| Step |
Hyp |
Ref |
Expression |
| 1 |
|
bdayon |
⊢ ( bday ‘ 𝐴 ) ∈ On |
| 2 |
|
bdayon |
⊢ ( bday ‘ 𝐵 ) ∈ On |
| 3 |
|
naddcl |
⊢ ( ( ( bday ‘ 𝐴 ) ∈ On ∧ ( bday ‘ 𝐵 ) ∈ On ) → ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∈ On ) |
| 4 |
1 2 3
|
mp2an |
⊢ ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∈ On |
| 5 |
|
bdayon |
⊢ ( bday ‘ 𝐶 ) ∈ On |
| 6 |
|
bdayon |
⊢ ( bday ‘ 𝐸 ) ∈ On |
| 7 |
|
naddcl |
⊢ ( ( ( bday ‘ 𝐶 ) ∈ On ∧ ( bday ‘ 𝐸 ) ∈ On ) → ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∈ On ) |
| 8 |
5 6 7
|
mp2an |
⊢ ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∈ On |
| 9 |
|
bdayon |
⊢ ( bday ‘ 𝐷 ) ∈ On |
| 10 |
|
bdayon |
⊢ ( bday ‘ 𝐹 ) ∈ On |
| 11 |
|
naddcl |
⊢ ( ( ( bday ‘ 𝐷 ) ∈ On ∧ ( bday ‘ 𝐹 ) ∈ On ) → ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ∈ On ) |
| 12 |
9 10 11
|
mp2an |
⊢ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ∈ On |
| 13 |
8 12
|
onun2i |
⊢ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∈ On |
| 14 |
|
naddcl |
⊢ ( ( ( bday ‘ 𝐶 ) ∈ On ∧ ( bday ‘ 𝐹 ) ∈ On ) → ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∈ On ) |
| 15 |
5 10 14
|
mp2an |
⊢ ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∈ On |
| 16 |
|
naddcl |
⊢ ( ( ( bday ‘ 𝐷 ) ∈ On ∧ ( bday ‘ 𝐸 ) ∈ On ) → ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ∈ On ) |
| 17 |
9 6 16
|
mp2an |
⊢ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ∈ On |
| 18 |
15 17
|
onun2i |
⊢ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ∈ On |
| 19 |
13 18
|
onun2i |
⊢ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ∈ On |
| 20 |
4 19
|
onun2i |
⊢ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ∈ On |
| 21 |
|
risset |
⊢ ( ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ∈ On ↔ ∃ 𝑥 ∈ On 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ) |
| 22 |
20 21
|
mpbi |
⊢ ∃ 𝑥 ∈ On 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) |
| 23 |
|
fveq2 |
⊢ ( 𝑎 = 𝑔 → ( bday ‘ 𝑎 ) = ( bday ‘ 𝑔 ) ) |
| 24 |
23
|
oveq1d |
⊢ ( 𝑎 = 𝑔 → ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) = ( ( bday ‘ 𝑔 ) +no ( bday ‘ 𝑏 ) ) ) |
| 25 |
24
|
uneq1d |
⊢ ( 𝑎 = 𝑔 → ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) |
| 26 |
25
|
eqeq2d |
⊢ ( 𝑎 = 𝑔 → ( 𝑥 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ↔ 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) ) |
| 27 |
|
oveq1 |
⊢ ( 𝑎 = 𝑔 → ( 𝑎 ·s 𝑏 ) = ( 𝑔 ·s 𝑏 ) ) |
| 28 |
27
|
eleq1d |
⊢ ( 𝑎 = 𝑔 → ( ( 𝑎 ·s 𝑏 ) ∈ No ↔ ( 𝑔 ·s 𝑏 ) ∈ No ) ) |
| 29 |
28
|
anbi1d |
⊢ ( 𝑎 = 𝑔 → ( ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑔 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 30 |
26 29
|
imbi12d |
⊢ ( 𝑎 = 𝑔 → ( ( 𝑥 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑔 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 31 |
|
fveq2 |
⊢ ( 𝑏 = ℎ → ( bday ‘ 𝑏 ) = ( bday ‘ ℎ ) ) |
| 32 |
31
|
oveq2d |
⊢ ( 𝑏 = ℎ → ( ( bday ‘ 𝑔 ) +no ( bday ‘ 𝑏 ) ) = ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ) |
| 33 |
32
|
uneq1d |
⊢ ( 𝑏 = ℎ → ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) |
| 34 |
33
|
eqeq2d |
⊢ ( 𝑏 = ℎ → ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ↔ 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) ) |
| 35 |
|
oveq2 |
⊢ ( 𝑏 = ℎ → ( 𝑔 ·s 𝑏 ) = ( 𝑔 ·s ℎ ) ) |
| 36 |
35
|
eleq1d |
⊢ ( 𝑏 = ℎ → ( ( 𝑔 ·s 𝑏 ) ∈ No ↔ ( 𝑔 ·s ℎ ) ∈ No ) ) |
| 37 |
36
|
anbi1d |
⊢ ( 𝑏 = ℎ → ( ( ( 𝑔 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 38 |
34 37
|
imbi12d |
⊢ ( 𝑏 = ℎ → ( ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑔 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 39 |
|
fveq2 |
⊢ ( 𝑐 = 𝑖 → ( bday ‘ 𝑐 ) = ( bday ‘ 𝑖 ) ) |
| 40 |
39
|
oveq1d |
⊢ ( 𝑐 = 𝑖 → ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) = ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ) |
| 41 |
40
|
uneq1d |
⊢ ( 𝑐 = 𝑖 → ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ) |
| 42 |
39
|
oveq1d |
