Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
⊢ ( 𝑥 = 1 → ( RePart ‘ 𝑥 ) = ( RePart ‘ 1 ) ) |
2 |
|
fveq2 |
⊢ ( 𝑥 = 1 → ( 𝑝 ‘ 𝑥 ) = ( 𝑝 ‘ 1 ) ) |
3 |
2
|
oveq2d |
⊢ ( 𝑥 = 1 → ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) = ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 1 ) ) ) |
4 |
3
|
eleq2d |
⊢ ( 𝑥 = 1 → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) ↔ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 1 ) ) ) ) |
5 |
|
oveq2 |
⊢ ( 𝑥 = 1 → ( 0 ..^ 𝑥 ) = ( 0 ..^ 1 ) ) |
6 |
|
fzo01 |
⊢ ( 0 ..^ 1 ) = { 0 } |
7 |
5 6
|
eqtrdi |
⊢ ( 𝑥 = 1 → ( 0 ..^ 𝑥 ) = { 0 } ) |
8 |
7
|
rexeqdv |
⊢ ( 𝑥 = 1 → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑥 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ ∃ 𝑖 ∈ { 0 } 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
9 |
4 8
|
imbi12d |
⊢ ( 𝑥 = 1 → ( ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑥 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 1 ) ) → ∃ 𝑖 ∈ { 0 } 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
10 |
1 9
|
raleqbidv |
⊢ ( 𝑥 = 1 → ( ∀ 𝑝 ∈ ( RePart ‘ 𝑥 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑥 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ∀ 𝑝 ∈ ( RePart ‘ 1 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 1 ) ) → ∃ 𝑖 ∈ { 0 } 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
11 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( RePart ‘ 𝑥 ) = ( RePart ‘ 𝑦 ) ) |
12 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑝 ‘ 𝑥 ) = ( 𝑝 ‘ 𝑦 ) ) |
13 |
12
|
oveq2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) = ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) |
14 |
13
|
eleq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) ↔ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) ) |
15 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 0 ..^ 𝑥 ) = ( 0 ..^ 𝑦 ) ) |
16 |
15
|
rexeqdv |
⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑥 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
17 |
14 16
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑥 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
18 |
11 17
|
raleqbidv |
⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑝 ∈ ( RePart ‘ 𝑥 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑥 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
19 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( RePart ‘ 𝑥 ) = ( RePart ‘ ( 𝑦 + 1 ) ) ) |
20 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑝 ‘ 𝑥 ) = ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) |
21 |
20
|
oveq2d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) = ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) |
22 |
21
|
eleq2d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) ↔ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) |
23 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 0 ..^ 𝑥 ) = ( 0 ..^ ( 𝑦 + 1 ) ) ) |
24 |
23
|
rexeqdv |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑥 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
25 |
22 24
|
imbi12d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑥 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
26 |
19 25
|
raleqbidv |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ∀ 𝑝 ∈ ( RePart ‘ 𝑥 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑥 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ∀ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
27 |
|
fveq2 |
⊢ ( 𝑥 = 𝑀 → ( RePart ‘ 𝑥 ) = ( RePart ‘ 𝑀 ) ) |
28 |
|
fveq2 |
⊢ ( 𝑥 = 𝑀 → ( 𝑝 ‘ 𝑥 ) = ( 𝑝 ‘ 𝑀 ) ) |
29 |
28
|
oveq2d |
⊢ ( 𝑥 = 𝑀 → ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) = ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑀 ) ) ) |
30 |
29
|
eleq2d |
⊢ ( 𝑥 = 𝑀 → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) ↔ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑀 ) ) ) ) |
31 |
|
oveq2 |
⊢ ( 𝑥 = 𝑀 → ( 0 ..^ 𝑥 ) = ( 0 ..^ 𝑀 ) ) |
32 |
31
|
rexeqdv |
⊢ ( 𝑥 = 𝑀 → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑥 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
33 |
30 32
|
imbi12d |
⊢ ( 𝑥 = 𝑀 → ( ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑥 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑀 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
34 |
27 33
|
raleqbidv |
