Proof of Theorem clwwnisshclwwsn
Step | Hyp | Ref
| Expression |
1 | | eqid 2622 |
. . . 4
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
2 | 1 | clwwlknbp0 26884 |
. . 3
⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁))) |
3 | | clwwlkclwwlkn 26891 |
. . . . . . 7
⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ (ClWWalks‘𝐺)) |
4 | 3 | 3ad2ant2 1083 |
. . . . . 6
⊢ ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → 𝑊 ∈ (ClWWalks‘𝐺)) |
5 | | oveq2 6658 |
. . . . . . . . . . . 12
⊢ (𝑁 = (#‘𝑊) → (0...𝑁) = (0...(#‘𝑊))) |
6 | 5 | eleq2d 2687 |
. . . . . . . . . . 11
⊢ (𝑁 = (#‘𝑊) → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...(#‘𝑊)))) |
7 | 6 | eqcoms 2630 |
. . . . . . . . . 10
⊢
((#‘𝑊) = 𝑁 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...(#‘𝑊)))) |
8 | 7 | ad2antll 765 |
. . . . . . . . 9
⊢ (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...(#‘𝑊)))) |
9 | 8 | biimpd 219 |
. . . . . . . 8
⊢ (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) → (𝑀 ∈ (0...𝑁) → 𝑀 ∈ (0...(#‘𝑊)))) |
10 | 9 | a1d 25 |
. . . . . . 7
⊢ (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑀 ∈ (0...𝑁) → 𝑀 ∈ (0...(#‘𝑊))))) |
11 | 10 | 3imp 1256 |
. . . . . 6
⊢ ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → 𝑀 ∈ (0...(#‘𝑊))) |
12 | | clwwisshclwwsn 26929 |
. . . . . 6
⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑀 ∈ (0...(#‘𝑊))) → (𝑊 cyclShift 𝑀) ∈ (ClWWalks‘𝐺)) |
13 | 4, 11, 12 | syl2anc 693 |
. . . . 5
⊢ ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑀) ∈ (ClWWalks‘𝐺)) |
14 | | elfzelz 12342 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℤ) |
15 | | cshwlen 13545 |
. . . . . . . . . . 11
⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑀 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊)) |
16 | 14, 15 | sylan2 491 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊)) |
17 | 16 | ex 450 |
. . . . . . . . 9
⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))) |
18 | 17 | ad2antrl 764 |
. . . . . . . 8
⊢ (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) → (𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))) |
19 | 18 | a1d 25 |
. . . . . . 7
⊢ (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊)))) |
20 | 19 | 3imp 1256 |
. . . . . 6
⊢ ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊)) |
21 | | simp1rr 1127 |
. . . . . 6
⊢ ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (#‘𝑊) = 𝑁) |
22 | 20, 21 | eqtrd 2656 |
. . . . 5
⊢ ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁) |
23 | | simp1lr 1125 |
. . . . . 6
⊢ ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ ℕ) |
24 | | isclwwlksn 26882 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → ((𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 cyclShift 𝑀) ∈ (ClWWalks‘𝐺) ∧ (#‘(𝑊 cyclShift 𝑀)) = 𝑁))) |
25 | 23, 24 | syl 17 |
. . . . 5
⊢ ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 cyclShift 𝑀) ∈ (ClWWalks‘𝐺) ∧ (#‘(𝑊 cyclShift 𝑀)) = 𝑁))) |
26 | 13, 22, 25 | mpbir2and 957 |
. . . 4
⊢ ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺)) |
27 | 26 | 3exp 1264 |
. . 3
⊢ (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑀 ∈ (0...𝑁) → (𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺)))) |
28 | 2, 27 | mpcom 38 |
. 2
⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑀 ∈ (0...𝑁) → (𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺))) |
29 | 28 | imp 445 |
1
⊢ ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺)) |