⊢ ( 𝑐 = 𝑖 → ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) = ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ) |
| 43 |
42
|
uneq1d |
⊢ ( 𝑐 = 𝑖 → ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) |
| 44 |
41 43
|
uneq12d |
⊢ ( 𝑐 = 𝑖 → ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) = ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) |
| 45 |
44
|
uneq2d |
⊢ ( 𝑐 = 𝑖 → ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) |
| 46 |
45
|
eqeq2d |
⊢ ( 𝑐 = 𝑖 → ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ↔ 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) ) |
| 47 |
|
breq1 |
⊢ ( 𝑐 = 𝑖 → ( 𝑐 <s 𝑑 ↔ 𝑖 <s 𝑑 ) ) |
| 48 |
47
|
anbi1d |
⊢ ( 𝑐 = 𝑖 → ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) ↔ ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) ) ) |
| 49 |
|
oveq1 |
⊢ ( 𝑐 = 𝑖 → ( 𝑐 ·s 𝑓 ) = ( 𝑖 ·s 𝑓 ) ) |
| 50 |
|
oveq1 |
⊢ ( 𝑐 = 𝑖 → ( 𝑐 ·s 𝑒 ) = ( 𝑖 ·s 𝑒 ) ) |
| 51 |
49 50
|
oveq12d |
⊢ ( 𝑐 = 𝑖 → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) = ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) ) |
| 52 |
51
|
breq1d |
⊢ ( 𝑐 = 𝑖 → ( ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ↔ ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) |
| 53 |
48 52
|
imbi12d |
⊢ ( 𝑐 = 𝑖 → ( ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ↔ ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) |
| 54 |
53
|
anbi2d |
⊢ ( 𝑐 = 𝑖 → ( ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 55 |
46 54
|
imbi12d |
⊢ ( 𝑐 = 𝑖 → ( ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 56 |
|
fveq2 |
⊢ ( 𝑑 = 𝑗 → ( bday ‘ 𝑑 ) = ( bday ‘ 𝑗 ) ) |
| 57 |
56
|
oveq1d |
⊢ ( 𝑑 = 𝑗 → ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) = ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) |
| 58 |
57
|
uneq2d |
⊢ ( 𝑑 = 𝑗 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ) |
| 59 |
56
|
oveq1d |
⊢ ( 𝑑 = 𝑗 → ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) = ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) |
| 60 |
59
|
uneq2d |
⊢ ( 𝑑 = 𝑗 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) |
| 61 |
58 60
|
uneq12d |
⊢ ( 𝑑 = 𝑗 → ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) = ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) ) |
| 62 |
61
|
uneq2d |
⊢ ( 𝑑 = 𝑗 → ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) |
| 63 |
62
|
eqeq2d |
⊢ ( 𝑑 = 𝑗 → ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ↔ 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) ) |
| 64 |
|
breq2 |
⊢ ( 𝑑 = 𝑗 → ( 𝑖 <s 𝑑 ↔ 𝑖 <s 𝑗 ) ) |
| 65 |
64
|
anbi1d |
⊢ ( 𝑑 = 𝑗 → ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) ↔ ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) ) ) |
| 66 |
|
oveq1 |
⊢ ( 𝑑 = 𝑗 → ( 𝑑 ·s 𝑓 ) = ( 𝑗 ·s 𝑓 ) ) |
| 67 |
|
oveq1 |
⊢ ( 𝑑 = 𝑗 → ( 𝑑 ·s 𝑒 ) = ( 𝑗 ·s 𝑒 ) ) |
| 68 |
66 67
|
oveq12d |
⊢ ( 𝑑 = 𝑗 → ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) = ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) |
| 69 |
68
|
breq2d |
⊢ ( 𝑑 = 𝑗 → ( ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ↔ ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ) |
| 70 |
65 69
|
imbi12d |
⊢ ( 𝑑 = 𝑗 → ( ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ↔ ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ) ) |
| 71 |
70
|
anbi2d |
⊢ ( 𝑑 = 𝑗 → ( ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ) ) ) |
| 72 |
63 71
|
imbi12d |
⊢ ( 𝑑 = 𝑗 → ( ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ) ) ) ) |
| 73 |
|
fveq2 |
⊢ ( 𝑒 = 𝑘 → ( bday ‘ 𝑒 ) = ( bday ‘ 𝑘 ) ) |
| 74 |
73
|
oveq2d |
⊢ ( 𝑒 = 𝑘 → ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) = ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ) |
| 75 |
74
|
uneq1d |
⊢ ( 𝑒 = 𝑘 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ) |
| 76 |
73
|
oveq2d |
⊢ ( 𝑒 = 𝑘 → ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) = ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) |
| 77 |