⊢ ( 𝑥 = 𝑀 → ( ∀ 𝑝 ∈ ( RePart ‘ 𝑥 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑥 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑥 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ∀ 𝑝 ∈ ( RePart ‘ 𝑀 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑀 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
35 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
36 |
|
fveq2 |
⊢ ( 𝑖 = 0 → ( 𝑝 ‘ 𝑖 ) = ( 𝑝 ‘ 0 ) ) |
37 |
|
fv0p1e1 |
⊢ ( 𝑖 = 0 → ( 𝑝 ‘ ( 𝑖 + 1 ) ) = ( 𝑝 ‘ 1 ) ) |
38 |
36 37
|
oveq12d |
⊢ ( 𝑖 = 0 → ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 1 ) ) ) |
39 |
38
|
eleq2d |
⊢ ( 𝑖 = 0 → ( 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 1 ) ) ) ) |
40 |
39
|
rexsng |
⊢ ( 0 ∈ ℕ0 → ( ∃ 𝑖 ∈ { 0 } 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 1 ) ) ) ) |
41 |
35 40
|
ax-mp |
⊢ ( ∃ 𝑖 ∈ { 0 } 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 1 ) ) ) |
42 |
41
|
biimpri |
⊢ ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 1 ) ) → ∃ 𝑖 ∈ { 0 } 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) |
43 |
42
|
rgenw |
⊢ ∀ 𝑝 ∈ ( RePart ‘ 1 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 1 ) ) → ∃ 𝑖 ∈ { 0 } 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) |
44 |
|
nfv |
⊢ Ⅎ 𝑝 𝑦 ∈ ℕ |
45 |
|
nfra1 |
⊢ Ⅎ 𝑝 ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) |
46 |
44 45
|
nfan |
⊢ Ⅎ 𝑝 ( 𝑦 ∈ ℕ ∧ ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
47 |
|
nnnn0 |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ℕ0 ) |
48 |
|
fzonn0p1 |
⊢ ( 𝑦 ∈ ℕ0 → 𝑦 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ) |
49 |
47 48
|
syl |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ) |
50 |
49
|
ad2antrr |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) → 𝑦 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ) |
51 |
|
fveq2 |
⊢ ( 𝑖 = 𝑦 → ( 𝑝 ‘ 𝑖 ) = ( 𝑝 ‘ 𝑦 ) ) |
52 |
|
fvoveq1 |
⊢ ( 𝑖 = 𝑦 → ( 𝑝 ‘ ( 𝑖 + 1 ) ) = ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) |
53 |
51 52
|
oveq12d |
⊢ ( 𝑖 = 𝑦 → ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑝 ‘ 𝑦 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) |
54 |
53
|
eleq2d |
⊢ ( 𝑖 = 𝑦 → ( 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ 𝑋 ∈ ( ( 𝑝 ‘ 𝑦 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) |
55 |
54
|
adantl |
⊢ ( ( ( ( 𝑦 ∈ ℕ ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) ∧ 𝑖 = 𝑦 ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ 𝑋 ∈ ( ( 𝑝 ‘ 𝑦 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) |
56 |
|
peano2nn |
⊢ ( 𝑦 ∈ ℕ → ( 𝑦 + 1 ) ∈ ℕ ) |
57 |
56
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑦 + 1 ) ∈ ℕ ) |
58 |
|
simpr |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) |
59 |
56
|
nnnn0d |
⊢ ( 𝑦 ∈ ℕ → ( 𝑦 + 1 ) ∈ ℕ0 ) |
60 |
|
0elfz |
⊢ ( ( 𝑦 + 1 ) ∈ ℕ0 → 0 ∈ ( 0 ... ( 𝑦 + 1 ) ) ) |
61 |
59 60
|
syl |
⊢ ( 𝑦 ∈ ℕ → 0 ∈ ( 0 ... ( 𝑦 + 1 ) ) ) |
62 |
61
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → 0 ∈ ( 0 ... ( 𝑦 + 1 ) ) ) |
63 |
57 58 62
|
iccpartxr |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑝 ‘ 0 ) ∈ ℝ* ) |
64 |
|
nn0fz0 |
⊢ ( ( 𝑦 + 1 ) ∈ ℕ0 ↔ ( 𝑦 + 1 ) ∈ ( 0 ... ( 𝑦 + 1 ) ) ) |
65 |
59 64
|
sylib |
⊢ ( 𝑦 ∈ ℕ → ( 𝑦 + 1 ) ∈ ( 0 ... ( 𝑦 + 1 ) ) ) |
66 |
65
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑦 + 1 ) ∈ ( 0 ... ( 𝑦 + 1 ) ) ) |
67 |
57 58 66
|
iccpartxr |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑝 ‘ ( 𝑦 + 1 ) ) ∈ ℝ* ) |
68 |
63 67
|
jca |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ( 𝑝 ‘ 0 ) ∈ ℝ* ∧ ( 𝑝 ‘ ( 𝑦 + 1 ) ) ∈ ℝ* ) ) |
69 |
68
|
adantlr |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ( 𝑝 ‘ 0 ) ∈ ℝ* ∧ ( 𝑝 ‘ ( 𝑦 + 1 ) ) ∈ ℝ* ) ) |
70 |
|
elico1 |
⊢ ( ( ( 𝑝 ‘ 0 ) ∈ ℝ* ∧ ( 𝑝 ‘ ( 𝑦 + 1 ) ) ∈ ℝ* ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ↔ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) |
71 |
69 70
|
syl |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ↔ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) |
72 |
|
simp1 |
⊢ ( ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → 𝑋 ∈ ℝ* ) |
73 |
72
|
adantl |
⊢ ( ( ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ∧ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) → 𝑋 ∈ ℝ* ) |
74 |
|
simpl |
⊢ ( ( ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ∧ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) → ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) |
75 |
|
simpr3 |
⊢ ( ( ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ∧ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) → 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) |
