s ) /su ( 2s ↑s 0s ) ) = ( ( 𝑔 +s 1s ) /su 1s ) |
| 489 |
294 348
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s ) |
| 490 |
489
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s ) |
| 491 |
490
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 +s 1s ) ∈ No ) |
| 492 |
|
divs1 |
⊢ ( ( 𝑔 +s 1s ) ∈ No → ( ( 𝑔 +s 1s ) /su 1s ) = ( 𝑔 +s 1s ) ) |
| 493 |
491 492
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( 𝑔 +s 1s ) /su 1s ) = ( 𝑔 +s 1s ) ) |
| 494 |
|
simpllr |
⊢ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) → ( 𝑔 +s 1s ) = 𝑁 ) |
| 495 |
494
|
adantl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 +s 1s ) = 𝑁 ) |
| 496 |
|
simplll |
⊢ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) → 𝑑 = 𝑁 ) |
| 497 |
496
|
adantl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑑 = 𝑁 ) |
| 498 |
495 497
|
eqtr4d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 +s 1s ) = 𝑑 ) |
| 499 |
493 498
|
eqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( 𝑔 +s 1s ) /su 1s ) = 𝑑 ) |
| 500 |
488 499
|
eqtrid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( 𝑔 +s 1s ) /su ( 2s ↑s 0s ) ) = 𝑑 ) |
| 501 |
500
|
sneqd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → { ( ( 𝑔 +s 1s ) /su ( 2s ↑s 0s ) ) } = { 𝑑 } ) |
| 502 |
487 501
|
oveq12d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( { ( 𝑔 /su ( 2s ↑s 0s ) ) } |s { ( ( 𝑔 +s 1s ) /su ( 2s ↑s 0s ) ) } ) = ( { 𝑐 } |s { 𝑑 } ) ) |
| 503 |
294
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑔 ∈ ℕ0s ) |
| 504 |
503
|
n0zsd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑔 ∈ ℤs ) |
| 505 |
31
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 0s ∈ ℕ0s ) |
| 506 |
504 505
|
pw2cutp1 |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( { ( 𝑔 /su ( 2s ↑s 0s ) ) } |s { ( ( 𝑔 +s 1s ) /su ( 2s ↑s 0s ) ) } ) = ( ( ( 2s ·s 𝑔 ) +s 1s ) /su ( 2s ↑s ( 0s +s 1s ) ) ) ) |
| 507 |
506
|
eqcomd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( ( 2s ·s 𝑔 ) +s 1s ) /su ( 2s ↑s ( 0s +s 1s ) ) ) = ( { ( 𝑔 /su ( 2s ↑s 0s ) ) } |s { ( ( 𝑔 +s 1s ) /su ( 2s ↑s 0s ) ) } ) ) |
| 508 |
|
simpl2 |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) |
| 509 |
502 507 508
|
3eqtr4rd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑤 = ( ( ( 2s ·s 𝑔 ) +s 1s ) /su ( 2s ↑s ( 0s +s 1s ) ) ) ) |
| 510 |
|
mulscl |
⊢ ( ( 2s ∈ No ∧ 𝑔 ∈ No ) → ( 2s ·s 𝑔 ) ∈ No ) |
| 511 |
54 481 510
|
sylancr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 2s ·s 𝑔 ) ∈ No ) |
| 512 |
330
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 1s ∈ No ) |
| 513 |
|
addslid |
⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s ) |
| 514 |
330 513
|
ax-mp |
⊢ ( 0s +s 1s ) = 1s |
| 515 |
514 319
|
eqeltri |
⊢ ( 0s +s 1s ) ∈ ℕ0s |
| 516 |
515
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 0s +s 1s ) ∈ ℕ0s ) |
| 517 |
511 512 516
|
pw2divsdird |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( ( 2s ·s 𝑔 ) +s 1s ) /su ( 2s ↑s ( 0s +s 1s ) ) ) = ( ( ( 2s ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) ) +s ( 1s /su ( 2s ↑s ( 0s +s 1s ) ) ) ) ) |
| 518 |
|
exps1 |
⊢ ( 2s ∈ No → ( 2s ↑s 1s ) = 2s ) |
| 519 |
54 518
|
ax-mp |
⊢ ( 2s ↑s 1s ) = 2s |
| 520 |
519
|
oveq1i |
⊢ ( ( 2s ↑s 1s ) ·s 𝑔 ) = ( 2s ·s 𝑔 ) |
| 521 |
520
|
oveq1i |
⊢ ( ( ( 2s ↑s 1s ) ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) ) = ( ( 2s ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) ) |
| 522 |
319
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 1s ∈ ℕ0s ) |
| 523 |
481 505 522
|
pw2divscan4d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 /su ( 2s ↑s 0s ) ) = ( ( ( 2s ↑s 1s ) ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) ) ) |
| 524 |
480 483
|
eqtrid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 /su ( 2s ↑s 0s ) ) = 𝑔 ) |
| 525 |
523 524
|
eqtr3d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( ( 2s ↑s 1s ) ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) ) = 𝑔 ) |
| 526 |
521 525
|
eqtr3id |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( 2s ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) ) = 𝑔 ) |
| 527 |
514
|
oveq2i |
⊢ ( 2s ↑s ( 0s +s 1s ) ) = ( 2s ↑s 1s ) |
| 528 |
527 519
|
eqtri |
⊢ ( 2s ↑s ( 0s +s 1s ) ) = 2s |
| 529 |
528
|
oveq2i |
⊢ ( 1s /su ( 2s ↑s ( 0s +s 1s ) ) ) = ( 1s /su 2s ) |
| 530 |
529
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 1s /su ( 2s ↑s ( 0s +s 1s ) ) ) = ( 1s /su 2s ) ) |
| 531 |
526 530
|
oveq12d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( ( 2s ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) ) +s ( 1s /su ( 2s ↑s ( 0s +s 1s ) ) ) ) = ( 𝑔 +s ( 1s /su 2s ) ) ) |
| 532 |
517 531
|
eqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( ( 2s ·s 𝑔 ) +s 1s ) /su ( 2s ↑s ( 0s +s 1s ) ) ) = ( 𝑔 +s ( 1s /su 2s ) ) ) |
| 533 |
532
|
eqeq2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑤 = ( ( ( 2s ·s 𝑔 ) +s 1s ) /su ( 2s ↑s ( 0s +s 1s ) ) ) ↔ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) |
| 534 |
294
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 𝑔 ∈ ℕ0s ) |
| 535 |
319
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 1s ∈ ℕ0s ) |
| 536 |
|
simprr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) |
| 537 |
|
sltadd1 |
⊢ ( ( 0s ∈ No ∧ 1s ∈ No ∧ 1s ∈ No ) → ( 0s <s 1s ↔ ( 0s +s 1s ) <s ( 1s +s 1s ) ) ) |
| 538 |
59 330 330 537
|
mp3an |
⊢ ( 0s <s 1s ↔ ( 0s +s 1s ) <s ( 1s +s 1s ) ) |
| 539 |
38 538
|
mpbi |
⊢ ( 0s +s 1s ) <s ( 1s +s 1s ) |
| 540 |
539 514 333
|
3brtr3i |
⊢ 1s <s 2s |
| 541 |
540
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 1s <s 2s ) |
| 542 |
|
simp-4r |
⊢ ( ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) → ( 𝑔 +s 1s ) = 𝑁 ) |
| 543 |
542
|
adantl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → ( 𝑔 +s 1s ) = 𝑁 ) |
| 544 |
302
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 𝜑 ) |
| 545 |
544 34
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 𝑁 ∈ No ) |
| 546 |
545
|
sltp1d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 𝑁 <s ( 𝑁 +s 1s ) ) |
| 547 |
543 546
|
eqbrtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → ( 𝑔 +s 1s ) <s ( 𝑁 +s 1s ) ) |
| 548 |
|
oveq1 |
⊢ ( 𝑎 = 𝑔 → ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) |
| 549 |
548
|
eqeq2d |
⊢ ( 𝑎 = 𝑔 → ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) ) |
| 550 |
|
oveq1 |
⊢ ( 𝑎 = 𝑔 → ( 𝑎 +s 𝑞 ) = ( 𝑔 +s 𝑞 ) ) |
| 551 |
550
|
breq1d |
⊢ ( 𝑎 = 𝑔 → ( ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 552 |
549 551
|
3anbi13d |
⊢ ( 𝑎 = 𝑔 → ( ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 553 |
|
oveq1 |
⊢ ( 𝑏 = 1s → ( 𝑏 /su ( 2s ↑s 𝑞 ) ) = ( 1s /su ( 2s ↑s 𝑞 ) ) ) |
| 554 |
553
|
oveq2d |
⊢ ( 𝑏 = 1s → ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( 1s /su ( 2s ↑s 𝑞 ) ) ) ) |
| 555 |
554
|
eqeq2d |
⊢ ( 𝑏 = 1s → ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( 1s /su ( 2s ↑s 𝑞 ) ) ) ) ) |
| 556 |
|
breq1 |
⊢ ( 𝑏 = 1s → ( 𝑏 <s ( 2s ↑s 𝑞 ) ↔ 1s <s ( 2s ↑s 𝑞 ) ) ) |
| 557 |
555 556
|
3anbi12d |
⊢ ( 𝑏 = 1s → ( ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( 1s /su ( 2s ↑s 𝑞 ) ) ) ∧ 1s <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 558 |
|
oveq2 |
⊢ ( 𝑞 = 1s → ( 2s ↑s 𝑞 ) = ( 2s ↑s 1s ) ) |
| 559 |
558 519
|
eqtrdi |
⊢ ( 𝑞 = 1s → ( 2s ↑s 𝑞 ) = 2s ) |
| 560 |
559
|
oveq2d |
⊢ ( 𝑞 = 1s → ( 1s /su ( 2s ↑s 𝑞 ) ) = ( 1s /su 2s ) ) |
| 561 |
560
|
oveq2d |
⊢ ( 𝑞 = 1s → ( 𝑔 +s ( 1s /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( 1s /su 2s ) ) ) |
| 562 |
561
|
eqeq2d |
⊢ ( 𝑞 = 1s → ( 𝑤 = ( 𝑔 +s ( 1s /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) |
| 563 |
559
|
breq2d |
⊢ ( 𝑞 = 1s → ( 1s <s ( 2s ↑s 𝑞 ) ↔ 1s <s 2s ) ) |
| 564 |
|
oveq2 |
⊢ ( 𝑞 = 1s → ( 𝑔 +s 𝑞 ) = ( 𝑔 +s 1s ) ) |
| 565 |
564
|
breq1d |
⊢ ( 𝑞 = 1s → ( ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑔 +s 1s ) <s ( 𝑁 +s 1s ) ) ) |
| 566 |
562 563 565
|
3anbi123d |
⊢ ( 𝑞 = 1s → ( ( 𝑤 = ( 𝑔 +s ( 1s /su ( 2s ↑s 𝑞 ) ) ) ∧ 1s <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ∧ 1s <s 2s ∧ ( 𝑔 +s 1s ) <s ( 𝑁 +s 1s ) ) ) ) |
| 567 |
552 557 566
|
rspc3ev |
⊢ ( ( ( 𝑔 ∈ ℕ0s ∧ 1s ∈ ℕ0s ∧ 1s ∈ ℕ0s ) ∧ ( 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ∧ 1s <s 2s ∧ ( 𝑔 +s 1s ) <s ( 𝑁 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 568 |
534 535 535 536 541 547 567
|
syl33anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 569 |
568
|
expr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 570 |
533 569
|
sylbid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑤 = ( ( ( 2s ·s 𝑔 ) +s 1s ) /su ( 2s ↑s ( 0s +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 571 |
509 570
|
mpd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 572 |
571
|
expr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( 𝑐 = 𝑔 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 573 |
479 572
|
mpd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 574 |
573
|
expr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ( 𝑖 = 0s → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 575 |
461 574
|
mpd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 576 |
575
|
expr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) = 𝑁 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 577 |
448 576
|
mpd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 578 |
577
|
ex |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( 𝑑 = 𝑁 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 579 |
|
simprr1 |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 580 |
|
simprr2 |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑘 <s ( 2s ↑s 𝑙 ) ) |
| 581 |
|
simprl3 |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑙 ∈ ℕ0s ) |
| 582 |
|
expscl |
⊢ ( ( 2s ∈ No ∧ 𝑙 ∈ ℕ0s ) → ( 2s ↑s 𝑙 ) ∈ No ) |
| 583 |
54 581 582
|
sylancr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 2s ↑s 𝑙 ) ∈ No ) |
| 584 |
583
|
mulslidd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 1s ·s ( 2s ↑s 𝑙 ) ) = ( 2s ↑s 𝑙 ) ) |
| 585 |
580 584
|
breqtrrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑘 <s ( 1s ·s ( 2s ↑s 𝑙 ) ) ) |
| 586 |
|
simprl2 |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑘 ∈ ℕ0s ) |
| 587 |
586
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑘 ∈ No ) |
| 588 |
330
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 1s ∈ No ) |
| 589 |
587 588 581
|
pw2sltdivmul2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( ( 𝑘 /su ( 2s ↑s 𝑙 ) ) <s 1s ↔ 𝑘 <s ( 1s ·s ( 2s ↑s 𝑙 ) ) ) ) |
| 590 |
585 589
|
mpbird |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑘 /su ( 2s ↑s 𝑙 ) ) <s 1s ) |
| 591 |
587 581
|
pw2divscld |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ∈ No ) |
| 592 |
|
simprl1 |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑗 ∈ ℕ0s ) |
| 593 |
592
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑗 ∈ No ) |
| 594 |
591 588 593
|
sltadd2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( ( 𝑘 /su ( 2s ↑s 𝑙 ) ) <s 1s ↔ ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) <s ( 𝑗 +s 1s ) ) ) |
| 595 |
590 594
|
mpbid |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) <s ( 𝑗 +s 1s ) ) |
| 596 |
579 595
|
eqbrtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑑 <s ( 𝑗 +s 1s ) ) |
| 597 |
294
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑔 ∈ ℕ0s ) |
| 598 |
597
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 𝑔 ∈ ℕ0s ) |
| 599 |
598
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 𝑔 ∈ No ) |
| 600 |
599
|
addsridd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( 𝑔 +s 0s ) = 𝑔 ) |
| 601 |
363
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ℎ ∈ ℕ0s ) |
| 602 |
601
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ℎ ∈ ℕ0s ) |
| 603 |
|
n0sge0 |
⊢ ( ℎ ∈ ℕ0s → 0s ≤s ℎ ) |
| 604 |
602 603
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 0s ≤s ℎ ) |
| 605 |
602
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ℎ ∈ No ) |
| 606 |
297
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑖 ∈ ℕ0s ) |
| 607 |
606
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 𝑖 ∈ ℕ0s ) |
| 608 |
605 607
|
pw2ge0divsd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( 0s ≤s ℎ ↔ 0s ≤s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 609 |
59
|
a1i |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 0s ∈ No ) |
| 610 |
605 607
|
pw2divscld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) ∈ No ) |
| 611 |
609 610 599
|
sleadd2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( 0s ≤s ( ℎ /su ( 2s ↑s 𝑖 ) ) ↔ ( 𝑔 +s 0s ) ≤s ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) ) |
| 612 |
608 611
|
bitrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( 0s ≤s ℎ ↔ ( 𝑔 +s 0s ) ≤s ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) ) |
| 613 |
604 612
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( 𝑔 +s 0s ) ≤s ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 614 |
600 613
|
eqbrtrrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 𝑔 ≤s ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 615 |
354
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 616 |
615
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 617 |
614 616
|
breqtrrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 𝑔 ≤s 𝑐 ) |
| 618 |
597
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑔 ∈ ℕ0s ) |
| 619 |
618
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑔 ∈ No ) |
| 620 |
345
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑐 ∈ No ) |
| 621 |
620
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑐 ∈ No ) |
| 622 |
394
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑑 ∈ No ) |
| 623 |
622
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑑 ∈ No ) |
| 624 |
|
simprr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑔 ≤s 𝑐 ) |
| 625 |
286
|
3adant3 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑐 <s 𝑑 ) |
| 626 |
625
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑐 <s 𝑑 ) |
| 627 |
626
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑐 <s 𝑑 ) |
| 628 |
619 621 623 624 627
|
slelttrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑔 <s 𝑑 ) |
| 629 |
597
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑔 ∈ ℕ0s ) |