76
|
uneq2d |
⊢ ( 𝑒 = 𝑘 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) |
| 78 |
75 77
|
uneq12d |
⊢ ( 𝑒 = 𝑘 → ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) = ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) |
| 79 |
78
|
uneq2d |
⊢ ( 𝑒 = 𝑘 → ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) |
| 80 |
79
|
eqeq2d |
⊢ ( 𝑒 = 𝑘 → ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ↔ 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) ) |
| 81 |
|
breq1 |
⊢ ( 𝑒 = 𝑘 → ( 𝑒 <s 𝑓 ↔ 𝑘 <s 𝑓 ) ) |
| 82 |
81
|
anbi2d |
⊢ ( 𝑒 = 𝑘 → ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) ↔ ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) ) ) |
| 83 |
|
oveq2 |
⊢ ( 𝑒 = 𝑘 → ( 𝑖 ·s 𝑒 ) = ( 𝑖 ·s 𝑘 ) ) |
| 84 |
83
|
oveq2d |
⊢ ( 𝑒 = 𝑘 → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) = ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) ) |
| 85 |
|
oveq2 |
⊢ ( 𝑒 = 𝑘 → ( 𝑗 ·s 𝑒 ) = ( 𝑗 ·s 𝑘 ) ) |
| 86 |
85
|
oveq2d |
⊢ ( 𝑒 = 𝑘 → ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) = ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) |
| 87 |
84 86
|
breq12d |
⊢ ( 𝑒 = 𝑘 → ( ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ↔ ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) |
| 88 |
82 87
|
imbi12d |
⊢ ( 𝑒 = 𝑘 → ( ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ↔ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) |
| 89 |
88
|
anbi2d |
⊢ ( 𝑒 = 𝑘 → ( ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 90 |
80 89
|
imbi12d |
⊢ ( 𝑒 = 𝑘 → ( ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ) ) ↔ ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) ) |
| 91 |
|
fveq2 |
⊢ ( 𝑓 = 𝑙 → ( bday ‘ 𝑓 ) = ( bday ‘ 𝑙 ) ) |
| 92 |
91
|
oveq2d |
⊢ ( 𝑓 = 𝑙 → ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) = ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) |
| 93 |
92
|
uneq2d |
⊢ ( 𝑓 = 𝑙 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ) |
| 94 |
91
|
oveq2d |
⊢ ( 𝑓 = 𝑙 → ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) = ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ) |
| 95 |
94
|
uneq1d |
⊢ ( 𝑓 = 𝑙 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) |
| 96 |
93 95
|
uneq12d |
⊢ ( 𝑓 = 𝑙 → ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) = ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) |
| 97 |
96
|
uneq2d |
⊢ ( 𝑓 = 𝑙 → ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) |
| 98 |
97
|
eqeq2d |
⊢ ( 𝑓 = 𝑙 → ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ↔ 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) ) |
| 99 |
|
breq2 |
⊢ ( 𝑓 = 𝑙 → ( 𝑘 <s 𝑓 ↔ 𝑘 <s 𝑙 ) ) |
| 100 |
99
|
anbi2d |
⊢ ( 𝑓 = 𝑙 → ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) ↔ ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) ) ) |
| 101 |
|
oveq2 |
⊢ ( 𝑓 = 𝑙 → ( 𝑖 ·s 𝑓 ) = ( 𝑖 ·s 𝑙 ) ) |
| 102 |
101
|
oveq1d |
⊢ ( 𝑓 = 𝑙 → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) = ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) ) |
| 103 |
|
oveq2 |
⊢ ( 𝑓 = 𝑙 → ( 𝑗 ·s 𝑓 ) = ( 𝑗 ·s 𝑙 ) ) |
| 104 |
103
|
oveq1d |
⊢ ( 𝑓 = 𝑙 → ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) = ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) |
| 105 |
102 104
|
breq12d |
⊢ ( 𝑓 = 𝑙 → ( ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ↔ ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) |
| 106 |
100 105
|
imbi12d |
⊢ ( 𝑓 = 𝑙 → ( ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ↔ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) |
| 107 |
106
|
anbi2d |
⊢ ( 𝑓 = 𝑙 → ( ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ↔ ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 108 |
98 107
|
imbi12d |
⊢ ( 𝑓 = 𝑙 → ( ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ↔ ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) ) |
| 109 |
30 38 55 72 90 108
|
cbvral6vw |