76 |
73 74 75
|
3jca |
⊢ ( ( ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ∧ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) → ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) |
77 |
76
|
ex |
⊢ ( ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) |
78 |
77
|
adantl |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) → ( ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) |
79 |
78
|
adantr |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) |
80 |
71 79
|
sylbid |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) |
81 |
80
|
impr |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) → ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) |
82 |
|
nn0fz0 |
⊢ ( 𝑦 ∈ ℕ0 ↔ 𝑦 ∈ ( 0 ... 𝑦 ) ) |
83 |
47 82
|
sylib |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ( 0 ... 𝑦 ) ) |
84 |
|
fzelp1 |
⊢ ( 𝑦 ∈ ( 0 ... 𝑦 ) → 𝑦 ∈ ( 0 ... ( 𝑦 + 1 ) ) ) |
85 |
83 84
|
syl |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ( 0 ... ( 𝑦 + 1 ) ) ) |
86 |
85
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → 𝑦 ∈ ( 0 ... ( 𝑦 + 1 ) ) ) |
87 |
57 58 86
|
iccpartxr |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑝 ‘ 𝑦 ) ∈ ℝ* ) |
88 |
87 67
|
jca |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ( 𝑝 ‘ 𝑦 ) ∈ ℝ* ∧ ( 𝑝 ‘ ( 𝑦 + 1 ) ) ∈ ℝ* ) ) |
89 |
88
|
ad2ant2r |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) → ( ( 𝑝 ‘ 𝑦 ) ∈ ℝ* ∧ ( 𝑝 ‘ ( 𝑦 + 1 ) ) ∈ ℝ* ) ) |
90 |
|
elico1 |
⊢ ( ( ( 𝑝 ‘ 𝑦 ) ∈ ℝ* ∧ ( 𝑝 ‘ ( 𝑦 + 1 ) ) ∈ ℝ* ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 𝑦 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ↔ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) |
91 |
89 90
|
syl |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 𝑦 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ↔ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) |
92 |
81 91
|
mpbird |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) → 𝑋 ∈ ( ( 𝑝 ‘ 𝑦 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) |
93 |
50 55 92
|
rspcedvd |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) |
94 |
93
|
exp43 |
⊢ ( 𝑦 ∈ ℕ → ( ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) |
95 |
94
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ ∧ ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) → ( ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) |
96 |
|
iccpartres |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ ( RePart ‘ 𝑦 ) ) |
97 |
|
rspsbca |
⊢ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ ( RePart ‘ 𝑦 ) ∧ ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) → [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
98 |
|
vex |
⊢ 𝑝 ∈ V |
99 |
98
|
resex |
⊢ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V |
100 |
|
sbcimg |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ( [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
101 |
|
sbcel2 |
⊢ ( [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ↔ 𝑋 ∈ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) |
102 |
|
csbov12g |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) = ( ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ 0 ) [,) ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ 𝑦 ) ) ) |
103 |
|
csbfv12 |
⊢ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ 0 ) = ( ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑝 ‘ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 0 ) |
104 |
|
csbvarg |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑝 = ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ) |
105 |
|
csbconstg |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 0 = 0 ) |
106 |
104 105
|
fveq12d |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑝 ‘ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 0 ) = ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) ) |
107 |
103 106
|
syl5eq |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ 0 ) = ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) ) |
108 |
|
csbfv12 |
⊢ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ 𝑦 ) = ( ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑝 ‘ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑦 ) |
109 |
|
csbconstg |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑦 = 𝑦 ) |
110 |
104 109
|
fveq12d |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑝 ‘ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑦 ) = ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) |
