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Theorem clwwlkinwwlk 29293

Description: If the initial vertex of a walk occurs another time in the walk, the walk starts with a closed walk. Since the walk is expressed as a word over vertices, the closed walk can be expressed as a subword of this word. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 23-Jan-2022.) (Revised by AV, 30-Oct-2022.)

**Assertion**
Ref	Expression
clwwlkinwwlk	⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) ∧ 𝑊 ∈ (𝑀 WWalksN 𝐺) ∧ (𝑊‘𝑁) = (𝑊‘0)) → (𝑊 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺))

Proof of Theorem clwwlkinwwlk Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2733	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2733	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	wwlknp 29097	. 2 ⊢ (𝑊 ∈ (𝑀 WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
4		pfxcl 14627	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺))
5	4	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) → (𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺))
6	5	adantr 482	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺))
7		simpll 766	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑊 ∈ Word (Vtx‘𝐺))
8		simprl 770	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ ℕ)
9		eluz2 12828	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ_≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ≤ 𝑀))
10		zre 12562	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
11		zre 12562	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
12		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ≤ 𝑀 → 𝑁 ≤ 𝑀)
13	10, 11, 12	3anim123i 1152	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ≤ 𝑀) → (𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ≤ 𝑀))
14	9, 13	sylbi 216	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ_≥‘𝑁) → (𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ≤ 𝑀))
15		letrp1 12058	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ≤ 𝑀) → 𝑁 ≤ (𝑀 + 1))
16	14, 15	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (ℤ_≥‘𝑁) → 𝑁 ≤ (𝑀 + 1))
17	16	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ≤ (𝑀 + 1))
18	17	adantl 483	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ≤ (𝑀 + 1))
19		breq2 5153	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (𝑁 ≤ (♯‘𝑊) ↔ 𝑁 ≤ (𝑀 + 1)))
20	19	ad2antlr 726	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑁 ≤ (♯‘𝑊) ↔ 𝑁 ≤ (𝑀 + 1)))
21	18, 20	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ≤ (♯‘𝑊))
22		pfxn0 14636	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ ∧ 𝑁 ≤ (♯‘𝑊)) → (𝑊 prefix 𝑁) ≠ ∅)
23	7, 8, 21, 22	syl3anc 1372	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑊 prefix 𝑁) ≠ ∅)
24	6, 23	jca 513	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → ((𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ (𝑊 prefix 𝑁) ≠ ∅))
25	24	3adantl3 1169	. . . . . . . 8 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → ((𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ (𝑊 prefix 𝑁) ≠ ∅))
26	25	adantr 482	. . . . . . 7 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ (𝑊‘𝑁) = (𝑊‘0)) → ((𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ (𝑊 prefix 𝑁) ≠ ∅))
27		nnz 12579	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
28		1nn0 12488	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ₀
29		eluzmn 12829	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
30	27, 28, 29	sylancl 587	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
31		uzss 12845	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘(𝑁 − 1)))
32	30, 31	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘(𝑁 − 1)))
33	32	sselda 3983	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑀 ∈ (ℤ_≥‘(𝑁 − 1)))
34		fzoss2 13660	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ_≥‘(𝑁 − 1)) → (0..^(𝑁 − 1)) ⊆ (0..^𝑀))
35	33, 34	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → (0..^(𝑁 − 1)) ⊆ (0..^𝑀))
36	35	3ad2ant3 1136	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (0..^(𝑁 − 1)) ⊆ (0..^𝑀))
37		ssralv 4051	. . . . . . . . . . . . 13 ⊢ ((0..^(𝑁 − 1)) ⊆ (0..^𝑀) → (∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
38	36, 37	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
39	38	3exp 1120	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((♯‘𝑊) = (𝑀 + 1) → ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → (∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))))
40	39	com34 91	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((♯‘𝑊) = (𝑀 + 1) → (∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))))
41	40	3imp1 1348	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺))
42	41	adantr 482	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ (𝑊‘𝑁) = (𝑊‘0)) → ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺))
43		nnnn0 12479	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