⊢ ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑥 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ∀ 𝑔 ∈ No ∀ ℎ ∈ No ∀ 𝑖 ∈ No ∀ 𝑗 ∈ No ∀ 𝑘 ∈ No ∀ 𝑙 ∈ No ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 110 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ↔ 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) ) |
| 111 |
110
|
imbi1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 112 |
111
|
6ralbidv |
⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑥 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 113 |
109 112
|
bitr3id |
⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑔 ∈ No ∀ ℎ ∈ No ∀ 𝑖 ∈ No ∀ 𝑗 ∈ No ∀ 𝑘 ∈ No ∀ 𝑙 ∈ No ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ↔ ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 114 |
|
raleq |
⊢ ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ∀ 𝑦 ∈ 𝑥 ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 115 |
|
ralrot3 |
⊢ ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 116 |
|
ralrot3 |
⊢ ( ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 117 |
|
ralrot3 |
⊢ ( ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 118 |
|
r19.23v |
⊢ ( ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( ∃ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 119 |
|
risset |
⊢ ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ↔ ∃ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) |
| 120 |
119
|
imbi1i |
⊢ ( ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( ∃ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 121 |
118 120
|
bitr4i |
⊢ ( ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 122 |
121
|
2ralbii |
⊢ ( ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 123 |
117 122
|
bitr3i |
⊢ ( ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 124 |
123
|
2ralbii |
⊢ ( ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 125 |
116 124
|
bitr3i |
⊢ ( ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 126 |
125
|
2ralbii |
⊢ ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 127 |
115 126
|
bitr3i |
⊢ ( ∀ 𝑦 ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 128 |
114 127
|
bitrdi |
⊢ ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ∀ 𝑦 ∈ 𝑥 ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 129 |
|
simpl |
⊢ ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) → ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 130 |
|
simprl1 |
⊢ ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) → 𝑔 ∈ No ) |
| 131 |
|
simprl2 |
⊢ ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) → ℎ ∈ No ) |
| 132 |
129 130 131
|
mulsproplem11 |
⊢ ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) → ( 𝑔 ·s ℎ ) ∈ No ) |
| 133 |
129
|
adantr |
⊢ ( ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) ∧ ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) ) → ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 134 |
|
simprl3 |
⊢ ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) → 𝑖 ∈ No ) |
| 135 |
134
|
adantr |
⊢ ( ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) ∧ ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) ) → 𝑖 ∈ No ) |
| 136 |
|
simprr1 |
⊢ ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) → 𝑗 ∈ No ) |
| 137 |
136
|
adantr |
⊢ ( ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) ∧ ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) ) → 𝑗 ∈ No ) |
| 138 |
|
simprr2 |
⊢ ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) → 𝑘 ∈ No ) |
| 139 |
138
|
adantr |
⊢ ( ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) ∧ ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) ) → 𝑘 ∈ No ) |
| 140 |
|
simprr3 |
⊢ ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) → 𝑙 ∈ No ) |
| 141 |
140
|
adantr |
⊢ ( ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) ∧ ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) ) → 𝑙 ∈ No ) |
| 142 |
|
simprl |
⊢ ( ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) ∧ ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) ) → 𝑖 <s 𝑗 ) |