| 630 |
629
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑔 ∈ No ) |
| 631 |
622
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑑 ∈ No ) |
| 632 |
592
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑗 ∈ ℕ0s ) |
| 633 |
|
peano2n0s |
⊢ ( 𝑗 ∈ ℕ0s → ( 𝑗 +s 1s ) ∈ ℕ0s ) |
| 634 |
632 633
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → ( 𝑗 +s 1s ) ∈ ℕ0s ) |
| 635 |
634
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → ( 𝑗 +s 1s ) ∈ No ) |
| 636 |
|
simprr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑔 <s 𝑑 ) |
| 637 |
|
simprll |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑑 <s ( 𝑗 +s 1s ) ) |
| 638 |
630 631 635 636 637
|
slttrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑔 <s ( 𝑗 +s 1s ) ) |
| 639 |
|
n0sleltp1 |
⊢ ( ( 𝑔 ∈ ℕ0s ∧ 𝑗 ∈ ℕ0s ) → ( 𝑔 ≤s 𝑗 ↔ 𝑔 <s ( 𝑗 +s 1s ) ) ) |
| 640 |
629 632 639
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → ( 𝑔 ≤s 𝑗 ↔ 𝑔 <s ( 𝑗 +s 1s ) ) ) |
| 641 |
638 640
|
mpbird |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑔 ≤s 𝑗 ) |
| 642 |
641
|
expr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 <s 𝑑 → 𝑔 ≤s 𝑗 ) ) |
| 643 |
628 642
|
mpd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑔 ≤s 𝑗 ) |
| 644 |
593
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑗 ∈ No ) |
| 645 |
619 644
|
sleloed |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 ≤s 𝑗 ↔ ( 𝑔 <s 𝑗 ∨ 𝑔 = 𝑗 ) ) ) |
| 646 |
592
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑗 ∈ ℕ0s ) |
| 647 |
|
n0sltp1le |
⊢ ( ( 𝑔 ∈ ℕ0s ∧ 𝑗 ∈ ℕ0s ) → ( 𝑔 <s 𝑗 ↔ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) |
| 648 |
618 646 647
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 <s 𝑗 ↔ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) |
| 649 |
648
|
biimpd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 <s 𝑗 → ( 𝑔 +s 1s ) ≤s 𝑗 ) ) |
| 650 |
649
|
impr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ) → ( 𝑔 +s 1s ) ≤s 𝑗 ) |
| 651 |
489
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s ) |
| 652 |
651
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s ) |
| 653 |
652
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 𝑔 +s 1s ) ∈ No ) |
| 654 |
592
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑗 ∈ ℕ0s ) |
| 655 |
654
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑗 ∈ No ) |
| 656 |
622
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑑 ∈ No ) |
| 657 |
|
simprr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 𝑔 +s 1s ) ≤s 𝑗 ) |
| 658 |
586
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑘 ∈ ℕ0s ) |
| 659 |
|
n0sge0 |
⊢ ( 𝑘 ∈ ℕ0s → 0s ≤s 𝑘 ) |
| 660 |
658 659
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 0s ≤s 𝑘 ) |
| 661 |
658
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑘 ∈ No ) |
| 662 |
581
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑙 ∈ ℕ0s ) |
| 663 |
661 662
|
pw2ge0divsd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 0s ≤s 𝑘 ↔ 0s ≤s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 664 |
661 662
|
pw2divscld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ∈ No ) |
| 665 |
655 664
|
addsge01d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 0s ≤s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ↔ 𝑗 ≤s ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ) |
| 666 |
663 665
|
bitrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 0s ≤s 𝑘 ↔ 𝑗 ≤s ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ) |
| 667 |
660 666
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑗 ≤s ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 668 |
579
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 669 |
667 668
|
breqtrrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑗 ≤s 𝑑 ) |
| 670 |
653 655 656 657 669
|
sletrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 𝑔 +s 1s ) ≤s 𝑑 ) |
| 671 |
592
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑗 ∈ ℕ0s ) |
| 672 |
581
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑙 ∈ ℕ0s ) |
| 673 |
|
n0addscl |
⊢ ( ( 𝑗 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) → ( 𝑗 +s 𝑙 ) ∈ ℕ0s ) |
| 674 |
671 672 673
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑗 +s 𝑙 ) ∈ ℕ0s ) |
| 675 |
674
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑗 +s 𝑙 ) ∈ No ) |
| 676 |
302
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝜑 ) |
| 677 |
676
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝜑 ) |
| 678 |
677 34
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑁 ∈ No ) |
| 679 |
677 325
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑁 +s 1s ) ∈ No ) |
| 680 |
|
simprr3 |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑗 +s 𝑙 ) <s 𝑁 ) |
| 681 |
680
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑗 +s 𝑙 ) <s 𝑁 ) |
| 682 |
678
|
sltp1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑁 <s ( 𝑁 +s 1s ) ) |
| 683 |
675 678 679 681 682
|
slttrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑗 +s 𝑙 ) <s ( 𝑁 +s 1s ) ) |
| 684 |
675 679
|
sltnled |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( ( 𝑗 +s 𝑙 ) <s ( 𝑁 +s 1s ) ↔ ¬ ( 𝑁 +s 1s ) ≤s ( 𝑗 +s 𝑙 ) ) ) |
| 685 |
683 684
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ¬ ( 𝑁 +s 1s ) ≤s ( 𝑗 +s 𝑙 ) ) |
| 686 |
651
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s ) |
| 687 |
686
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑔 +s 1s ) ∈ No ) |
| 688 |
622
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑑 ∈ No ) |
| 689 |
687 688
|
sltnled |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( ( 𝑔 +s 1s ) <s 𝑑 ↔ ¬ 𝑑 ≤s ( 𝑔 +s 1s ) ) ) |
| 690 |
679
|
adantrr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑁 +s 1s ) ∈ No ) |
| 691 |
593
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑗 ∈ No ) |
| 692 |
675
|
adantrr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑗 +s 𝑙 ) ∈ No ) |
| 693 |
651
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s ) |
| 694 |
693
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑔 +s 1s ) ∈ No ) |
| 695 |
341
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 696 |
695
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 697 |
|
simpll2 |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) |
| 698 |
620
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑐 ∈ No ) |
| 699 |
615
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 700 |
356
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ℎ <s ( 2s ↑s 𝑖 ) ) |
| 701 |
700
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ℎ <s ( 2s ↑s 𝑖 ) ) |
| 702 |
606
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑖 ∈ ℕ0s ) |
| 703 |
54 702 359
|
sylancr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 2s ↑s 𝑖 ) ∈ No ) |
| 704 |
703
|
mulslidd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 1s ·s ( 2s ↑s 𝑖 ) ) = ( 2s ↑s 𝑖 ) ) |
| 705 |
701 704
|
breqtrrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ℎ <s ( 1s ·s ( 2s ↑s 𝑖 ) ) ) |
| 706 |
601
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ℎ ∈ ℕ0s ) |
| 707 |
706
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ℎ ∈ No ) |
| 708 |
330
|
a1i |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 1s ∈ No ) |
| 709 |
707 708 702
|
pw2sltdivmul2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s 1s ↔ ℎ <s ( 1s ·s ( 2s ↑s 𝑖 ) ) ) ) |
| 710 |
705 709
|
mpbird |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s 1s ) |
| 711 |
707 702
|
pw2divscld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) ∈ No ) |
| 712 |
597
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑔 ∈ ℕ0s ) |
| 713 |
712
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑔 ∈ No ) |
| 714 |
711 708 713
|
sltadd2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s 1s ↔ ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) ) ) |
| 715 |
710 714
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) ) |
| 716 |
699 715
|
eqbrtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑐 <s ( 𝑔 +s 1s ) ) |
| 717 |
698 694 716
|
ssltsn |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → { 𝑐 } <<s { ( 𝑔 +s 1s ) } ) |
| 718 |
622
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑑 ∈ No ) |
| 719 |
|
simprr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑔 +s 1s ) <s 𝑑 ) |
| 720 |
694 718 719
|
ssltsn |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → { ( 𝑔 +s 1s ) } <<s { 𝑑 } ) |
| 721 |
697 694 717 720
|
ssltbday |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑔 +s 1s ) ) ) |
| 722 |
696 721
|
eqsstrrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ ( 𝑔 +s 1s ) ) ) |
| 723 |
676
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝜑 ) |
| 724 |
723 405
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑁 +s 1s ) ∈ Ons ) |
| 725 |
|
n0ons |
⊢ ( ( 𝑔 +s 1s ) ∈ ℕ0s → ( 𝑔 +s 1s ) ∈ Ons ) |
| 726 |
693 725
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑔 +s 1s ) ∈ Ons ) |
| 727 |
724 726
|
onsled |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( ( 𝑁 +s 1s ) ≤s ( 𝑔 +s 1s ) ↔ ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ ( 𝑔 +s 1s ) ) ) ) |
| 728 |
722 727
|
mpbird |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑁 +s 1s ) ≤s ( 𝑔 +s 1s ) ) |
| 729 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) → ( 𝑔 +s 1s ) ≤s 𝑗 ) |
| 730 |
729
|
adantl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑔 +s 1s ) ≤s 𝑗 ) |
| 731 |
690 694 691 728 730
|
sletrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑁 +s 1s ) ≤s 𝑗 ) |
| 732 |
581
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑙 ∈ ℕ0s ) |
| 733 |
|
n0sge0 |
⊢ ( 𝑙 ∈ ℕ0s → 0s ≤s 𝑙 ) |
| 734 |
732 733
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 0s ≤s 𝑙 ) |
| 735 |
732
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑙 ∈ No ) |
| 736 |
691 735
|
addsge01d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 0s ≤s 𝑙 ↔ 𝑗 ≤s ( 𝑗 +s 𝑙 ) ) ) |
| 737 |
734 736
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑗 ≤s ( 𝑗 +s 𝑙 ) ) |
| 738 |
690 691 692 731 737
|
sletrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑁 +s 1s ) ≤s ( 𝑗 +s 𝑙 ) ) |
| 739 |
738
|
expr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( ( 𝑔 +s 1s ) <s 𝑑 → ( 𝑁 +s 1s ) ≤s ( 𝑗 +s 𝑙 ) ) ) |
| 740 |
689 739
|
sylbird |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( ¬ 𝑑 ≤s ( 𝑔 +s 1s ) → ( 𝑁 +s 1s ) ≤s ( 𝑗 +s 𝑙 ) ) ) |
| 741 |
685 740
|
mt3d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑑 ≤s ( 𝑔 +s 1s ) ) |
| 742 |
|
simprr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑔 +s 1s ) ≤s 𝑑 ) |
| 743 |
688 687
|
sletri3d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑑 = ( 𝑔 +s 1s ) ↔ ( 𝑑 ≤s ( 𝑔 +s 1s ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) ) |
| 744 |
741 742 743
|
mpbir2and |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑑 = ( 𝑔 +s 1s ) ) |
| 745 |
700
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ℎ <s ( 2s ↑s 𝑖 ) ) |
| 746 |
601
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ℎ ∈ ℕ0s ) |
| 747 |
606
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → 𝑖 ∈ ℕ0s ) |
| 748 |
|
n0expscl |
⊢ ( ( 2s ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) → ( 2s ↑s 𝑖 ) ∈ ℕ0s ) |
| 749 |
422 747 748
|
sylancr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( 2s ↑s 𝑖 ) ∈ ℕ0s ) |
| 750 |
|
n0sltp1le |
⊢ ( ( ℎ ∈ ℕ0s ∧ ( 2s ↑s 𝑖 ) ∈ ℕ0s ) → ( ℎ <s ( 2s ↑s 𝑖 ) ↔ ( ℎ +s 1s ) ≤s ( 2s ↑s 𝑖 ) ) ) |
| 751 |
746 749 750
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ℎ <s ( 2s ↑s 𝑖 ) ↔ ( ℎ +s 1s ) ≤s ( 2s ↑s 𝑖 ) ) ) |
| 752 |
745 751
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ℎ +s 1s ) ≤s ( 2s ↑s 𝑖 ) ) |
| 753 |
|
peano2n0s |
⊢ ( ℎ ∈ ℕ0s → ( ℎ +s 1s ) ∈ ℕ0s ) |
| 754 |
363 753
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( ℎ +s 1s ) ∈ ℕ0s ) |
| 755 |
754
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( ℎ +s 1s ) ∈ ℕ0s ) |
| 756 |
755
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ℎ +s 1s ) ∈ ℕ0s ) |
| 757 |
756
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ℎ +s 1s ) ∈ No ) |
| 758 |
749
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( 2s ↑s 𝑖 ) ∈ No ) |
| 759 |
757 758
|
sleloed |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ( ℎ +s 1s ) ≤s ( 2s ↑s 𝑖 ) ↔ ( ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ∨ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) ) |
| 760 |
676
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → 𝜑 ) |
| 761 |
34 325
|
sltnled |
⊢ ( 𝜑 → ( 𝑁 <s ( 𝑁 +s 1s ) ↔ ¬ ( 𝑁 +s 1s ) ≤s 𝑁 ) ) |
| 762 |
40 761
|
mpbid |
⊢ ( 𝜑 → ¬ ( 𝑁 +s 1s ) ≤s 𝑁 ) |
| 763 |
760 762
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ¬ ( 𝑁 +s 1s ) ≤s 𝑁 ) |
| 764 |
695
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 765 |
|
simpll2 |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) |
| 766 |
597
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑔 ∈ ℕ0s ) |
| 767 |
766
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑔 ∈ No ) |
| 768 |
755
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ +s 1s ) ∈ ℕ0s ) |
| 769 |
768
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ +s 1s ) ∈ No ) |
| 770 |
606
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑖 ∈ ℕ0s ) |
| 771 |
769 770
|
pw2divscld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ∈ No ) |
| 772 |
767 771
|
addscld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ∈ No ) |
| 773 |
620
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑐 ∈ No ) |
| 774 |
615
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 775 |
601
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ℎ ∈ ℕ0s ) |
| 776 |
775
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ℎ ∈ No ) |
| 777 |
776
|
sltp1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ℎ <s ( ℎ +s 1s ) ) |
| 778 |
776 769 770
|
pw2sltdiv1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ <s ( ℎ +s 1s ) ↔ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ) |
| 779 |
777 778
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) |
| 780 |
776 770
|
pw2divscld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) ∈ No ) |
| 781 |
780 771 767
|