111 |
108 110
|
syl5eq |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ 𝑦 ) = ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) |
112 |
107 111
|
oveq12d |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ 0 ) [,) ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ 𝑦 ) ) = ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) ) |
113 |
102 112
|
eqtrd |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) = ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) ) |
114 |
113
|
eleq2d |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( 𝑋 ∈ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ↔ 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) ) ) |
115 |
101 114
|
syl5bb |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ↔ 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) ) ) |
116 |
|
sbcrex |
⊢ ( [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) |
117 |
|
sbcel2 |
⊢ ( [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ 𝑋 ∈ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) |
118 |
|
csbov12g |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) = ( ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ 𝑖 ) [,) ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) |
119 |
|
csbfv12 |
⊢ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ 𝑖 ) = ( ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑝 ‘ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑖 ) |
120 |
|
csbconstg |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑖 = 𝑖 ) |
121 |
104 120
|
fveq12d |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑝 ‘ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑖 ) = ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) ) |
122 |
119 121
|
syl5eq |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ 𝑖 ) = ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) ) |
123 |
|
csbfv12 |
⊢ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ ( 𝑖 + 1 ) ) = ( ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑝 ‘ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑖 + 1 ) ) |
124 |
|
csbconstg |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑖 + 1 ) = ( 𝑖 + 1 ) ) |
125 |
104 124
|
fveq12d |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ 𝑝 ‘ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑖 + 1 ) ) = ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) |
126 |
123 125
|
syl5eq |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ ( 𝑖 + 1 ) ) = ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) |
127 |
122 126
|
oveq12d |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ 𝑖 ) [,) ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) = ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) |
128 |
118 127
|
eqtrd |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) = ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) |
129 |
128
|
eleq2d |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( 𝑋 ∈ ⦋ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ⦌ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) ) |
130 |
117 129
|
syl5bb |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) ) |
131 |
130
|
rexbidv |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) ) |
132 |
116 131
|
syl5bb |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) ) |
133 |
115 132
|
imbi12d |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( ( [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ( 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
134 |
100 133
|
bitrd |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ V → ( [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ( 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
135 |
99 134
|
ax-mp |
⊢ ( [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ( 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) ) |
136 |
68 70
|
syl |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ↔ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) |
137 |
136
|
adantr |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ↔ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) ) |
138 |
72
|
adantl |
⊢ ( ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) → 𝑋 ∈ ℝ* ) |
139 |
|
simpr2 |
⊢ ( ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) → ( 𝑝 ‘ 0 ) ≤ 𝑋 ) |
140 |
|
xrltnle |
