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Theorem clwwisshclww 26335
 Description: Cyclically shifting a closed walk as word results in a closed walk as word (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Mar-2018.) (Revised by Alexander van der Vekens, 10-Jun-2018.)
Assertion
Ref Expression
clwwisshclww ((𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑁 ∈ (0..^(#‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸))

Proof of Theorem clwwisshclww
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwwlkprop 26298 . . . . 5 (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉))
2 cshw0 13391 . . . . . . . 8 (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)
323ad2ant3 1077 . . . . . . 7 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) → (𝑊 cyclShift 0) = 𝑊)
43eleq1d 2672 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) → ((𝑊 cyclShift 0) ∈ (𝑉 ClWWalks 𝐸) ↔ 𝑊 ∈ (𝑉 ClWWalks 𝐸)))
54biimprd 237 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) → (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑊 cyclShift 0) ∈ (𝑉 ClWWalks 𝐸)))
61, 5mpcom 37 . . . 4 (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑊 cyclShift 0) ∈ (𝑉 ClWWalks 𝐸))
7 oveq2 6557 . . . . 5 (𝑁 = 0 → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift 0))
87eleq1d 2672 . . . 4 (𝑁 = 0 → ((𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑊 cyclShift 0) ∈ (𝑉 ClWWalks 𝐸)))
96, 8syl5ibrcom 236 . . 3 (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑁 = 0 → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸)))
109adantr 480 . 2 ((𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑁 ∈ (0..^(#‘𝑊))) → (𝑁 = 0 → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸)))
11 fzo1fzo0n0 12386 . . . . . 6 (𝑁 ∈ (1..^(#‘𝑊)) ↔ (𝑁 ∈ (0..^(#‘𝑊)) ∧ 𝑁 ≠ 0))
12 isclwwlk 26296 . . . . . . . . . 10 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑊 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)))
13123adant3 1074 . . . . . . . . 9 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) → (𝑊 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)))
14 simp3 1056 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) → 𝑊 ∈ Word 𝑉)
1514adantr 480 . . . . . . . . . . . . . . 15 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → 𝑊 ∈ Word 𝑉)
16 cshwcl 13395 . . . . . . . . . . . . . . 15 (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑁) ∈ Word 𝑉)
1715, 16syl 17 . . . . . . . . . . . . . 14 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ Word 𝑉)
1817adantr 480 . . . . . . . . . . . . 13 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) ∧ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)) → (𝑊 cyclShift 𝑁) ∈ Word 𝑉)
1914anim1i 590 . . . . . . . . . . . . . 14 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → (𝑊 ∈ Word 𝑉𝑁 ∈ (1..^(#‘𝑊))))
20 3simpc 1053 . . . . . . . . . . . . . 14 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸))
21 clwwisshclwwlem 26334 . . . . . . . . . . . . . . 15 ((𝑊 ∈ Word 𝑉𝑁 ∈ (1..^(#‘𝑊))) → ((∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) → ∀𝑗 ∈ (0..^((#‘(𝑊 cyclShift 𝑁)) − 1)){((𝑊 cyclShift 𝑁)‘𝑗), ((𝑊 cyclShift 𝑁)‘(𝑗 + 1))} ∈ ran 𝐸))
2221imp 444 . . . . . . . . . . . . . 14 (((𝑊 ∈ Word 𝑉𝑁 ∈ (1..^(#‘𝑊))) ∧ (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)) → ∀𝑗 ∈ (0..^((#‘(𝑊 cyclShift 𝑁)) − 1)){((𝑊 cyclShift 𝑁)‘𝑗), ((𝑊 cyclShift 𝑁)‘(𝑗 + 1))} ∈ ran 𝐸)
2319, 20, 22syl2an 493 . . . . . . . . . . . . 13 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) ∧ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)) → ∀𝑗 ∈ (0..^((#‘(𝑊 cyclShift 𝑁)) − 1)){((𝑊 cyclShift 𝑁)‘𝑗), ((𝑊 cyclShift 𝑁)‘(𝑗 + 1))} ∈ ran 𝐸)
24 elfzofz 12354 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (1..^(#‘𝑊)) → 𝑁 ∈ (1...(#‘𝑊)))
25 lswcshw 13412 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word 𝑉𝑁 ∈ (1...(#‘𝑊))) → ( lastS ‘(𝑊 cyclShift 𝑁)) = (𝑊‘(𝑁 − 1)))
2624, 25sylan2 490 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ Word 𝑉𝑁 ∈ (1..^(#‘𝑊))) → ( lastS ‘(𝑊 cyclShift 𝑁)) = (𝑊‘(𝑁 − 1)))