44		elnn0uz 12867	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (ℤ_≥‘0))
45	43, 44	sylib 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘0))
46		eluzfz 13496	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (ℤ_≥‘0) ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ∈ (0...𝑀))
47	45, 46	sylan 581	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ∈ (0...𝑀))
48		fzelp1 13553	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (0...𝑀) → 𝑁 ∈ (0...(𝑀 + 1)))
49	47, 48	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ∈ (0...(𝑀 + 1)))
50	49	adantl 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (0...(𝑀 + 1)))
51		oveq2 7417	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (0...(♯‘𝑊)) = (0...(𝑀 + 1)))
52	51	eleq2d 2820	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (𝑁 ∈ (0...(♯‘𝑊)) ↔ 𝑁 ∈ (0...(𝑀 + 1))))
53	52	ad2antlr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑁 ∈ (0...(♯‘𝑊)) ↔ 𝑁 ∈ (0...(𝑀 + 1))))
54	50, 53	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (0...(♯‘𝑊)))
55		pfxlen 14633	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝑁)) = 𝑁)
56	7, 54, 55	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (♯‘(𝑊 prefix 𝑁)) = 𝑁)
57	56	oveq1d 7424	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → ((♯‘(𝑊 prefix 𝑁)) − 1) = (𝑁 − 1))
58	57	oveq2d 7425	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (0..^((♯‘(𝑊 prefix 𝑁)) − 1)) = (0..^(𝑁 − 1)))
59	58	raleqdv 3326	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (∀𝑖 ∈ (0..^((♯‘(𝑊 prefix 𝑁)) − 1)){((𝑊 prefix 𝑁)‘𝑖), ((𝑊 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑊 prefix 𝑁)‘𝑖), ((𝑊 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
60	7	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑊 ∈ Word (Vtx‘𝐺))
61	54	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑁 ∈ (0...(♯‘𝑊)))
62	30	ad2antrl 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
63		fzoss2 13660	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
64	62, 63	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
65	64	sselda 3983	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑖 ∈ (0..^𝑁))
66		pfxfv 14632	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑊 prefix 𝑁)‘𝑖) = (𝑊‘𝑖))
67	60, 61, 65, 66	syl3anc 1372	. . . . . . . . . . . . . 14 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → ((𝑊 prefix 𝑁)‘𝑖) = (𝑊‘𝑖))
68	27	ad2antrl 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ ℤ)
69		elfzom1elp1fzo 13699	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑖 + 1) ∈ (0..^𝑁))
70	68, 69	sylan 581	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑖 + 1) ∈ (0..^𝑁))
71		pfxfv 14632	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ (𝑖 + 1) ∈ (0..^𝑁)) → ((𝑊 prefix 𝑁)‘(𝑖 + 1)) = (𝑊‘(𝑖 + 1)))
72	60, 61, 70, 71	syl3anc 1372	. . . . . . . . . . . . . 14 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → ((𝑊 prefix 𝑁)‘(𝑖 + 1)) = (𝑊‘(𝑖 + 1)))
73	67, 72	preq12d 4746	. . . . . . . . . . . . 13 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → {((𝑊 prefix 𝑁)‘𝑖), ((𝑊 prefix 𝑁)‘(𝑖 + 1))} = {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))})
74	73	eleq1d 2819	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → ({((𝑊 prefix 𝑁)‘𝑖), ((𝑊 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
75	74	ralbidva 3176	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑊 prefix 𝑁)‘𝑖), ((𝑊 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
76	59, 75	bitrd 279	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (∀𝑖 ∈ (0..^((♯‘(𝑊 prefix 𝑁)) − 1)){((𝑊 prefix 𝑁)‘𝑖), ((𝑊 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
77	76	3adantl3 1169	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (∀𝑖 ∈ (0..^((♯‘(𝑊 prefix 𝑁)) − 1)){((𝑊 prefix 𝑁)‘𝑖), ((𝑊 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
78	77	adantr 482	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ (𝑊‘𝑁) = (𝑊‘0)) → (∀𝑖 ∈ (0..^((♯‘(𝑊 prefix 𝑁)) − 1)){((𝑊 prefix 𝑁)‘𝑖), ((𝑊 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
79	42, 78	mpbird 257	. . . . . . 7 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ (𝑊‘𝑁) = (𝑊‘0)) → ∀𝑖 ∈ (0..^((♯‘(𝑊 prefix 𝑁)) − 1)){((𝑊 prefix 𝑁)‘𝑖), ((𝑊 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
80		elfz1uz 13571	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ∈ (1...𝑀))
81		fzelp1 13553	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (1...𝑀) → 𝑁 ∈ (1...(𝑀 + 1)))
82	80, 81	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ∈ (1...(𝑀 + 1)))
83	82	adantl 483	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (1...(𝑀 + 1)))
84		oveq2 7417	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (1...(♯‘𝑊)) = (1...(𝑀 + 1)))
85	84	eleq2d 2820	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (𝑁 ∈ (1...(♯‘𝑊)) ↔ 𝑁 ∈ (1...(𝑀 + 1))))
86	85	ad2antlr 726	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑁 ∈ (1...(♯‘𝑊)) ↔ 𝑁 ∈ (1...(𝑀 + 1))))