sltadd2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ↔ ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ) ) |
| 782 |
779 781
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ) |
| 783 |
774 782
|
eqbrtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑐 <s ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ) |
| 784 |
773 772 783
|
ssltsn |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → { 𝑐 } <<s { ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) } ) |
| 785 |
622
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑑 ∈ No ) |
| 786 |
|
simprr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) |
| 787 |
54 770 359
|
sylancr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s 𝑖 ) ∈ No ) |
| 788 |
787
|
mulslidd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 1s ·s ( 2s ↑s 𝑖 ) ) = ( 2s ↑s 𝑖 ) ) |
| 789 |
786 788
|
breqtrrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ +s 1s ) <s ( 1s ·s ( 2s ↑s 𝑖 ) ) ) |
| 790 |
330
|
a1i |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 1s ∈ No ) |
| 791 |
769 790 770
|
pw2sltdivmul2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) <s 1s ↔ ( ℎ +s 1s ) <s ( 1s ·s ( 2s ↑s 𝑖 ) ) ) ) |
| 792 |
791
|
bicomd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( ℎ +s 1s ) <s ( 1s ·s ( 2s ↑s 𝑖 ) ) ↔ ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) <s 1s ) ) |
| 793 |
771 790 767
|
sltadd2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) <s 1s ↔ ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) ) ) |
| 794 |
792 793
|
bitrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( ℎ +s 1s ) <s ( 1s ·s ( 2s ↑s 𝑖 ) ) ↔ ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) ) ) |
| 795 |
789 794
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) ) |
| 796 |
|
simprlr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑑 = ( 𝑔 +s 1s ) ) |
| 797 |
795 796
|
breqtrrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) <s 𝑑 ) |
| 798 |
772 785 797
|
ssltsn |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → { ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) } <<s { 𝑑 } ) |
| 799 |
765 772 784 798
|
ssltbday |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ) ) |
| 800 |
764 799
|
eqsstrrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ) ) |
| 801 |
676
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝜑 ) |
| 802 |
801 1
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑁 ∈ ℕ0s ) |
| 803 |
310
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑔 +s 𝑖 ) <s 𝑁 ) |
| 804 |
803
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s 𝑖 ) <s 𝑁 ) |
| 805 |
802 766 768 770 786 804
|
bdaypw2bnd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( bday ‘ ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ) ⊆ ( bday ‘ 𝑁 ) ) |
| 806 |
800 805
|
sstrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ) |
| 807 |
|
onslt |
⊢ ( ( 𝑁 ∈ Ons ∧ ( 𝑁 +s 1s ) ∈ Ons ) → ( 𝑁 <s ( 𝑁 +s 1s ) ↔ ( bday ‘ 𝑁 ) ∈ ( bday ‘ ( 𝑁 +s 1s ) ) ) ) |
| 808 |
263 405 807
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 <s ( 𝑁 +s 1s ) ↔ ( bday ‘ 𝑁 ) ∈ ( bday ‘ ( 𝑁 +s 1s ) ) ) ) |
| 809 |
801 808
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑁 <s ( 𝑁 +s 1s ) ↔ ( bday ‘ 𝑁 ) ∈ ( bday ‘ ( 𝑁 +s 1s ) ) ) ) |
| 810 |
809
|
notbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ¬ 𝑁 <s ( 𝑁 +s 1s ) ↔ ¬ ( bday ‘ 𝑁 ) ∈ ( bday ‘ ( 𝑁 +s 1s ) ) ) ) |
| 811 |
325 34
|
slenltd |
⊢ ( 𝜑 → ( ( 𝑁 +s 1s ) ≤s 𝑁 ↔ ¬ 𝑁 <s ( 𝑁 +s 1s ) ) ) |
| 812 |
801 811
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( 𝑁 +s 1s ) ≤s 𝑁 ↔ ¬ 𝑁 <s ( 𝑁 +s 1s ) ) ) |
| 813 |
|
bdayelon |
⊢ ( bday ‘ ( 𝑁 +s 1s ) ) ∈ On |
| 814 |
|
ontri1 |
⊢ ( ( ( bday ‘ ( 𝑁 +s 1s ) ) ∈ On ∧ ( bday ‘ 𝑁 ) ∈ On ) → ( ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ↔ ¬ ( bday ‘ 𝑁 ) ∈ ( bday ‘ ( 𝑁 +s 1s ) ) ) ) |
| 815 |
813 8 814
|
mp2an |
⊢ ( ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ↔ ¬ ( bday ‘ 𝑁 ) ∈ ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 816 |
815
|
a1i |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ↔ ¬ ( bday ‘ 𝑁 ) ∈ ( bday ‘ ( 𝑁 +s 1s ) ) ) ) |
| 817 |
810 812 816
|
3bitr4d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( 𝑁 +s 1s ) ≤s 𝑁 ↔ ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ) ) |
| 818 |
806 817
|
mpbird |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑁 +s 1s ) ≤s 𝑁 ) |
| 819 |
818
|
expr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) → ( 𝑁 +s 1s ) ≤s 𝑁 ) ) |
| 820 |
763 819
|
mtod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ¬ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) |
| 821 |
|
orel1 |
⊢ ( ¬ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) → ( ( ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ∨ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) → ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) |
| 822 |
820 821
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ( ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ∨ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) → ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) |
| 823 |
597
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑔 ∈ ℕ0s ) |
| 824 |
601
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ℎ ∈ ℕ0s ) |
| 825 |
|
n0mulscl |
⊢ ( ( 2s ∈ ℕ0s ∧ ℎ ∈ ℕ0s ) → ( 2s ·s ℎ ) ∈ ℕ0s ) |
| 826 |
422 824 825
|
sylancr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ℎ ) ∈ ℕ0s ) |
| 827 |
|
n0addscl |
⊢ ( ( ( 2s ·s ℎ ) ∈ ℕ0s ∧ 1s ∈ ℕ0s ) → ( ( 2s ·s ℎ ) +s 1s ) ∈ ℕ0s ) |
| 828 |
826 319 827
|
sylancl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ℎ ) +s 1s ) ∈ ℕ0s ) |
| 829 |
|
peano2n0s |
⊢ ( 𝑖 ∈ ℕ0s → ( 𝑖 +s 1s ) ∈ ℕ0s ) |
| 830 |
606 829
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑖 +s 1s ) ∈ ℕ0s ) |
| 831 |
830
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 𝑖 +s 1s ) ∈ ℕ0s ) |
| 832 |
|
simpll2 |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) |
| 833 |
615
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 834 |
54 606 359
|
sylancr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 2s ↑s 𝑖 ) ∈ No ) |
| 835 |
834
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s 𝑖 ) ∈ No ) |
| 836 |
823
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑔 ∈ No ) |
| 837 |
835 836
|
mulscld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ∈ No ) |
| 838 |
601
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ℎ ∈ No ) |
| 839 |
838
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ℎ ∈ No ) |
| 840 |
606
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑖 ∈ ℕ0s ) |
| 841 |
837 839 840
|
pw2divsdird |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) /su ( 2s ↑s 𝑖 ) ) = ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) /su ( 2s ↑s 𝑖 ) ) +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 842 |
836 840
|
pw2divscan3d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) /su ( 2s ↑s 𝑖 ) ) = 𝑔 ) |
| 843 |
842
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) /su ( 2s ↑s 𝑖 ) ) +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 844 |
841 843
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) /su ( 2s ↑s 𝑖 ) ) = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 845 |
833 844
|
eqtr4d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑐 = ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) /su ( 2s ↑s 𝑖 ) ) ) |
| 846 |
845
|
sneqd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → { 𝑐 } = { ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) /su ( 2s ↑s 𝑖 ) ) } ) |
| 847 |
|
simprlr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑑 = ( 𝑔 +s 1s ) ) |
| 848 |
330
|
a1i |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 1s ∈ No ) |
| 849 |
837 839 848
|
addsassd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) = ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ( ℎ +s 1s ) ) ) |
| 850 |
849
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) /su ( 2s ↑s 𝑖 ) ) = ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ( ℎ +s 1s ) ) /su ( 2s ↑s 𝑖 ) ) ) |
| 851 |
|
simprr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) |
| 852 |
851 835
|
eqeltrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ℎ +s 1s ) ∈ No ) |
| 853 |
837 852 840
|
pw2divsdird |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ( ℎ +s 1s ) ) /su ( 2s ↑s 𝑖 ) ) = ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) /su ( 2s ↑s 𝑖 ) ) +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ) |
| 854 |
851
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) = ( ( 2s ↑s 𝑖 ) /su ( 2s ↑s 𝑖 ) ) ) |
| 855 |
840
|
pw2divsidd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s 𝑖 ) /su ( 2s ↑s 𝑖 ) ) = 1s ) |
| 856 |
854 855
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) = 1s ) |
| 857 |
842 856
|
oveq12d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) /su ( 2s ↑s 𝑖 ) ) +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) = ( 𝑔 +s 1s ) ) |
| 858 |
853 857
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ( ℎ +s 1s ) ) /su ( 2s ↑s 𝑖 ) ) = ( 𝑔 +s 1s ) ) |
| 859 |
850 858
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) /su ( 2s ↑s 𝑖 ) ) = ( 𝑔 +s 1s ) ) |
| 860 |
847 859
|
eqtr4d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑑 = ( ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) |
| 861 |
860
|
sneqd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → { 𝑑 } = { ( ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) /su ( 2s ↑s 𝑖 ) ) } ) |
| 862 |
846 861
|
oveq12d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( { 𝑐 } |s { 𝑑 } ) = ( { ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) /su ( 2s ↑s 𝑖 ) ) } |s { ( ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) /su ( 2s ↑s 𝑖 ) ) } ) ) |
| 863 |
422 840 748
|
sylancr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s 𝑖 ) ∈ ℕ0s ) |
| 864 |
|
n0mulscl |
⊢ ( ( ( 2s ↑s 𝑖 ) ∈ ℕ0s ∧ 𝑔 ∈ ℕ0s ) → ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ∈ ℕ0s ) |
| 865 |
863 823 864
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ∈ ℕ0s ) |
| 866 |
|
n0addscl |
⊢ ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ∈ ℕ0s ∧ ℎ ∈ ℕ0s ) → ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ∈ ℕ0s ) |
| 867 |
865 824 866
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ∈ ℕ0s ) |
| 868 |
867
|
n0zsd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ∈ ℤs ) |
| 869 |
868 840
|
pw2cutp1 |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( { ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) /su ( 2s ↑s 𝑖 ) ) } |s { ( ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) /su ( 2s ↑s 𝑖 ) ) } ) = ( ( ( 2s ·s ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) |
| 870 |
54
|
a1i |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 2s ∈ No ) |
| 871 |
870 837 839
|
addsdid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ) = ( ( 2s ·s ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ) +s ( 2s ·s ℎ ) ) ) |
| 872 |
|
expsp1 |
⊢ ( ( 2s ∈ No ∧ 𝑖 ∈ ℕ0s ) → ( 2s ↑s ( 𝑖 +s 1s ) ) = ( ( 2s ↑s 𝑖 ) ·s 2s ) ) |
| 873 |
54 840 872
|
sylancr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s ( 𝑖 +s 1s ) ) = ( ( 2s ↑s 𝑖 ) ·s 2s ) ) |
| 874 |
835 870
|
mulscomd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s 𝑖 ) ·s 2s ) = ( 2s ·s ( 2s ↑s 𝑖 ) ) ) |
| 875 |
873 874
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s ( 𝑖 +s 1s ) ) = ( 2s ·s ( 2s ↑s 𝑖 ) ) ) |
| 876 |
875
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) = ( ( 2s ·s ( 2s ↑s 𝑖 ) ) ·s 𝑔 ) ) |
| 877 |
870 835 836
|
mulsassd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ( 2s ↑s 𝑖 ) ) ·s 𝑔 ) = ( 2s ·s ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ) ) |
| 878 |
876 877
|
eqtr2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ) = ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) ) |
| 879 |
878
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ) +s ( 2s ·s ℎ ) ) = ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) ) |
| 880 |
871 879
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ) = ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) ) |
| 881 |
880
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ) +s 1s ) = ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) +s 1s ) ) |
| 882 |
881
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( 2s ·s ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) = ( ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) |
| 883 |
869 882
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( { ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) /su ( 2s ↑s 𝑖 ) ) } |s { ( ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) /su ( 2s ↑s 𝑖 ) ) } ) = ( ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) |
| 884 |
832 862 883
|
3eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑤 = ( ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) |
| 885 |
|
expscl |
⊢ ( ( 2s ∈ No ∧ ( 𝑖 +s 1s ) ∈ ℕ0s ) → ( 2s ↑s ( 𝑖 +s 1s ) ) ∈ No ) |
| 886 |
54 831 885
|
sylancr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s ( 𝑖 +s 1s ) ) ∈ No ) |
| 887 |
886 836
|
mulscld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) ∈ No ) |
| 888 |
826
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ℎ ) ∈ No ) |
| 889 |
887 888 848
|
addsassd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) +s 1s ) = ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s ℎ ) +s 1s ) ) ) |
| 890 |
889
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) = ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s ℎ ) +s 1s ) ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) |
| 891 |
884 890
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑤 = ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s ℎ ) +s 1s ) ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) |
| 892 |
828
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ℎ ) +s 1s ) ∈ No ) |
| 893 |
887 892 831
|
pw2divsdird |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s ℎ ) +s 1s ) ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) = ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) ) |
| 894 |
836 831
|
pw2divscan3d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) = 𝑔 ) |
| 895 |
894