27 fzo0ss1 12367 . . . . . . . . . . . . . . . . . . 19 (1..^(#‘𝑊)) ⊆ (0..^(#‘𝑊))
2827sseli 3564 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (1..^(#‘𝑊)) → 𝑁 ∈ (0..^(#‘𝑊)))
29 cshwidx0 13403 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word 𝑉𝑁 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift 𝑁)‘0) = (𝑊𝑁))
3028, 29sylan2 490 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ Word 𝑉𝑁 ∈ (1..^(#‘𝑊))) → ((𝑊 cyclShift 𝑁)‘0) = (𝑊𝑁))
3126, 30preq12d 4220 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ Word 𝑉𝑁 ∈ (1..^(#‘𝑊))) → {( lastS ‘(𝑊 cyclShift 𝑁)), ((𝑊 cyclShift 𝑁)‘0)} = {(𝑊‘(𝑁 − 1)), (𝑊𝑁)})
32313ad2antl3 1218 . . . . . . . . . . . . . . 15 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → {( lastS ‘(𝑊 cyclShift 𝑁)), ((𝑊 cyclShift 𝑁)‘0)} = {(𝑊‘(𝑁 − 1)), (𝑊𝑁)})
3332adantr 480 . . . . . . . . . . . . . 14 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) ∧ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)) → {( lastS ‘(𝑊 cyclShift 𝑁)), ((𝑊 cyclShift 𝑁)‘0)} = {(𝑊‘(𝑁 − 1)), (𝑊𝑁)})
34 elfzo1 12385 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (1..^(#‘𝑊)) ↔ (𝑁 ∈ ℕ ∧ (#‘𝑊) ∈ ℕ ∧ 𝑁 < (#‘𝑊)))
35 nnz 11276 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
36 nnz 11276 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℤ)
3735, 36anim12i 588 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ ∧ (#‘𝑊) ∈ ℕ) → (𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℤ))
38373adant3 1074 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ ∧ (#‘𝑊) ∈ ℕ ∧ 𝑁 < (#‘𝑊)) → (𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℤ))
3934, 38sylbi 206 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (1..^(#‘𝑊)) → (𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℤ))
40 elfzom1b 12433 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℤ) → (𝑁 ∈ (1..^(#‘𝑊)) ↔ (𝑁 − 1) ∈ (0..^((#‘𝑊) − 1))))
4139, 40syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (1..^(#‘𝑊)) → (𝑁 ∈ (1..^(#‘𝑊)) ↔ (𝑁 − 1) ∈ (0..^((#‘𝑊) − 1))))
4241ibi 255 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (1..^(#‘𝑊)) → (𝑁 − 1) ∈ (0..^((#‘𝑊) − 1)))
4342adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → (𝑁 − 1) ∈ (0..^((#‘𝑊) − 1)))
44 fveq2 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (𝑁 − 1) → (𝑊𝑖) = (𝑊‘(𝑁 − 1)))
4544adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) ∧ 𝑖 = (𝑁 − 1)) → (𝑊𝑖) = (𝑊‘(𝑁 − 1)))
46 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = (𝑁 − 1) → (𝑖 + 1) = ((𝑁 − 1) + 1))
4746fveq2d 6107 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (𝑁 − 1) → (𝑊‘(𝑖 + 1)) = (𝑊‘((𝑁 − 1) + 1)))
48 elfzoelz 12339 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ (1..^(#‘𝑊)) → 𝑁 ∈ ℤ)
4948zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ (1..^(#‘𝑊)) → 𝑁 ∈ ℂ)
5049adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → 𝑁 ∈ ℂ)
51 1cnd 9935 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → 1 ∈ ℂ)
5250, 51npcand 10275 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → ((𝑁 − 1) + 1) = 𝑁)
5352fveq2d 6107 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → (𝑊‘((𝑁 − 1) + 1)) = (𝑊𝑁))
5447, 53sylan9eqr 2666 . . . . . . . . . . . . . . . . . . . 20 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) ∧ 𝑖 = (𝑁 − 1)) → (𝑊‘(𝑖 + 1)) = (𝑊𝑁))
5545, 54preq12d 4220 . . . . . . . . . . . . . . . . . . 19 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) ∧ 𝑖 = (𝑁 − 1)) → {(𝑊𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊‘(𝑁 − 1)), (𝑊𝑁)})
5655eleq1d 2672 . . . . . . . . . . . . . . . . . 18 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) ∧ 𝑖 = (𝑁 − 1)) → ({(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑊‘(𝑁 − 1)), (𝑊𝑁)} ∈ ran 𝐸))
5743, 56rspcdv 3285 . . . . . . . . . . . . . . . . 17 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → {(𝑊‘(𝑁 − 1)), (𝑊𝑁)} ∈ ran 𝐸))
5857com12 32 . . . . . . . . . . . . . . . 16 (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → {(𝑊‘(𝑁 − 1)), (𝑊𝑁)} ∈ ran 𝐸))
59583ad2ant2 1076 . . . . . . . . . . . . . . 15 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) → (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → {(𝑊‘(𝑁 − 1)), (𝑊𝑁)} ∈ ran 𝐸))
