Proof of Theorem 1wlklnwwlkln2lem
Step | Hyp | Ref
| Expression |
1 | | eqid 2610 |
. . . 4
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
2 | 1 | wwlknbp 41044 |
. . 3
⊢ (𝑃 ∈ (𝑁 WWalkSN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑃 ∈ Word (Vtx‘𝐺))) |
3 | | iswwlksn 41041 |
. . . . . 6
⊢ (𝑁 ∈ ℕ0
→ (𝑃 ∈ (𝑁 WWalkSN 𝐺) ↔ (𝑃 ∈ (WWalkS‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1)))) |
4 | 3 | adantr 480 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
(𝑃 ∈ (𝑁 WWalkSN 𝐺) ↔ (𝑃 ∈ (WWalkS‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1)))) |
5 | | lencl 13179 |
. . . . . . . . . . . . . 14
⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (#‘𝑃) ∈
ℕ0) |
6 | 5 | nn0cnd 11230 |
. . . . . . . . . . . . 13
⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (#‘𝑃) ∈
ℂ) |
7 | 6 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
(#‘𝑃) ∈
ℂ) |
8 | | 1cnd 9935 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) → 1
∈ ℂ) |
9 | | nn0cn 11179 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℂ) |
10 | 9 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) → 𝑁 ∈
ℂ) |
11 | 7, 8, 10 | subadd2d 10290 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
(((#‘𝑃) − 1) =
𝑁 ↔ (𝑁 + 1) = (#‘𝑃))) |
12 | | eqcom 2617 |
. . . . . . . . . . 11
⊢ ((𝑁 + 1) = (#‘𝑃) ↔ (#‘𝑃) = (𝑁 + 1)) |
13 | 11, 12 | syl6rbb 276 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
((#‘𝑃) = (𝑁 + 1) ↔ ((#‘𝑃) − 1) = 𝑁)) |
14 | 13 | biimpcd 238 |
. . . . . . . . 9
⊢
((#‘𝑃) =
(𝑁 + 1) → ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
((#‘𝑃) − 1) =
𝑁)) |
15 | 14 | adantl 481 |
. . . . . . . 8
⊢ ((𝑃 ∈ (WWalkS‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1)) → ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → ((#‘𝑃) − 1) = 𝑁)) |
16 | 15 | impcom 445 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) ∧ (𝑃 ∈ (WWalkS‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1))) → ((#‘𝑃) − 1) = 𝑁) |
17 | | 1wlklnwwlkln2lem.1 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑃 ∈ (WWalkS‘𝐺) → ∃𝑓 𝑓(1Walks‘𝐺)𝑃)) |
18 | 17 | com12 32 |
. . . . . . . . . . . . 13
⊢ (𝑃 ∈ (WWalkS‘𝐺) → (𝜑 → ∃𝑓 𝑓(1Walks‘𝐺)𝑃)) |
19 | 18 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑃 ∈ (WWalkS‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1)) → (𝜑 → ∃𝑓 𝑓(1Walks‘𝐺)𝑃)) |
20 | 19 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) ∧ (𝑃 ∈ (WWalkS‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1))) → (𝜑 → ∃𝑓 𝑓(1Walks‘𝐺)𝑃)) |
21 | 20 | imp 444 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) ∧ (𝑃 ∈ (WWalkS‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1))) ∧ 𝜑) → ∃𝑓 𝑓(1Walks‘𝐺)𝑃) |
22 | | simpr 476 |
. . . . . . . . . . . . 13
⊢
(((((𝑁 ∈
ℕ0 ∧ 𝑃
∈ Word (Vtx‘𝐺))
∧ (𝑃 ∈
(WWalkS‘𝐺) ∧
(#‘𝑃) = (𝑁 + 1))) ∧ 𝜑) ∧ 𝑓(1Walks‘𝐺)𝑃) → 𝑓(1Walks‘𝐺)𝑃) |