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) ) |
| 896 |
893 895
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s ℎ ) +s 1s ) ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) ) |
| 897 |
891 896
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) ) |
| 898 |
848 870 888
|
sltadd2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 1s <s 2s ↔ ( ( 2s ·s ℎ ) +s 1s ) <s ( ( 2s ·s ℎ ) +s 2s ) ) ) |
| 899 |
540 898
|
mpbii |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ℎ ) +s 1s ) <s ( ( 2s ·s ℎ ) +s 2s ) ) |
| 900 |
851
|
oveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ( ℎ +s 1s ) ) = ( 2s ·s ( 2s ↑s 𝑖 ) ) ) |
| 901 |
870 839 848
|
addsdid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ( ℎ +s 1s ) ) = ( ( 2s ·s ℎ ) +s ( 2s ·s 1s ) ) ) |
| 902 |
|
mulsrid |
⊢ ( 2s ∈ No → ( 2s ·s 1s ) = 2s ) |
| 903 |
54 902
|
ax-mp |
⊢ ( 2s ·s 1s ) = 2s |
| 904 |
903
|
oveq2i |
⊢ ( ( 2s ·s ℎ ) +s ( 2s ·s 1s ) ) = ( ( 2s ·s ℎ ) +s 2s ) |
| 905 |
904
|
a1i |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ℎ ) +s ( 2s ·s 1s ) ) = ( ( 2s ·s ℎ ) +s 2s ) ) |
| 906 |
901 905
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ( ℎ +s 1s ) ) = ( ( 2s ·s ℎ ) +s 2s ) ) |
| 907 |
900 906
|
eqtr3d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ( 2s ↑s 𝑖 ) ) = ( ( 2s ·s ℎ ) +s 2s ) ) |
| 908 |
874 907
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s 𝑖 ) ·s 2s ) = ( ( 2s ·s ℎ ) +s 2s ) ) |
| 909 |
873 908
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s ( 𝑖 +s 1s ) ) = ( ( 2s ·s ℎ ) +s 2s ) ) |
| 910 |
899 909
|
breqtrrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s ( 𝑖 +s 1s ) ) ) |
| 911 |
840
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑖 ∈ No ) |
| 912 |
836 911 848
|
addsassd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 𝑔 +s 𝑖 ) +s 1s ) = ( 𝑔 +s ( 𝑖 +s 1s ) ) ) |
| 913 |
803
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s 𝑖 ) <s 𝑁 ) |
| 914 |
836 911
|
addscld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s 𝑖 ) ∈ No ) |
| 915 |
676
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝜑 ) |
| 916 |
915 34
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑁 ∈ No ) |
| 917 |
914 916 848
|
sltadd1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 𝑔 +s 𝑖 ) <s 𝑁 ↔ ( ( 𝑔 +s 𝑖 ) +s 1s ) <s ( 𝑁 +s 1s ) ) ) |
| 918 |
913 917
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 𝑔 +s 𝑖 ) +s 1s ) <s ( 𝑁 +s 1s ) ) |
| 919 |
912 918
|
eqbrtrrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s ( 𝑖 +s 1s ) ) <s ( 𝑁 +s 1s ) ) |
| 920 |
|
oveq1 |
⊢ ( 𝑏 = ( ( 2s ·s ℎ ) +s 1s ) → ( 𝑏 /su ( 2s ↑s 𝑞 ) ) = ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) |
| 921 |
920
|
oveq2d |
⊢ ( 𝑏 = ( ( 2s ·s ℎ ) +s 1s ) → ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ) |
| 922 |
921
|
eqeq2d |
⊢ ( 𝑏 = ( ( 2s ·s ℎ ) +s 1s ) → ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ) ) |
| 923 |
|
breq1 |
⊢ ( 𝑏 = ( ( 2s ·s ℎ ) +s 1s ) → ( 𝑏 <s ( 2s ↑s 𝑞 ) ↔ ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s 𝑞 ) ) ) |
| 924 |
922 923
|
3anbi12d |
⊢ ( 𝑏 = ( ( 2s ·s ℎ ) +s 1s ) → ( ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ∧ ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 925 |
|
oveq2 |
⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( 2s ↑s 𝑞 ) = ( 2s ↑s ( 𝑖 +s 1s ) ) ) |
| 926 |
925
|
oveq2d |
⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) = ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) |
| 927 |
926
|
oveq2d |
⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) ) |
| 928 |
927
|
eqeq2d |
⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) ) ) |
| 929 |
925
|
breq2d |
⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s 𝑞 ) ↔ ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) |
| 930 |
|
oveq2 |
⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( 𝑔 +s 𝑞 ) = ( 𝑔 +s ( 𝑖 +s 1s ) ) ) |
| 931 |
930
|
breq1d |
⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑔 +s ( 𝑖 +s 1s ) ) <s ( 𝑁 +s 1s ) ) ) |
| 932 |
928 929 931
|
3anbi123d |
⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ∧ ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) ∧ ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s ( 𝑖 +s 1s ) ) ∧ ( 𝑔 +s ( 𝑖 +s 1s ) ) <s ( 𝑁 +s 1s ) ) ) ) |
| 933 |
552 924 932
|
rspc3ev |
⊢ ( ( ( 𝑔 ∈ ℕ0s ∧ ( ( 2s ·s ℎ ) +s 1s ) ∈ ℕ0s ∧ ( 𝑖 +s 1s ) ∈ ℕ0s ) ∧ ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) ∧ ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s ( 𝑖 +s 1s ) ) ∧ ( 𝑔 +s ( 𝑖 +s 1s ) ) <s ( 𝑁 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 934 |
823 828 831 897 910 919 933
|
syl33anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 935 |
934
|
expr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 936 |
822 935
|
syld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ( ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ∨ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 937 |
759 936
|
sylbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ( ℎ +s 1s ) ≤s ( 2s ↑s 𝑖 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 938 |
752 937
|
mpd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 939 |
938
|
expr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 𝑑 = ( 𝑔 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 940 |
939
|
adantrr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑑 = ( 𝑔 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 941 |
744 940
|
mpd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 942 |
941
|
expr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( ( 𝑔 +s 1s ) ≤s 𝑑 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 943 |
670 942
|
mpd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 944 |
943
|
expr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ) → ( ( 𝑔 +s 1s ) ≤s 𝑗 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 945 |
650 944
|
mpd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 946 |
945
|
expr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 <s 𝑗 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 947 |
626
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑐 <s 𝑑 ) |
| 948 |
615
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 949 |
579
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 950 |
|
simprr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑔 = 𝑗 ) |
| 951 |
950
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 952 |
949 951
|
eqtr4d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 953 |
947 948 952
|
3brtr3d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 954 |
838
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ℎ ∈ No ) |
| 955 |
606
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑖 ∈ ℕ0s ) |
| 956 |
954 955
|
pw2divscld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) ∈ No ) |
| 957 |
587
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑘 ∈ No ) |
| 958 |
581
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑙 ∈ ℕ0s ) |
| 959 |
957 958
|
pw2divscld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ∈ No ) |
| 960 |
597
|
n0snod |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑔 ∈ No ) |
| 961 |
960
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑔 ∈ No ) |
| 962 |
956 959 961
|
sltadd2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ↔ ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ) |
| 963 |
953 962
|
mpbird |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) |
| 964 |
601
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ℎ ∈ ℕ0s ) |
| 965 |
|
simpl3 |
⊢ ( ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) → 𝑙 ∈ ℕ0s ) |
| 966 |
965
|
adantl |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑙 ∈ ℕ0s ) |
| 967 |
966
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑙 ∈ ℕ0s ) |
| 968 |
606
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑖 ∈ ℕ0s ) |
| 969 |
|
n0subs |
⊢ ( ( 𝑙 ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) → ( 𝑙 ≤s 𝑖 ↔ ( 𝑖 -s 𝑙 ) ∈ ℕ0s ) ) |
| 970 |
967 968 969
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑙 ≤s 𝑖 ↔ ( 𝑖 -s 𝑙 ) ∈ ℕ0s ) ) |
| 971 |
970
|
biimpd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑙 ≤s 𝑖 → ( 𝑖 -s 𝑙 ) ∈ ℕ0s ) ) |
| 972 |
971
|
impr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑖 -s 𝑙 ) ∈ ℕ0s ) |
| 973 |
|
n0expscl |
⊢ ( ( 2s ∈ ℕ0s ∧ ( 𝑖 -s 𝑙 ) ∈ ℕ0s ) → ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ∈ ℕ0s ) |
| 974 |
422 972 973
|
sylancr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ∈ ℕ0s ) |
| 975 |
586
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑘 ∈ ℕ0s ) |
| 976 |
|
n0mulscl |
⊢ ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ) → ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∈ ℕ0s ) |
| 977 |
974 975 976
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∈ ℕ0s ) |
| 978 |
606
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑖 ∈ ℕ0s ) |
| 979 |
|
simprlr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) |
| 980 |
966
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑙 ∈ ℕ0s ) |
| 981 |
980
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑙 ∈ No ) |
| 982 |
972
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑖 -s 𝑙 ) ∈ No ) |
| 983 |
981 982
|
addscomd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑙 +s ( 𝑖 -s 𝑙 ) ) = ( ( 𝑖 -s 𝑙 ) +s 𝑙 ) ) |
| 984 |
978
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑖 ∈ No ) |
| 985 |
|
npcans |
⊢ ( ( 𝑖 ∈ No ∧ 𝑙 ∈ No ) → ( ( 𝑖 -s 𝑙 ) +s 𝑙 ) = 𝑖 ) |
| 986 |
984 981 985
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( 𝑖 -s 𝑙 ) +s 𝑙 ) = 𝑖 ) |
| 987 |
983 986
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑙 +s ( 𝑖 -s 𝑙 ) ) = 𝑖 ) |
| 988 |
987
|
oveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s ( 𝑙 +s ( 𝑖 -s 𝑙 ) ) ) = ( 2s ↑s 𝑖 ) ) |
| 989 |
988
|
oveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s ( 𝑙 +s ( 𝑖 -s 𝑙 ) ) ) ) = ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) |
| 990 |
989
|
eqcomd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) = ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s ( 𝑙 +s ( 𝑖 -s 𝑙 ) ) ) ) ) |
| 991 |
975
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑘 ∈ No ) |
| 992 |
991 980 972
|
pw2divscan4d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑘 /su ( 2s ↑s 𝑙 ) ) = ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s ( 𝑙 +s ( 𝑖 -s 𝑙 ) ) ) ) ) |
| 993 |
990 992
|
eqtr4d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) = ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) |
| 994 |
993
|
eqcomd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑘 /su ( 2s ↑s 𝑙 ) ) = ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) |
| 995 |
979 994
|
breqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) |
| 996 |
964
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ℎ ∈ No ) |
| 997 |
977
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∈ No ) |
| 998 |
996 997 978
|
pw2sltdiv1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ↔ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) ) |
| 999 |
995 998
|
mpbird |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ) |
| 1000 |
700
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ℎ <s ( 2s ↑s 𝑖 ) ) |
| 1001 |
580
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑘 <s ( 2s ↑s 𝑙 ) ) |
| 1002 |
|
n0expscl |
⊢ ( ( 2s ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) → ( 2s ↑s 𝑙 ) ∈ ℕ0s ) |
| 1003 |
422 980 1002
|
sylancr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s 𝑙 ) ∈ ℕ0s ) |
| 1004 |
1003
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s 𝑙 ) ∈ No ) |
| 1005 |
974
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ∈ No ) |
| 1006 |
|
nnsgt0 |
⊢ ( 2s ∈ ℕs → 0s <s 2s ) |
| 1007 |
420 1006
|
ax-mp |
⊢ 0s <s 2s |
| 1008 |
|
expsgt0 |
⊢ ( ( 2s ∈ No ∧ ( 𝑖 -s 𝑙 ) ∈ ℕ0s ∧ 0s <s 2s ) → 0s <s ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ) |
| 1009 |
54 1007 1008
|
mp3an13 |
⊢ ( ( 𝑖 -s 𝑙 ) ∈ ℕ0s → 0s <s ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ) |
| 1010 |
972 1009
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 0s <s ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ) |
| 1011 |
991 1004 1005 1010
|
sltmul2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑘 <s ( 2s ↑s 𝑙 ) ↔ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s ( 2s ↑s 𝑙 ) ) ) ) |
| 1012 |
1001 1011
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s ( 2s ↑s 𝑙 ) ) ) |
| 1013 |
|
expadds |
⊢ ( ( 2s ∈ No ∧ ( 𝑖 -s 𝑙 ) ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) → ( 2s ↑s ( ( 𝑖 -s 𝑙 ) +s 𝑙 ) ) = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s ( 2s ↑s 𝑙 ) ) ) |
| 1014 |
54 1013
|
mp3an1 |
⊢ ( ( ( 𝑖 -s 𝑙 ) ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) → ( 2s ↑s ( ( 𝑖 -s 𝑙 ) +s 𝑙 ) ) = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s ( 2s ↑s 𝑙 ) ) ) |
| 1015 |
972 980 1014
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s ( ( 𝑖 -s 𝑙 ) +s 𝑙 ) ) = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s ( 2s ↑s 𝑙 ) ) ) |
| 1016 |
986
|
oveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s ( ( 𝑖 -s 𝑙 ) +s 𝑙 ) ) = ( 2s ↑s 𝑖 ) ) |
| 1017 |
1015 1016
|
eqtr3d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s ( 2s ↑s 𝑙 ) ) = ( 2s ↑s 𝑖 ) ) |
| 1018 |
1012 1017
|
breqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑖 ) ) |
| 1019 |
999 1000 1018
|
3jca |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑖 ) ) ) |
| 1020 |
615
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 1021 |
579
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 1022 |
|
simpllr |
⊢ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) → 𝑔 = 𝑗 ) |
| 1023 |
1022
|
adantl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑔 = 𝑗 ) |
| 1024 |
1023 993
|
oveq12d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 1025 |
1021 1024
|
eqtr4d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) ) |
| 1026 |
803
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑔 +s 𝑖 ) <s 𝑁 ) |
| 1027 |
1020 1025 1026
|
3jca |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) |
| 1028 |
|
breq1 |
⊢ ( 𝑚 = ℎ → ( 𝑚 <s 𝑛 ↔ ℎ <s 𝑛 ) ) |
| 1029 |
|
breq1 |