6059impcom 445 . . . . . . . . . . . . . 14 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) ∧ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)) → {(𝑊‘(𝑁 − 1)), (𝑊𝑁)} ∈ ran 𝐸)
6133, 60eqeltrd 2688 . . . . . . . . . . . . 13 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) ∧ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)) → {( lastS ‘(𝑊 cyclShift 𝑁)), ((𝑊 cyclShift 𝑁)‘0)} ∈ ran 𝐸)
6218, 23, 613jca 1235 . . . . . . . . . . . 12 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ 𝑁 ∈ (1..^(#‘𝑊))) ∧ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)) → ((𝑊 cyclShift 𝑁) ∈ Word 𝑉 ∧ ∀𝑗 ∈ (0..^((#‘(𝑊 cyclShift 𝑁)) − 1)){((𝑊 cyclShift 𝑁)‘𝑗), ((𝑊 cyclShift 𝑁)‘(𝑗 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑊 cyclShift 𝑁)), ((𝑊 cyclShift 𝑁)‘0)} ∈ ran 𝐸))
6362an32s 842 . . . . . . . . . . 11 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → ((𝑊 cyclShift 𝑁) ∈ Word 𝑉 ∧ ∀𝑗 ∈ (0..^((#‘(𝑊 cyclShift 𝑁)) − 1)){((𝑊 cyclShift 𝑁)‘𝑗), ((𝑊 cyclShift 𝑁)‘(𝑗 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑊 cyclShift 𝑁)), ((𝑊 cyclShift 𝑁)‘0)} ∈ ran 𝐸))
64 3simpa 1051 . . . . . . . . . . . . . 14 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
6564adantr 480 . . . . . . . . . . . . 13 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
6665adantr 480 . . . . . . . . . . . 12 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
67 isclwwlk 26296 . . . . . . . . . . . 12 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸) ↔ ((𝑊 cyclShift 𝑁) ∈ Word 𝑉 ∧ ∀𝑗 ∈ (0..^((#‘(𝑊 cyclShift 𝑁)) − 1)){((𝑊 cyclShift 𝑁)‘𝑗), ((𝑊 cyclShift 𝑁)‘(𝑗 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑊 cyclShift 𝑁)), ((𝑊 cyclShift 𝑁)‘0)} ∈ ran 𝐸)))
6866, 67syl 17 . . . . . . . . . . 11 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → ((𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸) ↔ ((𝑊 cyclShift 𝑁) ∈ Word 𝑉 ∧ ∀𝑗 ∈ (0..^((#‘(𝑊 cyclShift 𝑁)) − 1)){((𝑊 cyclShift 𝑁)‘𝑗), ((𝑊 cyclShift 𝑁)‘(𝑗 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑊 cyclShift 𝑁)), ((𝑊 cyclShift 𝑁)‘0)} ∈ ran 𝐸)))
6963, 68mpbird 246 . . . . . . . . . 10 ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)) ∧ 𝑁 ∈ (1..^(#‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸))
7069exp31 628 . . . . . . . . 9 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) → ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) → (𝑁 ∈ (1..^(#‘𝑊)) → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸))))
7113, 70sylbid 229 . . . . . . . 8 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) → (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑁 ∈ (1..^(#‘𝑊)) → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸))))
721, 71mpcom 37 . . . . . . 7 (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑁 ∈ (1..^(#‘𝑊)) → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸)))
7372com12 32 . . . . . 6 (𝑁 ∈ (1..^(#‘𝑊)) → (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸)))
7411, 73sylbir 224 . . . . 5 ((𝑁 ∈ (0..^(#‘𝑊)) ∧ 𝑁 ≠ 0) → (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸)))
7574expcom 450 . . . 4 (𝑁 ≠ 0 → (𝑁 ∈ (0..^(#‘𝑊)) → (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸))))
7675com13 86 . . 3 (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑁 ∈ (0..^(#‘𝑊)) → (𝑁 ≠ 0 → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸))))
7776imp 444 . 2 ((𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑁 ∈ (0..^(#‘𝑊))) → (𝑁 ≠ 0 → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸)))
7810, 77pm2.61dne 2868 1 ((𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑁 ∈ (0..^(#‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  {cpr 4127   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   − cmin 10145  ℕcn 10897  ℤcz 11254  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   cyclShift ccsh 13385   ClWWalks cclwwlk 26276 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-substr 13158  df-csh 13386  df-clwwlk 26279 This theorem is referenced by:  clwwisshclwwn  26336
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