| 143 |
|
simprr |
⊢ ( ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) ∧ ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) ) → 𝑘 <s 𝑙 ) |
| 144 |
133 135 137 139 141 142 143
|
mulsproplem14 |
⊢ ( ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) ∧ ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) |
| 145 |
144
|
ex |
⊢ ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) → ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) |
| 146 |
132 145
|
jca |
⊢ ( ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) |
| 147 |
146
|
ex |
⊢ ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) → ( ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 148 |
128 147
|
biimtrdi |
⊢ ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ∀ 𝑦 ∈ 𝑥 ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) → ( ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) ) |
| 149 |
148
|
impd |
⊢ ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( ∀ 𝑦 ∈ 𝑥 ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 150 |
149
|
com12 |
⊢ ( ( ∀ 𝑦 ∈ 𝑥 ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) ) → ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 151 |
150
|
anassrs |
⊢ ( ( ( ∀ 𝑦 ∈ 𝑥 ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ) ∧ ( 𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ) ) → ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 152 |
151
|
ralrimivvva |
⊢ ( ( ∀ 𝑦 ∈ 𝑥 ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ∧ ( 𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ) → ∀ 𝑗 ∈ No ∀ 𝑘 ∈ No ∀ 𝑙 ∈ No ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 153 |
152
|
ralrimivvva |
⊢ ( ∀ 𝑦 ∈ 𝑥 ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) → ∀ 𝑔 ∈ No ∀ ℎ ∈ No ∀ 𝑖 ∈ No ∀ 𝑗 ∈ No ∀ 𝑘 ∈ No ∀ 𝑙 ∈ No ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 154 |
153
|
a1i |
⊢ ( 𝑥 ∈ On → ( ∀ 𝑦 ∈ 𝑥 ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( 𝑦 = ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) → ∀ 𝑔 ∈ No ∀ ℎ ∈ No ∀ 𝑖 ∈ No ∀ 𝑗 ∈ No ∀ 𝑘 ∈ No ∀ 𝑙 ∈ No ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) ) |
| 155 |
113 154
|
tfis2 |
⊢ ( 𝑥 ∈ On → ∀ 𝑔 ∈ No ∀ ℎ ∈ No ∀ 𝑖 ∈ No ∀ 𝑗 ∈ No ∀ 𝑘 ∈ No ∀ 𝑙 ∈ No ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 156 |
|
fveq2 |
⊢ ( 𝑔 = 𝐴 → ( bday ‘ 𝑔 ) = ( bday ‘ 𝐴 ) ) |
| 157 |
156
|
oveq1d |
⊢ ( 𝑔 = 𝐴 → ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) = ( ( bday ‘ 𝐴 ) +no ( bday ‘ ℎ ) ) ) |
| 158 |
157
|
uneq1d |
⊢ ( 𝑔 = 𝐴 → ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) |
| 159 |
158
|
eqeq2d |
⊢ ( 𝑔 = 𝐴 → ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ↔ 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) ) |
| 160 |
|
oveq1 |
⊢ ( 𝑔 = 𝐴 → ( 𝑔 ·s ℎ ) = ( 𝐴 ·s ℎ ) ) |
| 161 |
160
|
eleq1d |
⊢ ( 𝑔 = 𝐴 → ( ( 𝑔 ·s ℎ ) ∈ No ↔ ( 𝐴 ·s ℎ ) ∈ No ) ) |
| 162 |
161
|
anbi1d |
⊢ ( 𝑔 = 𝐴 → ( ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ↔ ( ( 𝐴 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 163 |
159 162
|
imbi12d |
⊢ ( 𝑔 = 𝐴 → ( ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ↔ ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝐴 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) ) |
| 164 |
|
fveq2 |
⊢ ( ℎ = 𝐵 → ( bday ‘ ℎ ) = ( bday ‘ 𝐵 ) ) |
| 165 |
164
|
oveq2d |
⊢ ( ℎ = 𝐵 → ( ( bday ‘ 𝐴 ) +no ( bday ‘ ℎ ) ) = ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ) |
| 166 |
165
|
uneq1d |
⊢ ( ℎ = 𝐵 → ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) |
| 167 |
166
|
eqeq2d |
⊢ ( ℎ = 𝐵 → ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ↔ 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) ) |
| 168 |
|
oveq2 |
⊢ ( ℎ = 𝐵 → ( 𝐴 ·s ℎ ) = ( 𝐴 ·s 𝐵 ) ) |
| 169 |
168
|
eleq1d |
⊢ ( ℎ = 𝐵 → ( ( 𝐴 ·s ℎ ) ∈ No ↔ ( 𝐴 ·s 𝐵 ) ∈ No ) ) |
| 170 |
169
|
anbi1d |
⊢ ( ℎ = 𝐵 → ( ( ( 𝐴 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ↔ ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 171 |