⊢ ( 𝑚 = ℎ → ( 𝑚 <s ( 2s ↑s 𝑜 ) ↔ ℎ <s ( 2s ↑s 𝑜 ) ) ) |
| 1030 |
1028 1029
|
3anbi12d |
⊢ ( 𝑚 = ℎ → ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ↔ ( ℎ <s 𝑛 ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ) ) |
| 1031 |
|
oveq1 |
⊢ ( 𝑚 = ℎ → ( 𝑚 /su ( 2s ↑s 𝑜 ) ) = ( ℎ /su ( 2s ↑s 𝑜 ) ) ) |
| 1032 |
1031
|
oveq2d |
⊢ ( 𝑚 = ℎ → ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1033 |
1032
|
eqeq2d |
⊢ ( 𝑚 = ℎ → ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ) ) |
| 1034 |
1033
|
3anbi1d |
⊢ ( 𝑚 = ℎ → ( ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) |
| 1035 |
1030 1034
|
anbi12d |
⊢ ( 𝑚 = ℎ → ( ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ↔ ( ( ℎ <s 𝑛 ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) |
| 1036 |
|
breq2 |
⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( ℎ <s 𝑛 ↔ ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ) ) |
| 1037 |
|
breq1 |
⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( 𝑛 <s ( 2s ↑s 𝑜 ) ↔ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑜 ) ) ) |
| 1038 |
1036 1037
|
3anbi13d |
⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( ( ℎ <s 𝑛 ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ↔ ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑜 ) ) ) ) |
| 1039 |
|
oveq1 |
⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( 𝑛 /su ( 2s ↑s 𝑜 ) ) = ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) |
| 1040 |
1039
|
oveq2d |
⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1041 |
1040
|
eqeq2d |
⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) ) ) |
| 1042 |
1041
|
3anbi2d |
⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) |
| 1043 |
1038 1042
|
anbi12d |
⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( ( ( ℎ <s 𝑛 ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ↔ ( ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) |
| 1044 |
|
oveq2 |
⊢ ( 𝑜 = 𝑖 → ( 2s ↑s 𝑜 ) = ( 2s ↑s 𝑖 ) ) |
| 1045 |
1044
|
breq2d |
⊢ ( 𝑜 = 𝑖 → ( ℎ <s ( 2s ↑s 𝑜 ) ↔ ℎ <s ( 2s ↑s 𝑖 ) ) ) |
| 1046 |
1044
|
breq2d |
⊢ ( 𝑜 = 𝑖 → ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑜 ) ↔ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑖 ) ) ) |
| 1047 |
1045 1046
|
3anbi23d |
⊢ ( 𝑜 = 𝑖 → ( ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑜 ) ) ↔ ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑖 ) ) ) ) |
| 1048 |
1044
|
oveq2d |
⊢ ( 𝑜 = 𝑖 → ( ℎ /su ( 2s ↑s 𝑜 ) ) = ( ℎ /su ( 2s ↑s 𝑖 ) ) ) |
| 1049 |
1048
|
oveq2d |
⊢ ( 𝑜 = 𝑖 → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 1050 |
1049
|
eqeq2d |
⊢ ( 𝑜 = 𝑖 → ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) ) |
| 1051 |
1044
|
oveq2d |
⊢ ( 𝑜 = 𝑖 → ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) = ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) |
| 1052 |
1051
|
oveq2d |
⊢ ( 𝑜 = 𝑖 → ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) ) |
| 1053 |
1052
|
eqeq2d |
⊢ ( 𝑜 = 𝑖 → ( 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) ) ) |
| 1054 |
|
oveq2 |
⊢ ( 𝑜 = 𝑖 → ( 𝑔 +s 𝑜 ) = ( 𝑔 +s 𝑖 ) ) |
| 1055 |
1054
|
breq1d |
⊢ ( 𝑜 = 𝑖 → ( ( 𝑔 +s 𝑜 ) <s 𝑁 ↔ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) |
| 1056 |
1050 1053 1055
|
3anbi123d |
⊢ ( 𝑜 = 𝑖 → ( ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) |
| 1057 |
1047 1056
|
anbi12d |
⊢ ( 𝑜 = 𝑖 → ( ( ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ↔ ( ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑖 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ) |
| 1058 |
1035 1043 1057
|
rspc3ev |
⊢ ( ( ( ℎ ∈ ℕ0s ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑖 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) |
| 1059 |
964 977 978 1019 1027 1058
|
syl32anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) |
| 1060 |
1059
|
expr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑙 ≤s 𝑖 → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) |
| 1061 |
|
n0subs |
⊢ ( ( 𝑖 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) → ( 𝑖 ≤s 𝑙 ↔ ( 𝑙 -s 𝑖 ) ∈ ℕ0s ) ) |
| 1062 |
968 967 1061
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑖 ≤s 𝑙 ↔ ( 𝑙 -s 𝑖 ) ∈ ℕ0s ) ) |
| 1063 |
1062
|
biimpd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑖 ≤s 𝑙 → ( 𝑙 -s 𝑖 ) ∈ ℕ0s ) ) |
| 1064 |
1063
|
impr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑙 -s 𝑖 ) ∈ ℕ0s ) |
| 1065 |
|
n0expscl |
⊢ ( ( 2s ∈ ℕ0s ∧ ( 𝑙 -s 𝑖 ) ∈ ℕ0s ) → ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ∈ ℕ0s ) |
| 1066 |
422 1064 1065
|
sylancr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ∈ ℕ0s ) |
| 1067 |
601
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ℎ ∈ ℕ0s ) |
| 1068 |
|
n0mulscl |
⊢ ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ∈ ℕ0s ∧ ℎ ∈ ℕ0s ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) ∈ ℕ0s ) |
| 1069 |
1066 1067 1068
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) ∈ ℕ0s ) |
| 1070 |
586
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑘 ∈ ℕ0s ) |
| 1071 |
966
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑙 ∈ ℕ0s ) |
| 1072 |
1067
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ℎ ∈ No ) |
| 1073 |
606
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑖 ∈ ℕ0s ) |
| 1074 |
1072 1073 1064
|
pw2divscan4d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) = ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s ( 𝑖 +s ( 𝑙 -s 𝑖 ) ) ) ) ) |
| 1075 |
1073
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑖 ∈ No ) |
| 1076 |
1064
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑙 -s 𝑖 ) ∈ No ) |
| 1077 |
1075 1076
|
addscomd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑖 +s ( 𝑙 -s 𝑖 ) ) = ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) |
| 1078 |
1077
|
oveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 2s ↑s ( 𝑖 +s ( 𝑙 -s 𝑖 ) ) ) = ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) ) |
| 1079 |
1078
|
oveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s ( 𝑖 +s ( 𝑙 -s 𝑖 ) ) ) ) = ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) ) ) |
| 1080 |
1074 1079
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) = ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) ) ) |
| 1081 |
1071
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑙 ∈ No ) |
| 1082 |
|
npcans |
⊢ ( ( 𝑙 ∈ No ∧ 𝑖 ∈ No ) → ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) = 𝑙 ) |
| 1083 |
1081 1075 1082
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) = 𝑙 ) |
| 1084 |
1083
|
oveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) = ( 2s ↑s 𝑙 ) ) |
| 1085 |
1084
|
oveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) ) = ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) |
| 1086 |
1080 1085
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) = ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) |
| 1087 |
|
simprlr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) |
| 1088 |
1086 1087
|
eqbrtrrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) |
| 1089 |
|
expscl |
⊢ ( ( 2s ∈ No ∧ ( 𝑙 -s 𝑖 ) ∈ ℕ0s ) → ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ∈ No ) |
| 1090 |
54 1064 1089
|
sylancr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ∈ No ) |
| 1091 |
1090 1072
|
mulscld |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) ∈ No ) |
| 1092 |
1070
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑘 ∈ No ) |
| 1093 |
1091 1092 1071
|
pw2sltdiv1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ↔ ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 1094 |
1088 1093
|
mpbird |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ) |
| 1095 |
700
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ℎ <s ( 2s ↑s 𝑖 ) ) |
| 1096 |
54 1073 359
|
sylancr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 2s ↑s 𝑖 ) ∈ No ) |
| 1097 |
|
expsgt0 |
⊢ ( ( 2s ∈ No ∧ ( 𝑙 -s 𝑖 ) ∈ ℕ0s ∧ 0s <s 2s ) → 0s <s ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ) |
| 1098 |
54 1007 1097
|
mp3an13 |
⊢ ( ( 𝑙 -s 𝑖 ) ∈ ℕ0s → 0s <s ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ) |
| 1099 |
1064 1098
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 0s <s ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ) |
| 1100 |
1072 1096 1090 1099
|
sltmul2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ℎ <s ( 2s ↑s 𝑖 ) ↔ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ( 2s ↑s 𝑖 ) ) ) ) |
| 1101 |
1095 1100
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ( 2s ↑s 𝑖 ) ) ) |
| 1102 |
|
expadds |
⊢ ( ( 2s ∈ No ∧ ( 𝑙 -s 𝑖 ) ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) → ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ( 2s ↑s 𝑖 ) ) ) |
| 1103 |
54 1102
|
mp3an1 |
⊢ ( ( ( 𝑙 -s 𝑖 ) ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) → ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ( 2s ↑s 𝑖 ) ) ) |
| 1104 |
1064 1073 1103
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ( 2s ↑s 𝑖 ) ) ) |
| 1105 |
1104 1084
|
eqtr3d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ( 2s ↑s 𝑖 ) ) = ( 2s ↑s 𝑙 ) ) |
| 1106 |
1101 1105
|
breqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑙 ) ) |
| 1107 |
580
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑘 <s ( 2s ↑s 𝑙 ) ) |
| 1108 |
1094 1106 1107
|
3jca |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑙 ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ) ) |
| 1109 |
615
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) |
| 1110 |
1086
|
oveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) ) |
| 1111 |
1109 1110
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) ) |
| 1112 |
579
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 1113 |
|
simpllr |
⊢ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) → 𝑔 = 𝑗 ) |
| 1114 |
1113
|
adantl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑔 = 𝑗 ) |
| 1115 |
1114
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 1116 |
1112 1115
|
eqtr4d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 1117 |
1114
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑔 +s 𝑙 ) = ( 𝑗 +s 𝑙 ) ) |
| 1118 |
680
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑗 +s 𝑙 ) <s 𝑁 ) |
| 1119 |
1117 1118
|
eqbrtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑔 +s 𝑙 ) <s 𝑁 ) |
| 1120 |
1111 1116 1119
|
3jca |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ ( 𝑔 +s 𝑙 ) <s 𝑁 ) ) |
| 1121 |
|
breq1 |
⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( 𝑚 <s 𝑛 ↔ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑛 ) ) |
| 1122 |
|
breq1 |
⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( 𝑚 <s ( 2s ↑s 𝑜 ) ↔ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ) ) |
| 1123 |
1121 1122
|
3anbi12d |
⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ↔ ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑛 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ) ) |
| 1124 |
|
oveq1 |
⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( 𝑚 /su ( 2s ↑s 𝑜 ) ) = ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) |
| 1125 |
1124
|
oveq2d |
⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1126 |
1125
|
eqeq2d |
⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ) ) |
| 1127 |
1126
|
3anbi1d |
⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) |
| 1128 |
1123 1127
|
anbi12d |
⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ↔ ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑛 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) |
| 1129 |
|
breq2 |
⊢ ( 𝑛 = 𝑘 → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑛 ↔ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ) ) |
| 1130 |
|
breq1 |
⊢ ( 𝑛 = 𝑘 → ( 𝑛 <s ( 2s ↑s 𝑜 ) ↔ 𝑘 <s ( 2s ↑s 𝑜 ) ) ) |
| 1131 |
1129 1130
|
3anbi13d |
⊢ ( 𝑛 = 𝑘 → ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑛 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ↔ ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑘 <s ( 2s ↑s 𝑜 ) ) ) ) |
| 1132 |
|
oveq1 |
⊢ ( 𝑛 = 𝑘 → ( 𝑛 /su ( 2s ↑s 𝑜 ) ) = ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) |
| 1133 |
1132
|
oveq2d |
⊢ ( 𝑛 = 𝑘 → ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1134 |
1133
|
eqeq2d |
⊢ ( 𝑛 = 𝑘 → ( 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) ) ) |
| 1135 |
1134
|
3anbi2d |
⊢ ( 𝑛 = 𝑘 → ( ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) |
| 1136 |
1131 1135
|
anbi12d |
⊢ ( 𝑛 = 𝑘 → ( ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑛 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ↔ ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑘 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) |
| 1137 |
|
oveq2 |
⊢ ( 𝑜 = 𝑙 → ( 2s ↑s 𝑜 ) = ( 2s ↑s 𝑙 ) ) |
| 1138 |
1137
|
breq2d |
⊢ ( 𝑜 = 𝑙 → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ↔ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑙 ) ) ) |
| 1139 |
1137
|
breq2d |
⊢ ( 𝑜 = 𝑙 → ( 𝑘 <s ( 2s ↑s 𝑜 ) ↔ 𝑘 <s ( 2s ↑s 𝑙 ) ) ) |
| 1140 |
1138 1139
|
3anbi23d |
⊢ ( 𝑜 = 𝑙 → ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑘 <s ( 2s ↑s 𝑜 ) ) ↔ ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑙 ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ) ) ) |
| 1141 |
1137
|
oveq2d |
⊢ ( 𝑜 = 𝑙 → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) = ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) |
| 1142 |
1141
|
oveq2d |
⊢ ( 𝑜 = 𝑙 → ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) ) |
| 1143 |
1142
|
eqeq2d |
⊢ ( 𝑜 = 𝑙 → ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) ) ) |
| 1144 |
1137
|
oveq2d |
⊢ ( 𝑜 = 𝑙 → ( 𝑘 /su ( 2s ↑s 𝑜 ) ) = ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) |
| 1145 |
1144
|
oveq2d |
⊢ ( 𝑜 = 𝑙 → ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 1146 |
1145
|
eqeq2d |
⊢ ( 𝑜 = 𝑙 → ( 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ) |
| 1147 |
|
oveq2 |
⊢ ( 𝑜 = 𝑙 → ( 𝑔 +s 𝑜 ) = ( 𝑔 +s 𝑙 ) ) |
| 1148 |
1147
|
breq1d |
⊢ ( 𝑜 = 𝑙 → ( ( 𝑔 +s 𝑜 ) <s 𝑁 ↔ ( 𝑔 +s 𝑙 ) <s 𝑁 ) ) |
| 1149 |
1143 1146 1148
|
3anbi123d |
⊢ ( 𝑜 = 𝑙 → ( ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ ( 𝑔 +s 𝑙 ) <s 𝑁 ) ) ) |
| 1150 |
1140 1149
|
anbi12d |
⊢ ( 𝑜 = 𝑙 → ( ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑘 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ↔ ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑙 ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ ( 𝑔 +s 𝑙 ) <s 𝑁 ) ) ) ) |