⊢ ( ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 𝑦 ) ∈ ℝ* ) → ( 𝑋 < ( 𝑝 ‘ 𝑦 ) ↔ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ) |
141 |
72 87 140
|
syl2anr |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) → ( 𝑋 < ( 𝑝 ‘ 𝑦 ) ↔ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ) |
142 |
141
|
exbiri |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ( ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → 𝑋 < ( 𝑝 ‘ 𝑦 ) ) ) ) |
143 |
142
|
com23 |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → 𝑋 < ( 𝑝 ‘ 𝑦 ) ) ) ) |
144 |
143
|
imp31 |
⊢ ( ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) → 𝑋 < ( 𝑝 ‘ 𝑦 ) ) |
145 |
138 139 144
|
3jca |
⊢ ( ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) → ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ 𝑦 ) ) ) |
146 |
63 87
|
jca |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ( 𝑝 ‘ 0 ) ∈ ℝ* ∧ ( 𝑝 ‘ 𝑦 ) ∈ ℝ* ) ) |
147 |
146
|
ad2antrr |
⊢ ( ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) → ( ( 𝑝 ‘ 0 ) ∈ ℝ* ∧ ( 𝑝 ‘ 𝑦 ) ∈ ℝ* ) ) |
148 |
|
elico1 |
⊢ ( ( ( 𝑝 ‘ 0 ) ∈ ℝ* ∧ ( 𝑝 ‘ 𝑦 ) ∈ ℝ* ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ↔ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ 𝑦 ) ) ) ) |
149 |
147 148
|
syl |
⊢ ( ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ↔ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ 𝑦 ) ) ) ) |
150 |
145 149
|
mpbird |
⊢ ( ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) ∧ ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) ) → 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) |
151 |
150
|
ex |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) → ( ( 𝑋 ∈ ℝ* ∧ ( 𝑝 ‘ 0 ) ≤ 𝑋 ∧ 𝑋 < ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) ) |
152 |
137 151
|
sylbid |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) ) |
153 |
|
0elfz |
⊢ ( 𝑦 ∈ ℕ0 → 0 ∈ ( 0 ... 𝑦 ) ) |
154 |
47 153
|
syl |
⊢ ( 𝑦 ∈ ℕ → 0 ∈ ( 0 ... 𝑦 ) ) |
155 |
154
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → 0 ∈ ( 0 ... 𝑦 ) ) |
156 |
|
fvres |
⊢ ( 0 ∈ ( 0 ... 𝑦 ) → ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) = ( 𝑝 ‘ 0 ) ) |
157 |
155 156
|
syl |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) = ( 𝑝 ‘ 0 ) ) |
158 |
157
|
eqcomd |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑝 ‘ 0 ) = ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) ) |
159 |
83
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → 𝑦 ∈ ( 0 ... 𝑦 ) ) |
160 |
|
fvres |
⊢ ( 𝑦 ∈ ( 0 ... 𝑦 ) → ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) = ( 𝑝 ‘ 𝑦 ) ) |
161 |
159 160
|
syl |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) = ( 𝑝 ‘ 𝑦 ) ) |
162 |
161
|
eqcomd |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑝 ‘ 𝑦 ) = ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) |
163 |
158 162
|
oveq12d |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) = ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) ) |
164 |
163
|
eleq2d |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ↔ 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) ) ) |
165 |
164
|
biimpa |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) → 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) ) |
166 |
|
elfzofz |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → 𝑖 ∈ ( 0 ... 𝑦 ) ) |
167 |
166
|
adantl |
⊢ ( ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → 𝑖 ∈ ( 0 ... 𝑦 ) ) |
168 |
|
fvres |
⊢ ( 𝑖 ∈ ( 0 ... 𝑦 ) → ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) = ( 𝑝 ‘ 𝑖 ) ) |
169 |
167 168
|
syl |
⊢ ( ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) = ( 𝑝 ‘ 𝑖 ) ) |
170 |
|
fzofzp1 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑦 ) ) |
171 |
170
|
adantl |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑦 ) ) |
172 |
|
fvres |
⊢ ( ( 𝑖 + 1 ) ∈ ( 0 ... 𝑦 ) → ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) = ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) |
173 |
171 172
|
syl |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) = ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) |
174 |
173
|
adantlr |
⊢ ( ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) = ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) |
175 |
169 174
|
oveq12d |
⊢ ( ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) |
176 |
175
|
eleq2d |
⊢ ( ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ↔ 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