23 | | 1wlklenvm1 40826 |
. . . . . . . . . . . . 13
⊢ (𝑓(1Walks‘𝐺)𝑃 → (#‘𝑓) = ((#‘𝑃) − 1)) |
24 | 22, 23 | jccir 560 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈
ℕ0 ∧ 𝑃
∈ Word (Vtx‘𝐺))
∧ (𝑃 ∈
(WWalkS‘𝐺) ∧
(#‘𝑃) = (𝑁 + 1))) ∧ 𝜑) ∧ 𝑓(1Walks‘𝐺)𝑃) → (𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = ((#‘𝑃) − 1))) |
25 | 24 | ex 449 |
. . . . . . . . . . 11
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) ∧ (𝑃 ∈ (WWalkS‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1))) ∧ 𝜑) → (𝑓(1Walks‘𝐺)𝑃 → (𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = ((#‘𝑃) − 1)))) |
26 | 25 | eximdv 1833 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) ∧ (𝑃 ∈ (WWalkS‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1))) ∧ 𝜑) → (∃𝑓 𝑓(1Walks‘𝐺)𝑃 → ∃𝑓(𝑓(1Walks‘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(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = ((#‘𝑃) − 1))) |
28 | | eqeq2 2621 |
. . . . . . . . . . 11
⊢
(((#‘𝑃)
− 1) = 𝑁 →
((#‘𝑓) =
((#‘𝑃) − 1)
↔ (#‘𝑓) = 𝑁)) |
29 | 28 | anbi2d 736 |
. . . . . . . . . 10
⊢
(((#‘𝑃)
− 1) = 𝑁 →
((𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = ((#‘𝑃) − 1)) ↔ (𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁))) |
30 | 29 | exbidv 1837 |
. . . . . . . . 9
⊢
(((#‘𝑃)
− 1) = 𝑁 →
(∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = ((#‘𝑃) − 1)) ↔ ∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁))) |
31 | 27, 30 | syl5ib 233 |
. . . . . . . 8
⊢
(((#‘𝑃)
− 1) = 𝑁 →
((((𝑁 ∈
ℕ0 ∧ 𝑃
∈ Word (Vtx‘𝐺))
∧ (𝑃 ∈
(WWalkS‘𝐺) ∧
(#‘𝑃) = (𝑁 + 1))) ∧ 𝜑) → ∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁))) |
32 | 31 | expd 451 |
. . . . . . 7
⊢
(((#‘𝑃)
− 1) = 𝑁 →
(((𝑁 ∈
ℕ0 ∧ 𝑃
∈ Word (Vtx‘𝐺))
∧ (𝑃 ∈
(WWalkS‘𝐺) ∧
(#‘𝑃) = (𝑁 + 1))) → (𝜑 → ∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁)))) |
33 | 16, 32 | mpcom 37 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) ∧ (𝑃 ∈ (WWalkS‘𝐺) ∧ (#‘𝑃) = (𝑁 + 1))) → (𝜑 → ∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁))) |
34 | 33 | ex 449 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
((𝑃 ∈
(WWalkS‘𝐺) ∧
(#‘𝑃) = (𝑁 + 1)) → (𝜑 → ∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁)))) |
35 | 4, 34 | sylbid 229 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
(𝑃 ∈ (𝑁 WWalkSN 𝐺) → (𝜑 → ∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁)))) |
36 | 35 | 3adant1 1072 |
. . 3
⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0
∧ 𝑃 ∈ Word
(Vtx‘𝐺)) →
(𝑃 ∈ (𝑁 WWalkSN 𝐺) → (𝜑 → ∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁)))) |
37 | 2, 36 | mpcom 37 |
. 2
⊢ (𝑃 ∈ (𝑁 WWalkSN 𝐺) → (𝜑 → ∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁))) |
38 | 37 | com12 32 |
1
⊢ (𝜑 → (𝑃 ∈ (𝑁 WWalkSN 𝐺) → ∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁))) |