| 1151 |
1128 1136 1150
|
rspc3ev |
⊢ ( ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑙 ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ ( 𝑔 +s 𝑙 ) <s 𝑁 ) ) ) → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) |
| 1152 |
1069 1070 1071 1108 1120 1151
|
syl32anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) |
| 1153 |
1152
|
expr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑖 ≤s 𝑙 → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) |
| 1154 |
967
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑙 ∈ No ) |
| 1155 |
968
|
n0snod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑖 ∈ No ) |
| 1156 |
|
sletric |
⊢ ( ( 𝑙 ∈ No ∧ 𝑖 ∈ No ) → ( 𝑙 ≤s 𝑖 ∨ 𝑖 ≤s 𝑙 ) ) |
| 1157 |
1154 1155 1156
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑙 ≤s 𝑖 ∨ 𝑖 ≤s 𝑙 ) ) |
| 1158 |
1060 1153 1157
|
mpjaod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) |
| 1159 |
597
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑔 ∈ ℕ0s ) |
| 1160 |
1159
|
adantr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑔 ∈ ℕ0s ) |
| 1161 |
|
simprl1 |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑚 ∈ ℕ0s ) |
| 1162 |
|
n0mulscl |
⊢ ( ( 2s ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s ) → ( 2s ·s 𝑚 ) ∈ ℕ0s ) |
| 1163 |
422 1161 1162
|
sylancr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s 𝑚 ) ∈ ℕ0s ) |
| 1164 |
|
peano2n0s |
⊢ ( ( 2s ·s 𝑚 ) ∈ ℕ0s → ( ( 2s ·s 𝑚 ) +s 1s ) ∈ ℕ0s ) |
| 1165 |
1163 1164
|
syl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s 𝑚 ) +s 1s ) ∈ ℕ0s ) |
| 1166 |
|
simprl3 |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑜 ∈ ℕ0s ) |
| 1167 |
|
peano2n0s |
⊢ ( 𝑜 ∈ ℕ0s → ( 𝑜 +s 1s ) ∈ ℕ0s ) |
| 1168 |
1166 1167
|
syl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑜 +s 1s ) ∈ ℕ0s ) |
| 1169 |
|
simpl2 |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) |
| 1170 |
1169
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) |
| 1171 |
1170
|
adantr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) |
| 1172 |
|
simprr1 |
⊢ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) → 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1173 |
1172
|
adantl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1174 |
|
n0expscl |
⊢ ( ( 2s ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) → ( 2s ↑s 𝑜 ) ∈ ℕ0s ) |
| 1175 |
422 1166 1174
|
sylancr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ↑s 𝑜 ) ∈ ℕ0s ) |
| 1176 |
|
n0mulscl |
⊢ ( ( ( 2s ↑s 𝑜 ) ∈ ℕ0s ∧ 𝑔 ∈ ℕ0s ) → ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ∈ ℕ0s ) |
| 1177 |
1175 1160 1176
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ∈ ℕ0s ) |
| 1178 |
1177
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ∈ No ) |
| 1179 |
1161
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑚 ∈ No ) |
| 1180 |
1178 1179 1166
|
pw2divsdird |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) /su ( 2s ↑s 𝑜 ) ) = ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) /su ( 2s ↑s 𝑜 ) ) +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1181 |
1160
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑔 ∈ No ) |
| 1182 |
1181 1166
|
pw2divscan3d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) /su ( 2s ↑s 𝑜 ) ) = 𝑔 ) |
| 1183 |
1182
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) /su ( 2s ↑s 𝑜 ) ) +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1184 |
1180 1183
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) /su ( 2s ↑s 𝑜 ) ) = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1185 |
1173 1184
|
eqtr4d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑐 = ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) /su ( 2s ↑s 𝑜 ) ) ) |
| 1186 |
1185
|
sneqd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → { 𝑐 } = { ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) /su ( 2s ↑s 𝑜 ) ) } ) |
| 1187 |
|
simprr2 |
⊢ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) → 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1188 |
1187
|
adantl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1189 |
676
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝜑 ) |
| 1190 |
1189
|
adantr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝜑 ) |
| 1191 |
1190 762
|
syl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ¬ ( 𝑁 +s 1s ) ≤s 𝑁 ) |
| 1192 |
330
|
a1i |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 1s ∈ No ) |
| 1193 |
|
simprl2 |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑛 ∈ ℕ0s ) |
| 1194 |
1193
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑛 ∈ No ) |
| 1195 |
1194 1179
|
subscld |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑛 -s 𝑚 ) ∈ No ) |
| 1196 |
1192 1195
|
sltnled |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 1s <s ( 𝑛 -s 𝑚 ) ↔ ¬ ( 𝑛 -s 𝑚 ) ≤s 1s ) ) |
| 1197 |
695
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 1198 |
1197
|
adantr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) |
| 1199 |
1170
|
adantr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) |
| 1200 |
1159
|
adantr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑔 ∈ ℕ0s ) |
| 1201 |
1200
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑔 ∈ No ) |
| 1202 |
1161
|
adantrr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑚 ∈ ℕ0s ) |
| 1203 |
|
peano2n0s |
⊢ ( 𝑚 ∈ ℕ0s → ( 𝑚 +s 1s ) ∈ ℕ0s ) |
| 1204 |
1202 1203
|
syl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 +s 1s ) ∈ ℕ0s ) |
| 1205 |
1204
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 +s 1s ) ∈ No ) |
| 1206 |
1166
|
adantrr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑜 ∈ ℕ0s ) |
| 1207 |
1205 1206
|
pw2divscld |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ∈ No ) |
| 1208 |
1201 1207
|
addscld |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ∈ No ) |
| 1209 |
620
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑐 ∈ No ) |
| 1210 |
1209
|
adantr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑐 ∈ No ) |
| 1211 |
1173
|
adantrr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1212 |
1179
|
adantrr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑚 ∈ No ) |
| 1213 |
1212
|
sltp1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑚 <s ( 𝑚 +s 1s ) ) |
| 1214 |
1212 1205 1206
|
pw2sltdiv1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 <s ( 𝑚 +s 1s ) ↔ ( 𝑚 /su ( 2s ↑s 𝑜 ) ) <s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1215 |
1212 1206
|
pw2divscld |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ∈ No ) |
| 1216 |
1215 1207 1201
|
sltadd2d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( ( 𝑚 /su ( 2s ↑s 𝑜 ) ) <s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ↔ ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) <s ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) ) |
| 1217 |
1214 1216
|
bitrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 <s ( 𝑚 +s 1s ) ↔ ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) <s ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) ) |
| 1218 |
1213 1217
|
mpbid |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) <s ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1219 |
1211 1218
|
eqbrtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑐 <s ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1220 |
1210 1208 1219
|
ssltsn |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → { 𝑐 } <<s { ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) } ) |
| 1221 |
622
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑑 ∈ No ) |
| 1222 |
1221
|
adantr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑑 ∈ No ) |
| 1223 |
1179 1192
|
addscomd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 +s 1s ) = ( 1s +s 𝑚 ) ) |
| 1224 |
1223
|
breq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑚 +s 1s ) <s 𝑛 ↔ ( 1s +s 𝑚 ) <s 𝑛 ) ) |
| 1225 |
1192 1179 1194
|
sltaddsubd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 1s +s 𝑚 ) <s 𝑛 ↔ 1s <s ( 𝑛 -s 𝑚 ) ) ) |
| 1226 |
1224 1225
|
bitrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑚 +s 1s ) <s 𝑛 ↔ 1s <s ( 𝑛 -s 𝑚 ) ) ) |
| 1227 |
1226
|
biimprd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 1s <s ( 𝑛 -s 𝑚 ) → ( 𝑚 +s 1s ) <s 𝑛 ) ) |
| 1228 |
1227
|
impr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 +s 1s ) <s 𝑛 ) |
| 1229 |
1193
|
adantrr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑛 ∈ ℕ0s ) |
| 1230 |
1229
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑛 ∈ No ) |
| 1231 |
1205 1230 1206
|
pw2sltdiv1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( ( 𝑚 +s 1s ) <s 𝑛 ↔ ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) <s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1232 |
1230 1206
|
pw2divscld |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ∈ No ) |
| 1233 |
1207 1232 1201
|
sltadd2d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) <s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ↔ ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) <s ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ) ) |
| 1234 |
1231 1233
|
bitrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( ( 𝑚 +s 1s ) <s 𝑛 ↔ ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) <s ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ) ) |
| 1235 |
1228 1234
|
mpbid |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) <s ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1236 |
1188
|
adantrr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1237 |
1235 1236
|
breqtrrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) <s 𝑑 ) |
| 1238 |
1208 1222 1237
|
ssltsn |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → { ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) } <<s { 𝑑 } ) |
| 1239 |
1199 1208 1220 1238
|
ssltbday |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) ) |
| 1240 |
1198 1239
|
eqsstrrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) ) |
| 1241 |
1189
|
adantr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝜑 ) |
| 1242 |
1241 1
|
syl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑁 ∈ ℕ0s ) |
| 1243 |
|
expscl |
⊢ ( ( 2s ∈ No ∧ 𝑜 ∈ ℕ0s ) → ( 2s ↑s 𝑜 ) ∈ No ) |
| 1244 |
54 1166 1243
|
sylancr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ↑s 𝑜 ) ∈ No ) |
| 1245 |
1244
|
adantrr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 2s ↑s 𝑜 ) ∈ No ) |
| 1246 |
|
simprl1 |
⊢ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) → 𝑚 <s 𝑛 ) |
| 1247 |
1246
|
adantl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑚 <s 𝑛 ) |
| 1248 |
1247
|
adantrr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑚 <s 𝑛 ) |
| 1249 |
|
n0sltp1le |
⊢ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ) → ( 𝑚 <s 𝑛 ↔ ( 𝑚 +s 1s ) ≤s 𝑛 ) ) |
| 1250 |
1202 1229 1249
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 <s 𝑛 ↔ ( 𝑚 +s 1s ) ≤s 𝑛 ) ) |
| 1251 |
1248 1250
|
mpbid |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 +s 1s ) ≤s 𝑛 ) |
| 1252 |
|
simprl3 |
⊢ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) → 𝑛 <s ( 2s ↑s 𝑜 ) ) |
| 1253 |
1252
|
adantl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑛 <s ( 2s ↑s 𝑜 ) ) |
| 1254 |
1253
|
adantrr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑛 <s ( 2s ↑s 𝑜 ) ) |
| 1255 |
1205 1230 1245 1251 1254
|
slelttrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 +s 1s ) <s ( 2s ↑s 𝑜 ) ) |
| 1256 |
|
simprr3 |
⊢ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) → ( 𝑔 +s 𝑜 ) <s 𝑁 ) |
| 1257 |
1256
|
adantr |
⊢ ( ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) → ( 𝑔 +s 𝑜 ) <s 𝑁 ) |
| 1258 |
1257
|
adantl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑔 +s 𝑜 ) <s 𝑁 ) |
| 1259 |
1242 1200 1204 1206 1255 1258
|
bdaypw2bnd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( bday ‘ ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) ⊆ ( bday ‘ 𝑁 ) ) |
| 1260 |
1240 1259
|
sstrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ) |
| 1261 |
405 263
|
onsled |
⊢ ( 𝜑 → ( ( 𝑁 +s 1s ) ≤s 𝑁 ↔ ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ) ) |
| 1262 |
1241 1261
|
syl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( ( 𝑁 +s 1s ) ≤s 𝑁 ↔ ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ) ) |
| 1263 |
1260 1262
|
mpbird |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑁 +s 1s ) ≤s 𝑁 ) |
| 1264 |
1263
|
expr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 1s <s ( 𝑛 -s 𝑚 ) → ( 𝑁 +s 1s ) ≤s 𝑁 ) ) |
| 1265 |
1196 1264
|
sylbird |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ¬ ( 𝑛 -s 𝑚 ) ≤s 1s → ( 𝑁 +s 1s ) ≤s 𝑁 ) ) |
| 1266 |
1191 1265
|
mt3d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑛 -s 𝑚 ) ≤s 1s ) |
| 1267 |
1161 1193 1249
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 <s 𝑛 ↔ ( 𝑚 +s 1s ) ≤s 𝑛 ) ) |
| 1268 |
|
npcans |
⊢ ( ( 𝑛 ∈ No ∧ 1s ∈ No ) → ( ( 𝑛 -s 1s ) +s 1s ) = 𝑛 ) |
| 1269 |
1194 330 1268
|
sylancl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑛 -s 1s ) +s 1s ) = 𝑛 ) |
| 1270 |
1269
|
breq2d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑚 +s 1s ) ≤s ( ( 𝑛 -s 1s ) +s 1s ) ↔ ( 𝑚 +s 1s ) ≤s 𝑛 ) ) |
| 1271 |
1267 1270
|
bitr4d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 <s 𝑛 ↔ ( 𝑚 +s 1s ) ≤s ( ( 𝑛 -s 1s ) +s 1s ) ) ) |
| 1272 |
1194 1192
|
subscld |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑛 -s 1s ) ∈ No ) |
| 1273 |
1179 1272 1192
|
sleadd1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 ≤s ( 𝑛 -s 1s ) ↔ ( 𝑚 +s 1s ) ≤s ( ( 𝑛 -s 1s ) +s 1s ) ) ) |
| 1274 |
1271 1273
|
bitr4d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 <s 𝑛 ↔ 𝑚 ≤s ( 𝑛 -s 1s ) ) ) |
| 1275 |
1179 1194 1192
|
slesubd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 ≤s ( 𝑛 -s 1s ) ↔ 1s ≤s ( 𝑛 -s 𝑚 ) ) ) |
| 1276 |
1274 1275
|
bitrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 <s 𝑛 ↔ 1s ≤s ( 𝑛 -s 𝑚 ) ) ) |
| 1277 |
1247 1276
|
mpbid |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 1s ≤s ( 𝑛 -s 𝑚 ) ) |
| 1278 |
1266 1277
|
jca |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑛 -s 𝑚 ) ≤s 1s ∧ 1s ≤s ( 𝑛 -s 𝑚 ) ) ) |
