Step |
Hyp |
Ref |
Expression |
1 |
|
wlkd.p |
⊢ ( 𝜑 → 𝑃 ∈ Word V ) |
2 |
|
wlkd.f |
⊢ ( 𝜑 → 𝐹 ∈ Word V ) |
3 |
|
wlkd.l |
⊢ ( 𝜑 → ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) ) |
4 |
|
wlkd.e |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
5 |
|
fzo0end |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
6 |
|
fveq2 |
⊢ ( 𝑘 = ( ( ♯ ‘ 𝐹 ) − 1 ) → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) |
7 |
|
fvoveq1 |
⊢ ( 𝑘 = ( ( ♯ ‘ 𝐹 ) − 1 ) → ( 𝑃 ‘ ( 𝑘 + 1 ) ) = ( 𝑃 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) ) |
8 |
6 7
|
preq12d |
⊢ ( 𝑘 = ( ( ♯ ‘ 𝐹 ) − 1 ) → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } = { ( 𝑃 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) , ( 𝑃 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) } ) |
9 |
|
2fveq3 |
⊢ ( 𝑘 = ( ( ♯ ‘ 𝐹 ) − 1 ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) |
10 |
8 9
|
sseq12d |
⊢ ( 𝑘 = ( ( ♯ ‘ 𝐹 ) − 1 ) → ( { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ { ( 𝑃 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) , ( 𝑃 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ) |
11 |
10
|
rspcv |
⊢ ( ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) → { ( 𝑃 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) , ( 𝑃 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ) |
12 |
5 11
|
syl |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) → { ( 𝑃 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) , ( 𝑃 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ) |
13 |
|
fvex |
⊢ ( 𝑃 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ∈ V |
14 |
|
fvex |
⊢ ( 𝑃 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) ∈ V |
15 |
13 14
|
prss |
⊢ ( ( ( 𝑃 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ∧ ( 𝑃 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ↔ { ( 𝑃 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) , ( 𝑃 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) |
16 |
|
nncn |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ♯ ‘ 𝐹 ) ∈ ℂ ) |
17 |
|
npcan1 |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℂ → ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) = ( ♯ ‘ 𝐹 ) ) |
18 |
16 17
|
syl |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) = ( ♯ ‘ 𝐹 ) ) |
19 |
18
|
fveq2d |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( 𝑃 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) |
20 |
19
|
eleq1d |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ( 𝑃 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ↔ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ) |
21 |
20
|
biimpd |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ( 𝑃 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ) |
22 |
21
|
adantld |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ( ( 𝑃 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ∧ ( 𝑃 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) → ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ) |
23 |
15 22
|
syl5bir |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( { ( 𝑃 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) , ( 𝑃 ‘ ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ) |
24 |
12 23
|
syld |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) → ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ) |
25 |
4 24
|
syl5com |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ) |
26 |
|
fvex |
⊢ ( 𝑃 ‘ 𝑘 ) ∈ V |
27 |
|
fvex |
⊢ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ V |
28 |
26 27
|
prss |
⊢ ( ( ( 𝑃 ‘ 𝑘 ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∧ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
29 |
|
simpl |
⊢ ( ( ( 𝑃 ‘ 𝑘 ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∧ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝑃 ‘ 𝑘 ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
30 |
28 29
|
sylbir |
⊢ ( { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) → ( 𝑃 ‘ 𝑘 ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
31 |
30
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) → ( 𝑃 ‘ 𝑘 ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
32 |
31
|
ralimdva |
⊢ ( 𝜑 → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
33 |
4 32
|
mpd |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
34 |
25 33
|
jca |
⊢ ( 𝜑 → ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ∈ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |