Proof of Theorem wlklnwwlkln2lem
Step | Hyp | Ref
| Expression |
1 | | eqid 2622 |
. . . 4
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
2 | 1 | wwlknbp 26733 |
. . 3
⊢ (𝑃 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑃 ∈ Word (Vtx‘𝐺))) |
3 | | iswwlksn 26730 |
. . . . . 6
⊢ (𝑁 ∈ ℕ0
→ (𝑃 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑃 ∈ (WWalks‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1)))) |
4 | 3 | adantr 481 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
(𝑃 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑃 ∈ (WWalks‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1)))) |
5 | | lencl 13324 |
. . . . . . . . . . . . . 14
⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (#‘𝑃) ∈
ℕ0) |
6 | 5 | nn0cnd 11353 |
. . . . . . . . . . . . 13
⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (#‘𝑃) ∈
ℂ) |
7 | 6 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
(#‘𝑃) ∈
ℂ) |
8 | | 1cnd 10056 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) → 1
∈ ℂ) |
9 | | nn0cn 11302 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℂ) |
10 | 9 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) → 𝑁 ∈
ℂ) |
11 | 7, 8, 10 | subadd2d 10411 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
(((#‘𝑃) − 1) =
𝑁 ↔ (𝑁 + 1) = (#‘𝑃))) |
12 | | eqcom 2629 |
. . . . . . . . . . 11
⊢ ((𝑁 + 1) = (#‘𝑃) ↔ (#‘𝑃) = (𝑁 + 1)) |
13 | 11, 12 | syl6rbb 277 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
((#‘𝑃) = (𝑁 + 1) ↔ ((#‘𝑃) − 1) = 𝑁)) |
14 | 13 | biimpcd 239 |
. . . . . . . . 9
⊢
((#‘𝑃) =
(𝑁 + 1) → ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
((#‘𝑃) − 1) =
𝑁)) |
15 | 14 | adantl 482 |
. . . . . . . 8
⊢ ((𝑃 ∈ (WWalks‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1)) → ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → ((#‘𝑃) − 1) = 𝑁)) |
16 | 15 | impcom 446 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) ∧ (𝑃 ∈ (WWalks‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1))) → ((#‘𝑃) − 1) = 𝑁) |
17 | | wlklnwwlkln2lem.1 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃)) |
18 | 17 | com12 32 |
. . . . . . . . . . . . 13
⊢ (𝑃 ∈ (WWalks‘𝐺) → (𝜑 → ∃𝑓 𝑓(Walks‘𝐺)𝑃)) |
19 | 18 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝑃 ∈ (WWalks‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1)) → (𝜑 → ∃𝑓 𝑓(Walks‘𝐺)𝑃)) |
20 | 19 | adantl 482 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) ∧ (𝑃 ∈ (WWalks‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1))) → (𝜑 → ∃𝑓 𝑓(Walks‘𝐺)𝑃)) |
21 | 20 | imp 445 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) ∧ (𝑃 ∈ (WWalks‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1))) ∧ 𝜑) → ∃𝑓 𝑓(Walks‘𝐺)𝑃) |
22 | | simpr 477 |
. . . . . . . . . . . . 13
⊢
(((((𝑁 ∈
ℕ0 ∧ 𝑃
∈ Word (Vtx‘𝐺))
∧ (𝑃 ∈
(WWalks‘𝐺) ∧
(#‘𝑃) = (𝑁 + 1))) ∧ 𝜑) ∧ 𝑓(Walks‘𝐺)𝑃) → 𝑓(Walks‘𝐺)𝑃) |
23 | | wlklenvm1 26517 |
. . . . . . . . . . . . 13
⊢ (𝑓(Walks‘𝐺)𝑃 → (#‘𝑓) = ((#‘𝑃) − 1)) |
24 | 22, 23 | jccir 562 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈
ℕ0 ∧ 𝑃
∈ Word (Vtx‘𝐺))
∧ (𝑃 ∈
(WWalks‘𝐺) ∧
(#‘𝑃) = (𝑁 + 1))) ∧ 𝜑) ∧ 𝑓(Walks‘𝐺)𝑃) → (𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = ((#‘𝑃) − 1))) |
25 | 24 | ex 450 |
. . . . . . . . . . 11
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) ∧ (𝑃 ∈ (WWalks‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1))) ∧ 𝜑) → (𝑓(Walks‘𝐺)𝑃 → (𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = ((#‘𝑃) − 1)))) |
26 | 25 | eximdv 1846 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) ∧ (𝑃 ∈ (WWalks‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1))) ∧ 𝜑) → (∃𝑓 𝑓(Walks‘𝐺)𝑃 → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = ((#‘𝑃) − 1)))) |
27 | 21, 26 | mpd 15 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) ∧ (𝑃 ∈ (WWalks‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1))) ∧ 𝜑) → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = ((#‘𝑃) − 1))) |
28 | | eqeq2 2633 |
. . . . . . . . . . 11
⊢
(((#‘𝑃)
− 1) = 𝑁 →
((#‘𝑓) =
((#‘𝑃) − 1)
↔ (#‘𝑓) = 𝑁)) |
29 | 28 | anbi2d 740 |
. . . . . . . . . 10
⊢
(((#‘𝑃)
− 1) = 𝑁 →
((𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = ((#‘𝑃) − 1)) ↔ (𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁))) |
30 | 29 | exbidv 1850 |
. . . . . . . . 9
⊢
(((#‘𝑃)
− 1) = 𝑁 →
(∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = ((#‘𝑃) − 1)) ↔ ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁))) |
31 | 27, 30 | syl5ib 234 |
. . . . . . . 8
⊢
(((#‘𝑃)
− 1) = 𝑁 →
((((𝑁 ∈
ℕ0 ∧ 𝑃
∈ Word (Vtx‘𝐺))
∧ (𝑃 ∈
(WWalks‘𝐺) ∧
(#‘𝑃) = (𝑁 + 1))) ∧ 𝜑) → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁))) |
32 | 31 | expd 452 |
. . . . . . 7
⊢
(((#‘𝑃)
− 1) = 𝑁 →
(((𝑁 ∈
ℕ0 ∧ 𝑃
∈ Word (Vtx‘𝐺))
∧ (𝑃 ∈
(WWalks‘𝐺) ∧
(#‘𝑃) = (𝑁 + 1))) → (𝜑 → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁)))) |
33 | 16, 32 | mpcom 38 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) ∧ (𝑃 ∈ (WWalks‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1))) → (𝜑 → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁))) |
34 | 33 | ex 450 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
((𝑃 ∈
(WWalks‘𝐺) ∧
(#‘𝑃) = (𝑁 + 1)) → (𝜑 → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁)))) |
35 | 4, 34 | sylbid 230 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
(𝑃 ∈ (𝑁 WWalksN 𝐺) → (𝜑 → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁)))) |
36 | 35 | 3adant1 1079 |
. . 3
⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
(𝑃 ∈ (𝑁 WWalksN 𝐺) → (𝜑 → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁)))) |
37 | 2, 36 | mpcom 38 |
. 2
⊢ (𝑃 ∈ (𝑁 WWalksN 𝐺) → (𝜑 → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁))) |
38 | 37 | com12 32 |
1
⊢ (𝜑 → (𝑃 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁))) |