| 1279 |
1195 1192
|
sletri3d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑛 -s 𝑚 ) = 1s ↔ ( ( 𝑛 -s 𝑚 ) ≤s 1s ∧ 1s ≤s ( 𝑛 -s 𝑚 ) ) ) ) |
| 1280 |
1278 1279
|
mpbird |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑛 -s 𝑚 ) = 1s ) |
| 1281 |
1194 1179 1192
|
subaddsd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑛 -s 𝑚 ) = 1s ↔ ( 𝑚 +s 1s ) = 𝑛 ) ) |
| 1282 |
1280 1281
|
mpbid |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 +s 1s ) = 𝑛 ) |
| 1283 |
1282
|
eqcomd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑛 = ( 𝑚 +s 1s ) ) |
| 1284 |
1283
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑛 /su ( 2s ↑s 𝑜 ) ) = ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) |
| 1285 |
1284
|
oveq2d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1286 |
1188 1285
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑑 = ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1287 |
1182
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) /su ( 2s ↑s 𝑜 ) ) +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1288 |
1287
|
eqcomd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) = ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) /su ( 2s ↑s 𝑜 ) ) +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1289 |
1161 1203
|
syl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 +s 1s ) ∈ ℕ0s ) |
| 1290 |
1289
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 +s 1s ) ∈ No ) |
| 1291 |
1178 1290 1166
|
pw2divsdird |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s ( 𝑚 +s 1s ) ) /su ( 2s ↑s 𝑜 ) ) = ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) /su ( 2s ↑s 𝑜 ) ) +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) |
| 1292 |
1288 1291
|
eqtr4d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) = ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s ( 𝑚 +s 1s ) ) /su ( 2s ↑s 𝑜 ) ) ) |
| 1293 |
1178 1179 1192
|
addsassd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) +s 1s ) = ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s ( 𝑚 +s 1s ) ) ) |
| 1294 |
1293
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑜 ) ) = ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s ( 𝑚 +s 1s ) ) /su ( 2s ↑s 𝑜 ) ) ) |
| 1295 |
1294
|
eqcomd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s ( 𝑚 +s 1s ) ) /su ( 2s ↑s 𝑜 ) ) = ( ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) |
| 1296 |
1286 1292 1295
|
3eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑑 = ( ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) |
| 1297 |
1296
|
sneqd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → { 𝑑 } = { ( ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑜 ) ) } ) |
| 1298 |
1186 1297
|
oveq12d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( { 𝑐 } |s { 𝑑 } ) = ( { ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) /su ( 2s ↑s 𝑜 ) ) } |s { ( ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑜 ) ) } ) ) |
| 1299 |
|
n0addscl |
⊢ ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s ) → ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ∈ ℕ0s ) |
| 1300 |
1177 1161 1299
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ∈ ℕ0s ) |
| 1301 |
1300
|
n0zsd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ∈ ℤs ) |
| 1302 |
1301 1166
|
pw2cutp1 |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( { ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) /su ( 2s ↑s 𝑜 ) ) } |s { ( ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑜 ) ) } ) = ( ( ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) |
| 1303 |
1171 1298 1302
|
3eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑤 = ( ( ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) |
| 1304 |
54
|
a1i |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 2s ∈ No ) |
| 1305 |
1304 1178 1179
|
addsdid |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) = ( ( 2s ·s ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ) +s ( 2s ·s 𝑚 ) ) ) |
| 1306 |
|
expsp1 |
⊢ ( ( 2s ∈ No ∧ 𝑜 ∈ ℕ0s ) → ( 2s ↑s ( 𝑜 +s 1s ) ) = ( ( 2s ↑s 𝑜 ) ·s 2s ) ) |
| 1307 |
54 1166 1306
|
sylancr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ↑s ( 𝑜 +s 1s ) ) = ( ( 2s ↑s 𝑜 ) ·s 2s ) ) |
| 1308 |
1244 1304
|
mulscomd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s 𝑜 ) ·s 2s ) = ( 2s ·s ( 2s ↑s 𝑜 ) ) ) |
| 1309 |
1307 1308
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ↑s ( 𝑜 +s 1s ) ) = ( 2s ·s ( 2s ↑s 𝑜 ) ) ) |
| 1310 |
1309
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) = ( ( 2s ·s ( 2s ↑s 𝑜 ) ) ·s 𝑔 ) ) |
| 1311 |
1304 1244 1181
|
mulsassd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s ( 2s ↑s 𝑜 ) ) ·s 𝑔 ) = ( 2s ·s ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ) ) |
| 1312 |
1310 1311
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) = ( 2s ·s ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ) ) |
| 1313 |
1312
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s 𝑚 ) ) = ( ( 2s ·s ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ) +s ( 2s ·s 𝑚 ) ) ) |
| 1314 |
1305 1313
|
eqtr4d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) = ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s 𝑚 ) ) ) |
| 1315 |
1314
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) +s 1s ) = ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s 𝑚 ) ) +s 1s ) ) |
| 1316 |
|
n0expscl |
⊢ ( ( 2s ∈ ℕ0s ∧ ( 𝑜 +s 1s ) ∈ ℕ0s ) → ( 2s ↑s ( 𝑜 +s 1s ) ) ∈ ℕ0s ) |
| 1317 |
422 1168 1316
|
sylancr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ↑s ( 𝑜 +s 1s ) ) ∈ ℕ0s ) |
| 1318 |
|
n0mulscl |
⊢ ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ∈ ℕ0s ∧ 𝑔 ∈ ℕ0s ) → ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) ∈ ℕ0s ) |
| 1319 |
1317 1160 1318
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) ∈ ℕ0s ) |
| 1320 |
1319
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) ∈ No ) |
| 1321 |
1163
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s 𝑚 ) ∈ No ) |
| 1322 |
1320 1321 1192
|
addsassd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s 𝑚 ) ) +s 1s ) = ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s 𝑚 ) +s 1s ) ) ) |
| 1323 |
1315 1322
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) +s 1s ) = ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s 𝑚 ) +s 1s ) ) ) |
| 1324 |
1323
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) = ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s 𝑚 ) +s 1s ) ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) |
| 1325 |
1303 1324
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑤 = ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s 𝑚 ) +s 1s ) ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) |
| 1326 |
1165
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s 𝑚 ) +s 1s ) ∈ No ) |
| 1327 |
1320 1326 1168
|
pw2divsdird |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s 𝑚 ) +s 1s ) ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) = ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) ) |
| 1328 |
1181 1168
|
pw2divscan3d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) = 𝑔 ) |
| 1329 |
1328
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) ) |
| 1330 |
1325 1327 1329
|
3eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) ) |
| 1331 |
|
n0mulscl |
⊢ ( ( 2s ∈ ℕ0s ∧ ( 𝑚 +s 1s ) ∈ ℕ0s ) → ( 2s ·s ( 𝑚 +s 1s ) ) ∈ ℕ0s ) |
| 1332 |
422 1289 1331
|
sylancr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( 𝑚 +s 1s ) ) ∈ ℕ0s ) |
| 1333 |
1332
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( 𝑚 +s 1s ) ) ∈ No ) |
| 1334 |
1317
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ↑s ( 𝑜 +s 1s ) ) ∈ No ) |
| 1335 |
1192 1304 1321
|
sltadd2d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 1s <s 2s ↔ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( ( 2s ·s 𝑚 ) +s 2s ) ) ) |
| 1336 |
540 1335
|
mpbii |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s 𝑚 ) +s 1s ) <s ( ( 2s ·s 𝑚 ) +s 2s ) ) |
| 1337 |
1304 1179 1192
|
addsdid |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( 𝑚 +s 1s ) ) = ( ( 2s ·s 𝑚 ) +s ( 2s ·s 1s ) ) ) |
| 1338 |
903
|
oveq2i |
⊢ ( ( 2s ·s 𝑚 ) +s ( 2s ·s 1s ) ) = ( ( 2s ·s 𝑚 ) +s 2s ) |
| 1339 |
1337 1338
|
eqtrdi |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( 𝑚 +s 1s ) ) = ( ( 2s ·s 𝑚 ) +s 2s ) ) |
| 1340 |
1336 1339
|
breqtrrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ·s ( 𝑚 +s 1s ) ) ) |
| 1341 |
|
simprl2 |
⊢ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) → 𝑚 <s ( 2s ↑s 𝑜 ) ) |
| 1342 |
1341
|
adantl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑚 <s ( 2s ↑s 𝑜 ) ) |
| 1343 |
|
n0sltp1le |
⊢ ( ( 𝑚 ∈ ℕ0s ∧ ( 2s ↑s 𝑜 ) ∈ ℕ0s ) → ( 𝑚 <s ( 2s ↑s 𝑜 ) ↔ ( 𝑚 +s 1s ) ≤s ( 2s ↑s 𝑜 ) ) ) |
| 1344 |
1161 1175 1343
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 <s ( 2s ↑s 𝑜 ) ↔ ( 𝑚 +s 1s ) ≤s ( 2s ↑s 𝑜 ) ) ) |
| 1345 |
1342 1344
|
mpbid |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 +s 1s ) ≤s ( 2s ↑s 𝑜 ) ) |
| 1346 |
1007
|
a1i |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 0s <s 2s ) |
| 1347 |
1290 1244 1304 1346
|
slemul2d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑚 +s 1s ) ≤s ( 2s ↑s 𝑜 ) ↔ ( 2s ·s ( 𝑚 +s 1s ) ) ≤s ( 2s ·s ( 2s ↑s 𝑜 ) ) ) ) |
| 1348 |
1345 1347
|
mpbid |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( 𝑚 +s 1s ) ) ≤s ( 2s ·s ( 2s ↑s 𝑜 ) ) ) |
| 1349 |
1348 1309
|
breqtrrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( 𝑚 +s 1s ) ) ≤s ( 2s ↑s ( 𝑜 +s 1s ) ) ) |
| 1350 |
1326 1333 1334 1340 1349
|
sltletrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s ( 𝑜 +s 1s ) ) ) |
| 1351 |
1166
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑜 ∈ No ) |
| 1352 |
1181 1351 1192
|
addsassd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑔 +s 𝑜 ) +s 1s ) = ( 𝑔 +s ( 𝑜 +s 1s ) ) ) |
| 1353 |
1256
|
adantl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑔 +s 𝑜 ) <s 𝑁 ) |
| 1354 |
|
n0addscl |
⊢ ( ( 𝑔 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) → ( 𝑔 +s 𝑜 ) ∈ ℕ0s ) |
| 1355 |
1160 1166 1354
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑔 +s 𝑜 ) ∈ ℕ0s ) |
| 1356 |
1355
|
n0snod |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑔 +s 𝑜 ) ∈ No ) |
| 1357 |
1190 34
|
syl |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑁 ∈ No ) |
| 1358 |
1356 1357 1192
|
sltadd1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑔 +s 𝑜 ) <s 𝑁 ↔ ( ( 𝑔 +s 𝑜 ) +s 1s ) <s ( 𝑁 +s 1s ) ) ) |
| 1359 |
1353 1358
|
mpbid |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑔 +s 𝑜 ) +s 1s ) <s ( 𝑁 +s 1s ) ) |
| 1360 |
1352 1359
|
eqbrtrrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑔 +s ( 𝑜 +s 1s ) ) <s ( 𝑁 +s 1s ) ) |
| 1361 |
|
oveq1 |
⊢ ( 𝑏 = ( ( 2s ·s 𝑚 ) +s 1s ) → ( 𝑏 /su ( 2s ↑s 𝑞 ) ) = ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) |
| 1362 |
1361
|
oveq2d |
⊢ ( 𝑏 = ( ( 2s ·s 𝑚 ) +s 1s ) → ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ) |
| 1363 |
1362
|
eqeq2d |
⊢ ( 𝑏 = ( ( 2s ·s 𝑚 ) +s 1s ) → ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ) ) |
| 1364 |
|
breq1 |
⊢ ( 𝑏 = ( ( 2s ·s 𝑚 ) +s 1s ) → ( 𝑏 <s ( 2s ↑s 𝑞 ) ↔ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s 𝑞 ) ) ) |
| 1365 |
1363 1364
|
3anbi12d |
⊢ ( 𝑏 = ( ( 2s ·s 𝑚 ) +s 1s ) → ( ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ∧ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1366 |
|
oveq2 |
⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( 2s ↑s 𝑞 ) = ( 2s ↑s ( 𝑜 +s 1s ) ) ) |
| 1367 |
1366
|
oveq2d |
⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) = ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) |
| 1368 |
1367
|
oveq2d |
⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) ) |
| 1369 |
1368
|
eqeq2d |
⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) ) ) |
| 1370 |
1366
|
breq2d |
⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s 𝑞 ) ↔ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) |
| 1371 |
|
oveq2 |
⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( 𝑔 +s 𝑞 ) = ( 𝑔 +s ( 𝑜 +s 1s ) ) ) |
| 1372 |
1371
|
breq1d |
⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑔 +s ( 𝑜 +s 1s ) ) <s ( 𝑁 +s 1s ) ) ) |
| 1373 |
1369 1370 1372
|
3anbi123d |
⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ∧ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) ∧ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s ( 𝑜 +s 1s ) ) ∧ ( 𝑔 +s ( 𝑜 +s 1s ) ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1374 |
552 1365 1373
|
rspc3ev |
⊢ ( ( ( 𝑔 ∈ ℕ0s ∧ ( ( 2s ·s 𝑚 ) +s 1s ) ∈ ℕ0s ∧ ( 𝑜 +s 1s ) ∈ ℕ0s ) ∧ ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) ∧ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s ( 𝑜 +s 1s ) ) ∧ ( 𝑔 +s ( 𝑜 +s 1s ) ) <s ( 𝑁 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1375 |
1160 1165 1168 1330 1350 1360 1374
|
syl33anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1376 |
1375
|
expr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ) → ( ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1377 |
1376
|
rexlimdvvva |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1378 |
1158 1377
|
mpd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1379 |
1378
|
expr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1380 |
963 1379
|
mpd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1381 |
1380
|
expr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 = 𝑗 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1382 |
946 1381
|
jaod |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( ( 𝑔 <s 𝑗 ∨ 𝑔 = 𝑗 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1383 |
645 1382
|
sylbid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 ≤s 𝑗 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1384 |
643 1383
|
mpd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1385 |
1384
|
expr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( 𝑔 ≤s 𝑐 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1386 |
617 1385
|
mpd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1387 |
1386
|
ex |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑑 <s ( 𝑗 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1388 |
596 1387
|