167 170
|
imbi12d |
⊢ ( ℎ = 𝐵 → ( ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝐴 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ↔ ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) ) |
| 172 |
|
fveq2 |
⊢ ( 𝑖 = 𝐶 → ( bday ‘ 𝑖 ) = ( bday ‘ 𝐶 ) ) |
| 173 |
172
|
oveq1d |
⊢ ( 𝑖 = 𝐶 → ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) = ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ) |
| 174 |
173
|
uneq1d |
⊢ ( 𝑖 = 𝐶 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) = ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ) |
| 175 |
172
|
oveq1d |
⊢ ( 𝑖 = 𝐶 → ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) = ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ) |
| 176 |
175
|
uneq1d |
⊢ ( 𝑖 = 𝐶 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) = ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) |
| 177 |
174 176
|
uneq12d |
⊢ ( 𝑖 = 𝐶 → ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) = ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) |
| 178 |
177
|
uneq2d |
⊢ ( 𝑖 = 𝐶 → ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) |
| 179 |
178
|
eqeq2d |
⊢ ( 𝑖 = 𝐶 → ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ↔ 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) ) |
| 180 |
|
breq1 |
⊢ ( 𝑖 = 𝐶 → ( 𝑖 <s 𝑗 ↔ 𝐶 <s 𝑗 ) ) |
| 181 |
180
|
anbi1d |
⊢ ( 𝑖 = 𝐶 → ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) ↔ ( 𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙 ) ) ) |
| 182 |
|
oveq1 |
⊢ ( 𝑖 = 𝐶 → ( 𝑖 ·s 𝑙 ) = ( 𝐶 ·s 𝑙 ) ) |
| 183 |
|
oveq1 |
⊢ ( 𝑖 = 𝐶 → ( 𝑖 ·s 𝑘 ) = ( 𝐶 ·s 𝑘 ) ) |
| 184 |
182 183
|
oveq12d |
⊢ ( 𝑖 = 𝐶 → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) = ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) ) |
| 185 |
184
|
breq1d |
⊢ ( 𝑖 = 𝐶 → ( ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ↔ ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) |
| 186 |
181 185
|
imbi12d |
⊢ ( 𝑖 = 𝐶 → ( ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ↔ ( ( 𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) |
| 187 |
186
|
anbi2d |
⊢ ( 𝑖 = 𝐶 → ( ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ↔ ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 188 |
179 187
|
imbi12d |
⊢ ( 𝑖 = 𝐶 → ( ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ↔ ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) ) |
| 189 |
|
fveq2 |
⊢ ( 𝑗 = 𝐷 → ( bday ‘ 𝑗 ) = ( bday ‘ 𝐷 ) ) |
| 190 |
189
|
oveq1d |
⊢ ( 𝑗 = 𝐷 → ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) = ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) |
| 191 |
190
|
uneq2d |
⊢ ( 𝑗 = 𝐷 → ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) = ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ) |
| 192 |
189
|
oveq1d |
⊢ ( 𝑗 = 𝐷 → ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) = ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑘 ) ) ) |
| 193 |
192
|
uneq2d |
⊢ ( 𝑗 = 𝐷 → ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) = ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑘 ) ) ) ) |
| 194 |
191 193
|
uneq12d |
⊢ ( 𝑗 = 𝐷 → ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) = ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑘 ) ) ) ) ) |
| 195 |
194
|
uneq2d |
⊢ ( 𝑗 = 𝐷 → ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) |
| 196 |
195
|
eqeq2d |
⊢ ( 𝑗 = 𝐷 → ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ↔ 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) ) |
| 197 |
|
breq2 |
⊢ ( 𝑗 = 𝐷 → ( 𝐶 <s 𝑗 ↔ 𝐶 <s 𝐷 ) ) |
| 198 |
197
|
anbi1d |
⊢ ( 𝑗 = 𝐷 → ( ( 𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙 ) ↔ ( 𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙 ) ) ) |
| 199 |
|
oveq1 |
⊢ ( 𝑗 = 𝐷 → ( 𝑗 ·s 𝑙 ) = ( 𝐷 ·s 𝑙 ) ) |
| 200 |
|
oveq1 |
⊢ ( 𝑗 = 𝐷 → ( 𝑗 ·s 𝑘 ) = ( 𝐷 ·s 𝑘 ) ) |
| 201 |
199 200
|
oveq12d |