177 |
176
|
rexbidva |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ↔ ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
178 |
|
nnz |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ℤ ) |
179 |
|
uzid |
⊢ ( 𝑦 ∈ ℤ → 𝑦 ∈ ( ℤ≥ ‘ 𝑦 ) ) |
180 |
178 179
|
syl |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ( ℤ≥ ‘ 𝑦 ) ) |
181 |
|
peano2uz |
⊢ ( 𝑦 ∈ ( ℤ≥ ‘ 𝑦 ) → ( 𝑦 + 1 ) ∈ ( ℤ≥ ‘ 𝑦 ) ) |
182 |
|
fzoss2 |
⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ≥ ‘ 𝑦 ) → ( 0 ..^ 𝑦 ) ⊆ ( 0 ..^ ( 𝑦 + 1 ) ) ) |
183 |
180 181 182
|
3syl |
⊢ ( 𝑦 ∈ ℕ → ( 0 ..^ 𝑦 ) ⊆ ( 0 ..^ ( 𝑦 + 1 ) ) ) |
184 |
183
|
ad2antrr |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) → ( 0 ..^ 𝑦 ) ⊆ ( 0 ..^ ( 𝑦 + 1 ) ) ) |
185 |
|
ssrexv |
⊢ ( ( 0 ..^ 𝑦 ) ⊆ ( 0 ..^ ( 𝑦 + 1 ) ) → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
186 |
184 185
|
syl |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
187 |
177 186
|
sylbid |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
188 |
165 187
|
embantd |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) ) → ( ( 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
189 |
188
|
ex |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ( ( 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
190 |
189
|
adantr |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ( ( 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
191 |
152 190
|
syld |
⊢ ( ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) ∧ ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ( ( 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
192 |
191
|
ex |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ( ( 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) |
193 |
192
|
com34 |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( ( 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) |
194 |
193
|
com13 |
⊢ ( ( 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 0 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ 𝑖 ) [,) ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ‘ ( 𝑖 + 1 ) ) ) ) → ( ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) |
195 |
135 194
|
sylbi |
⊢ ( [ ( 𝑝 ↾ ( 0 ... 𝑦 ) ) / 𝑝 ] ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) → ( ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) |
196 |
97 195
|
syl |
⊢ ( ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ ( RePart ‘ 𝑦 ) ∧ ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) → ( ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) |
197 |
196
|
ex |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ ( RePart ‘ 𝑦 ) → ( ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) → ( ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) ) |
198 |
197
|
com24 |
⊢ ( ( 𝑝 ↾ ( 0 ... 𝑦 ) ) ∈ ( RePart ‘ 𝑦 ) → ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) ) |
199 |
96 198
|
mpcom |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ) → ( ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) |
200 |
199
|
ex |
⊢ ( 𝑦 ∈ ℕ → ( 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) → ( ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) ) |
201 |
200
|
com24 |
⊢ ( 𝑦 ∈ ℕ → ( ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) → ( ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) ) |
202 |
201
|
imp |
⊢ ( ( 𝑦 ∈ ℕ ∧ ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) → ( ¬ ( 𝑝 ‘ 𝑦 ) ≤ 𝑋 → ( 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) |
203 |
95 202
|
pm2.61d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) → ( 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) → ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
204 |
46 203
|
ralrimi |
⊢ ( ( 𝑦 ∈ ℕ ∧ ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) → ∀ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |
205 |
204
|
ex |
⊢ ( 𝑦 ∈ ℕ → ( ∀ 𝑝 ∈ ( RePart ‘ 𝑦 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑦 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑦 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) → ∀ 𝑝 ∈ ( RePart ‘ ( 𝑦 + 1 ) ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ ( 𝑦 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
206 |
10 18 26 34 43 205
|
nnind |
⊢ ( 𝑀 ∈ ℕ → ∀ 𝑝 ∈ ( RePart ‘ 𝑀 ) ( 𝑋 ∈ ( ( 𝑝 ‘ 0 ) [,) ( 𝑝 ‘ 𝑀 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) 𝑋 ∈ ( ( 𝑝 ‘ 𝑖 ) [,) ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ) ) |