mpd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1389 |
1388
|
expr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ) → ( ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1390 |
1389
|
rexlimdvvva |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1391 |
273
|
3adant3 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ) |
| 1392 |
59
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 0s ∈ No ) |
| 1393 |
124
|
3ad2ant1 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑤 ∈ No ) |
| 1394 |
|
simp1r |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 0s <s 𝑤 ) |
| 1395 |
1393 1394
|
0elleft |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 0s ∈ ( L ‘ 𝑤 ) ) |
| 1396 |
215 192 1395
|
rspcdva |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 0s ≤s 𝑐 ) |
| 1397 |
1392 344 393 1396 285
|
slelttrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 0s <s 𝑑 ) |
| 1398 |
1392 393 1397
|
sltled |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 0s ≤s 𝑑 ) |
| 1399 |
1398
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 0s ≤s 𝑑 ) |
| 1400 |
|
fveq2 |
⊢ ( 𝑧 = 𝑑 → ( bday ‘ 𝑧 ) = ( bday ‘ 𝑑 ) ) |
| 1401 |
1400
|
sseq1d |
⊢ ( 𝑧 = 𝑑 → ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ) ) |
| 1402 |
|
breq2 |
⊢ ( 𝑧 = 𝑑 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑑 ) ) |
| 1403 |
1401 1402
|
anbi12d |
⊢ ( 𝑧 = 𝑑 → ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) ↔ ( ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑑 ) ) ) |
| 1404 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑑 → ( 𝑧 = 𝑁 ↔ 𝑑 = 𝑁 ) ) |
| 1405 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑑 → ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) ) |
| 1406 |
1405
|
3anbi1d |
⊢ ( 𝑧 = 𝑑 → ( ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) |
| 1407 |
1406
|
rexbidv |
⊢ ( 𝑧 = 𝑑 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) |
| 1408 |
1407
|
2rexbidv |
⊢ ( 𝑧 = 𝑑 → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) |
| 1409 |
|
oveq1 |
⊢ ( 𝑥 = 𝑗 → ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) |
| 1410 |
1409
|
eqeq2d |
⊢ ( 𝑥 = 𝑗 → ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) ) |
| 1411 |
|
oveq1 |
⊢ ( 𝑥 = 𝑗 → ( 𝑥 +s 𝑝 ) = ( 𝑗 +s 𝑝 ) ) |
| 1412 |
1411
|
breq1d |
⊢ ( 𝑥 = 𝑗 → ( ( 𝑥 +s 𝑝 ) <s 𝑁 ↔ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ) |
| 1413 |
1410 1412
|
3anbi13d |
⊢ ( 𝑥 = 𝑗 → ( ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ) ) |
| 1414 |
1413
|
rexbidv |
⊢ ( 𝑥 = 𝑗 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ) ) |
| 1415 |
|
oveq1 |
⊢ ( 𝑦 = 𝑘 → ( 𝑦 /su ( 2s ↑s 𝑝 ) ) = ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) |
| 1416 |
1415
|
oveq2d |
⊢ ( 𝑦 = 𝑘 → ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) ) |
| 1417 |
1416
|
eqeq2d |
⊢ ( 𝑦 = 𝑘 → ( 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) ) ) |
| 1418 |
|
breq1 |
⊢ ( 𝑦 = 𝑘 → ( 𝑦 <s ( 2s ↑s 𝑝 ) ↔ 𝑘 <s ( 2s ↑s 𝑝 ) ) ) |
| 1419 |
1417 1418
|
3anbi12d |
⊢ ( 𝑦 = 𝑘 → ( ( 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ) ) |
| 1420 |
1419
|
rexbidv |
⊢ ( 𝑦 = 𝑘 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ) ) |
| 1421 |
|
oveq2 |
⊢ ( 𝑝 = 𝑙 → ( 2s ↑s 𝑝 ) = ( 2s ↑s 𝑙 ) ) |
| 1422 |
1421
|
oveq2d |
⊢ ( 𝑝 = 𝑙 → ( 𝑘 /su ( 2s ↑s 𝑝 ) ) = ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) |
| 1423 |
1422
|
oveq2d |
⊢ ( 𝑝 = 𝑙 → ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) |
| 1424 |
1423
|
eqeq2d |
⊢ ( 𝑝 = 𝑙 → ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ) |
| 1425 |
1421
|
breq2d |
⊢ ( 𝑝 = 𝑙 → ( 𝑘 <s ( 2s ↑s 𝑝 ) ↔ 𝑘 <s ( 2s ↑s 𝑙 ) ) ) |
| 1426 |
|
oveq2 |
⊢ ( 𝑝 = 𝑙 → ( 𝑗 +s 𝑝 ) = ( 𝑗 +s 𝑙 ) ) |
| 1427 |
1426
|
breq1d |
⊢ ( 𝑝 = 𝑙 → ( ( 𝑗 +s 𝑝 ) <s 𝑁 ↔ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) |
| 1428 |
1424 1425 1427
|
3anbi123d |
⊢ ( 𝑝 = 𝑙 → ( ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) |
| 1429 |
1428
|
cbvrexvw |
⊢ ( ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) |
| 1430 |
1420 1429
|
bitrdi |
⊢ ( 𝑦 = 𝑘 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) |
| 1431 |
1414 1430
|
cbvrex2vw |
⊢ ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) |
| 1432 |
1408 1431
|
bitrdi |
⊢ ( 𝑧 = 𝑑 → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) |
| 1433 |
1404 1432
|
orbi12d |
⊢ ( 𝑧 = 𝑑 → ( ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ↔ ( 𝑑 = 𝑁 ∨ ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ) |
| 1434 |
1403 1433
|
imbi12d |
⊢ ( 𝑧 = 𝑑 → ( ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ↔ ( ( ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑑 ) → ( 𝑑 = 𝑁 ∨ ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ) ) |
| 1435 |
302 2
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ) |
| 1436 |
1434 1435 394
|
rspcdva |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( ( ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑑 ) → ( 𝑑 = 𝑁 ∨ ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ) |
| 1437 |
1391 1399 1436
|
mp2and |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( 𝑑 = 𝑁 ∨ ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) |
| 1438 |
578 1390 1437
|
mpjaod |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1439 |
1438
|
3expa |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1440 |
1439
|
expr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) ∧ ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ) → ( ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1441 |
1440
|
rexlimdvvva |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1442 |
293 1441
|
syld |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑐 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1443 |
214 219 1442
|
mp2and |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1444 |
1443
|
ex |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ( 𝑤 = ( { 𝑐 } |s { 𝑑 } ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1445 |
195 1444
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1446 |
1445
|
3expa |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1447 |
1446
|
expr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑑 ∈ ( R ‘ 𝑤 ) ) → ( ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1448 |
190 1447
|
sylbird |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑑 ∈ ( R ‘ 𝑤 ) ) → ( ∀ 𝑓 ∈ ( R ‘ 𝑤 ) ¬ 𝑓 <s 𝑑 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1449 |
1448
|
rexlimdva |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ( ∃ 𝑑 ∈ ( R ‘ 𝑤 ) ∀ 𝑓 ∈ ( R ‘ 𝑤 ) ¬ 𝑓 <s 𝑑 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1450 |
184 1449
|
syl5 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ( ( ( R ‘ 𝑤 ) ∈ Fin ∧ ( R ‘ 𝑤 ) ≠ ∅ ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1451 |
158 179 1450
|
mp2and |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1452 |
1451
|
expr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ 𝑐 ∈ ( L ‘ 𝑤 ) ) → ( ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1453 |
154 1452
|
sylbird |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ 𝑐 ∈ ( L ‘ 𝑤 ) ) → ( ∀ 𝑒 ∈ ( L ‘ 𝑤 ) ¬ 𝑐 <s 𝑒 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1454 |
1453
|
rexlimdva |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( ∃ 𝑐 ∈ ( L ‘ 𝑤 ) ∀ 𝑒 ∈ ( L ‘ 𝑤 ) ¬ 𝑐 <s 𝑒 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1455 |
147 1454
|
syl5 |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( ( ( L ‘ 𝑤 ) ∈ Fin ∧ ( L ‘ 𝑤 ) ≠ ∅ ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1456 |
141 1455
|
mpand |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( ( L ‘ 𝑤 ) ≠ ∅ → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1457 |
127 1456
|
mpd |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1458 |
1457
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( 0s <s 𝑤 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1459 |
|
id |
⊢ ( 𝜑 → 𝜑 ) |
| 1460 |
|
n0p1nns |
⊢ ( 𝑁 ∈ ℕ0s → ( 𝑁 +s 1s ) ∈ ℕs ) |
| 1461 |
1 1460
|
syl |
⊢ ( 𝜑 → ( 𝑁 +s 1s ) ∈ ℕs ) |
| 1462 |
|
nnsgt0 |
⊢ ( ( 𝑁 +s 1s ) ∈ ℕs → 0s <s ( 𝑁 +s 1s ) ) |
| 1463 |
1461 1462
|
syl |
⊢ ( 𝜑 → 0s <s ( 𝑁 +s 1s ) ) |
| 1464 |
|
addslid |
⊢ ( 0s ∈ No → ( 0s +s 0s ) = 0s ) |
| 1465 |
59 1464
|
ax-mp |
⊢ ( 0s +s 0s ) = 0s |
| 1466 |
1465
|
eqcomi |
⊢ 0s = ( 0s +s 0s ) |
| 1467 |
31 31 31
|
3pm3.2i |
⊢ ( 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s ) |
| 1468 |
|
oveq1 |
⊢ ( 𝑎 = 0s → ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 0s +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) |
| 1469 |
1468
|
eqeq2d |
⊢ ( 𝑎 = 0s → ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 0s = ( 0s +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) ) |
| 1470 |
|
oveq1 |
⊢ ( 𝑎 = 0s → ( 𝑎 +s 𝑞 ) = ( 0s +s 𝑞 ) ) |
| 1471 |
1470
|
breq1d |
⊢ ( 𝑎 = 0s → ( ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 0s +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1472 |
1469 1471
|
3anbi13d |
⊢ ( 𝑎 = 0s → ( ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 0s = ( 0s +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 0s +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1473 |
48
|
oveq2d |
⊢ ( 𝑏 = 0s → ( 0s +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 0s +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ) |
| 1474 |
1473
|
eqeq2d |
⊢ ( 𝑏 = 0s → ( 0s = ( 0s +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 0s = ( 0s +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ) ) |
| 1475 |
1474 51
|
3anbi12d |
⊢ ( 𝑏 = 0s → ( ( 0s = ( 0s +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 0s +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 0s = ( 0s +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ∧ 0s <s ( 2s ↑s 𝑞 ) ∧ ( 0s +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1476 |
62
|
oveq2d |
⊢ ( 𝑞 = 0s → ( 0s +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) = ( 0s +s 0s ) ) |
| 1477 |
1476
|
eqeq2d |
⊢ ( 𝑞 = 0s → ( 0s = ( 0s +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ↔ 0s = ( 0s +s 0s ) ) ) |
| 1478 |
|
oveq2 |
⊢ ( 𝑞 = 0s → ( 0s +s 𝑞 ) = ( 0s +s 0s ) ) |
| 1479 |
1478 1465
|
eqtrdi |
⊢ ( 𝑞 = 0s → ( 0s +s 𝑞 ) = 0s ) |
| 1480 |
1479
|
breq1d |
⊢ ( 𝑞 = 0s → ( ( 0s +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ 0s <s ( 𝑁 +s 1s ) ) ) |
| 1481 |
1477 65 1480
|
3anbi123d |
⊢ ( 𝑞 = 0s → ( ( 0s = ( 0s +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ∧ 0s <s ( 2s ↑s 𝑞 ) ∧ ( 0s +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 0s = ( 0s +s 0s ) ∧ 0s <s 1s ∧ 0s <s ( 𝑁 +s 1s ) ) ) ) |
| 1482 |
1472 1475 1481
|
rspc3ev |
⊢ ( ( ( 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s ) ∧ ( 0s = ( 0s +s 0s ) ∧ 0s <s 1s ∧ 0s <s ( 𝑁 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1483 |
1467 1482
|
mpan |
⊢ ( ( 0s = ( 0s +s 0s ) ∧ 0s <s 1s ∧ 0s <s ( 𝑁 +s 1s ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1484 |
1466 38 1483
|
mp3an12 |
⊢ ( 0s <s ( 𝑁 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1485 |
1463 1484
|
syl |
⊢ ( 𝜑 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1486 |
1459 1485
|
syl |
⊢ ( 𝜑 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) |
| 1487 |
|
eqeq1 |
⊢ ( 0s = 𝑤 → ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) ) |
| 1488 |
1487
|
3anbi1d |
⊢ ( 0s = 𝑤 → ( ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1489 |
1488
|
rexbidv |
⊢ ( 0s = 𝑤 → ( ∃ 𝑞 ∈ ℕ0s ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1490 |
1489
|
2rexbidv |
⊢ ( 0s = 𝑤 → ( ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1491 |
1486 1490
|
syl5ibcom |
⊢ ( 𝜑 → ( 0s = 𝑤 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1492 |
1491
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( 0s = 𝑤 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1493 |
1458 1492
|
jaod |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( ( 0s <s 𝑤 ∨ 0s = 𝑤 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1494 |
123 1493
|
sylbid |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( 0s ≤s 𝑤 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1495 |
1494
|
expr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) → ( 0s ≤s 𝑤 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) |
| 1496 |
1495
|
expd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) → ( 𝑤 ≠ ( 𝑁 +s 1s ) → ( 0s ≤s 𝑤 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) ) |
| 1497 |
1496
|
com34 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) → ( 0s ≤s 𝑤 → ( 𝑤 ≠ ( 𝑁 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) ) |
| 1498 |
1497
|
impd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 ≠ ( 𝑁 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) |
| 1499 |
1498
|
impr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) ) ) → ( 𝑤 ≠ ( 𝑁 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1500 |
120 1499
|
biimtrrid |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) ) ) → ( ¬ 𝑤 = ( 𝑁 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1501 |
1500
|
orrd |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) ) ) → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) |
| 1502 |
1501
|
expr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) |
| 1503 |
1502
|
expd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) → ( 0s ≤s 𝑤 → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) ) |
| 1504 |
119 1503
|
sylbird |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) → ( 0s ≤s 𝑤 → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) ) |
| 1505 |
118 1504
|
jaod |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∨ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) → ( 0s ≤s 𝑤 → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) ) |
| 1506 |
16 1505
|
biimtrid |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) ⊆ suc ( bday ‘ 𝑁 ) → ( 0s ≤s 𝑤 → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) ) |
| 1507 |
6 1506
|
sylbid |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑁 +s 1s ) ) → ( 0s ≤s 𝑤 → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) ) |
| 1508 |
1507
|
impd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) |
| 1509 |
1508
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑤 ∈ No ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) |