⊢ ( 𝑗 = 𝐷 → ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) = ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝑘 ) ) ) |
| 202 |
201
|
breq2d |
⊢ ( 𝑗 = 𝐷 → ( ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ↔ ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝑘 ) ) ) ) |
| 203 |
198 202
|
imbi12d |
⊢ ( 𝑗 = 𝐷 → ( ( ( 𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ↔ ( ( 𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝑘 ) ) ) ) ) |
| 204 |
203
|
anbi2d |
⊢ ( 𝑗 = 𝐷 → ( ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ↔ ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝑘 ) ) ) ) ) ) |
| 205 |
196 204
|
imbi12d |
⊢ ( 𝑗 = 𝐷 → ( ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ↔ ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝑘 ) ) ) ) ) ) ) |
| 206 |
|
fveq2 |
⊢ ( 𝑘 = 𝐸 → ( bday ‘ 𝑘 ) = ( bday ‘ 𝐸 ) ) |
| 207 |
206
|
oveq2d |
⊢ ( 𝑘 = 𝐸 → ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) = ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ) |
| 208 |
207
|
uneq1d |
⊢ ( 𝑘 = 𝐸 → ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) = ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ) |
| 209 |
206
|
oveq2d |
⊢ ( 𝑘 = 𝐸 → ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑘 ) ) = ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) |
| 210 |
209
|
uneq2d |
⊢ ( 𝑘 = 𝐸 → ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑘 ) ) ) = ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) |
| 211 |
208 210
|
uneq12d |
⊢ ( 𝑘 = 𝐸 → ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑘 ) ) ) ) = ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) |
| 212 |
211
|
uneq2d |
⊢ ( 𝑘 = 𝐸 → ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑘 ) ) ) ) ) = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ) |
| 213 |
212
|
eqeq2d |
⊢ ( 𝑘 = 𝐸 → ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ↔ 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ) ) |
| 214 |
|
breq1 |
⊢ ( 𝑘 = 𝐸 → ( 𝑘 <s 𝑙 ↔ 𝐸 <s 𝑙 ) ) |
| 215 |
214
|
anbi2d |
⊢ ( 𝑘 = 𝐸 → ( ( 𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙 ) ↔ ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙 ) ) ) |
| 216 |
|
oveq2 |
⊢ ( 𝑘 = 𝐸 → ( 𝐶 ·s 𝑘 ) = ( 𝐶 ·s 𝐸 ) ) |
| 217 |
216
|
oveq2d |
⊢ ( 𝑘 = 𝐸 → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) = ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝐸 ) ) ) |
| 218 |
|
oveq2 |
⊢ ( 𝑘 = 𝐸 → ( 𝐷 ·s 𝑘 ) = ( 𝐷 ·s 𝐸 ) ) |
| 219 |
218
|
oveq2d |
⊢ ( 𝑘 = 𝐸 → ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝑘 ) ) = ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝐸 ) ) ) |
| 220 |
217 219
|
breq12d |
⊢ ( 𝑘 = 𝐸 → ( ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝑘 ) ) ↔ ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) |
| 221 |
215 220
|
imbi12d |
⊢ ( 𝑘 = 𝐸 → ( ( ( 𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝑘 ) ) ) ↔ ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) ) |
| 222 |
221
|
anbi2d |
⊢ ( 𝑘 = 𝐸 → ( ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝑘 ) ) ) ) ↔ ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) ) ) |
| 223 |
213 222
|
imbi12d |
⊢ ( 𝑘 = 𝐸 → ( ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝑘 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝑘 ) ) ) ) ) ↔ ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) ) ) ) |
| 224 |
|
fveq2 |
⊢ ( 𝑙 = 𝐹 → ( bday ‘ 𝑙 ) = ( bday ‘ 𝐹 ) ) |
| 225 |
224
|
oveq2d |
⊢ ( 𝑙 = 𝐹 → ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) = ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) |
| 226 |
225
|
uneq2d |
⊢ ( 𝑙 = 𝐹 → ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) = ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ) |
| 227 |
224
|
oveq2d |
⊢ ( 𝑙 = 𝐹 → ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) = ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ) |
| 228 |
227
|
uneq1d |
⊢ ( 𝑙 = 𝐹 → ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) = ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) |
| 229 |
226 228
|
uneq12d |
⊢ ( 𝑙 = 𝐹 → ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) = ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) |
| 230 |
229
|
uneq2d |
⊢ ( 𝑙 = 𝐹 → ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ) |
| 231 |
230
|
eqeq2d |
⊢ ( 𝑙 = 𝐹 → ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ↔ 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ) ) |
| 232 |
|
breq2 |
⊢ ( 𝑙 = 𝐹 → ( 𝐸 <s 𝑙 ↔ 𝐸 <s 𝐹 ) ) |
| 233 |
232
|
anbi2d |
⊢ ( 𝑙 = 𝐹 → ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙 ) ↔ ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹 ) ) ) |
| 234 |
|
oveq2 |
⊢ ( 𝑙 = 𝐹 → ( 𝐶 ·s 𝑙 ) = ( 𝐶 ·s 𝐹 ) ) |
| 235 |
234
|
oveq1d |
⊢ ( 𝑙 = 𝐹 → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝐸 ) ) = ( ( 𝐶 ·s 𝐹 ) -s ( 𝐶 ·s 𝐸 ) ) ) |
| 236 |
|
oveq2 |
⊢ ( 𝑙 = 𝐹 → ( 𝐷 ·s 𝑙 ) = ( 𝐷 ·s 𝐹 ) ) |
| 237 |
236
|
oveq1d |
⊢ ( 𝑙 = 𝐹 → ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝐸 ) ) = ( ( 𝐷 ·s 𝐹 ) -s ( 𝐷 ·s 𝐸 ) ) ) |
| 238 |
235 237
|
breq12d |
⊢ ( 𝑙 = 𝐹 → ( ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝐸 ) ) ↔ ( ( 𝐶 ·s 𝐹 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝐹 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) |
| 239 |
233 238
|
imbi12d |
⊢ ( 𝑙 = 𝐹 → ( ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝐸 ) ) ) ↔ ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹 ) → ( ( 𝐶 ·s 𝐹 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝐹 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) ) |
| 240 |
239
|
anbi2d |
⊢ ( 𝑙 = 𝐹 → ( ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) ↔ ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹 ) → ( ( 𝐶 ·s 𝐹 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝐹 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) ) ) |
| 241 |
231 240
|
imbi12d |
⊢ ( 𝑙 = 𝐹 → ( ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙 ) → ( ( 𝐶 ·s 𝑙 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝑙 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) ) ↔ ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹 ) → ( ( 𝐶 ·s 𝐹 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝐹 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) ) ) ) |
| 242 |
163 171 188 205 223 241
|
rspc6v |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ ( 𝐸 ∈ No ∧ 𝐹 ∈ No ) ) → ( ∀ 𝑔 ∈ No ∀ ℎ ∈ No ∀ 𝑖 ∈ No ∀ 𝑗 ∈ No ∀ 𝑘 ∈ No ∀ 𝑙 ∈ No ( 𝑥 = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) → ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹 ) → ( ( 𝐶 ·s 𝐹 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝐹 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) ) ) ) |
| 243 |
155 242
|
syl5com |
⊢ ( 𝑥 ∈ On → ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ ( 𝐸 ∈ No ∧ 𝐹 ∈ No ) ) → ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹 ) → ( ( 𝐶 ·s 𝐹 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝐹 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) ) ) ) |
| 244 |
243
|
com23 |
⊢ ( 𝑥 ∈ On → ( 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ ( 𝐸 ∈ No ∧ 𝐹 ∈ No ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹 ) → ( ( 𝐶 ·s 𝐹 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝐹 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) ) ) ) |
| 245 |
244
|
rexlimiv |
⊢ ( ∃ 𝑥 ∈ On 𝑥 = ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ ( 𝐸 ∈ No ∧ 𝐹 ∈ No ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹 ) → ( ( 𝐶 ·s 𝐹 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝐹 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) ) ) |
| 246 |
22 245
|
ax-mp |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ ( 𝐸 ∈ No ∧ 𝐹 ∈ No ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ( 𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹 ) → ( ( 𝐶 ·s 𝐹 ) -s ( 𝐶 ·s 𝐸 ) ) <s ( ( 𝐷 ·s 𝐹 ) -s ( 𝐷 ·s 𝐸 ) ) ) ) ) |