87	83, 86	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (1...(♯‘𝑊)))
88		pfxfvlsw 14645	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑊))) → (lastS‘(𝑊 prefix 𝑁)) = (𝑊‘(𝑁 − 1)))
89		pfxfv0 14642	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((𝑊 prefix 𝑁)‘0) = (𝑊‘0))
90	88, 89	preq12d 4746	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑊))) → {(lastS‘(𝑊 prefix 𝑁)), ((𝑊 prefix 𝑁)‘0)} = {(𝑊‘(𝑁 − 1)), (𝑊‘0)})
91	7, 87, 90	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → {(lastS‘(𝑊 prefix 𝑁)), ((𝑊 prefix 𝑁)‘0)} = {(𝑊‘(𝑁 − 1)), (𝑊‘0)})
92	91	3adantl3 1169	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → {(lastS‘(𝑊 prefix 𝑁)), ((𝑊 prefix 𝑁)‘0)} = {(𝑊‘(𝑁 − 1)), (𝑊‘0)})
93	92	adantr 482	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ (𝑊‘𝑁) = (𝑊‘0)) → {(lastS‘(𝑊 prefix 𝑁)), ((𝑊 prefix 𝑁)‘0)} = {(𝑊‘(𝑁 − 1)), (𝑊‘0)})
94		fz1fzo0m1 13680	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (1...𝑀) → (𝑁 − 1) ∈ (0..^𝑀))
95	80, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → (𝑁 − 1) ∈ (0..^𝑀))
96	95	3ad2ant3 1136	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑁 − 1) ∈ (0..^𝑀))
97		simpr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → 𝑖 = (𝑁 − 1))
98	97	fveq2d 6896	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → (𝑊‘𝑖) = (𝑊‘(𝑁 − 1)))
99		oveq1 7416	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = (𝑁 − 1) → (𝑖 + 1) = ((𝑁 − 1) + 1))
100		nncn 12220	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
101		npcan1 11639	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
102	100, 101	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁)
103	99, 102	sylan9eqr 2795	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → (𝑖 + 1) = 𝑁)
104	103	fveq2d 6896	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → (𝑊‘(𝑖 + 1)) = (𝑊‘𝑁))
105	98, 104	preq12d 4746	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)})
106	105	eleq1d 2819	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺)))
107	106	ex 414	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑖 = (𝑁 − 1) → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺))))
108	107	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → (𝑖 = (𝑁 − 1) → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺))))
109	108	3ad2ant3 1136	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑖 = (𝑁 − 1) → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺))))
110	109	imp 408	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 = (𝑁 − 1)) → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺)))
111	96, 110	rspcdv 3605	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺)))
112	111	3exp 1120	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((♯‘𝑊) = (𝑀 + 1) → ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → (∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺)))))
113	112	com34 91	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((♯‘𝑊) = (𝑀 + 1) → (∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺)))))
114	113	3imp1 1348	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺))
115	114	adantr 482	. . . . . . . . 9 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ (𝑊‘𝑁) = (𝑊‘0)) → {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺))
116		preq2 4739	. . . . . . . . . . 11 ⊢ ((𝑊‘𝑁) = (𝑊‘0) → {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} = {(𝑊‘(𝑁 − 1)), (𝑊‘0)})
117	116	eleq1d 2819	. . . . . . . . . 10 ⊢ ((𝑊‘𝑁) = (𝑊‘0) → ({(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘0)} ∈ (Edg‘𝐺)))
118	117	adantl 483	. . . . . . . . 9 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ (𝑊‘𝑁) = (𝑊‘0)) → ({(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘0)} ∈ (Edg‘𝐺)))
119	115, 118	mpbid 231	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ (𝑊‘𝑁) = (𝑊‘0)) → {(𝑊‘(𝑁 − 1)), (𝑊‘0)} ∈ (Edg‘𝐺))
120	93, 119	eqeltrd 2834	. . . . . . 7 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ (𝑊‘𝑁) = (𝑊‘0)) → {(lastS‘(𝑊 prefix 𝑁)), ((𝑊 prefix 𝑁)‘0)} ∈ (Edg‘𝐺))
121	26, 79, 120	3jca 1129	. . . . . 6 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ (𝑊‘𝑁) = (𝑊‘0)) → (((𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ (𝑊 prefix 𝑁) ≠ ∅) ∧ ∀𝑖 ∈ (0..^((♯‘(𝑊 prefix 𝑁)) − 1)){((𝑊 prefix 𝑁)‘𝑖), ((𝑊 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘(𝑊 prefix 𝑁)), ((𝑊 prefix 𝑁)‘0)} ∈ (Edg‘𝐺)))
122	121	exp31 421	. . . . 5 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → ((𝑊‘𝑁) = (𝑊‘0) → (((𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ (𝑊 prefix 𝑁) ≠ ∅) ∧ ∀𝑖 ∈ (0..^((♯‘(𝑊 prefix 𝑁)) − 1)){((𝑊 prefix 𝑁)‘𝑖), ((𝑊 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘(𝑊 prefix 𝑁)), ((𝑊 prefix 𝑁)‘0)} ∈ (Edg‘𝐺)))))
123	122	3imp21 1115	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑊‘𝑁) = (𝑊‘0)) → (((𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ (𝑊 prefix 𝑁) ≠ ∅) ∧ ∀𝑖 ∈ (0..^((♯‘(𝑊 prefix 𝑁)) − 1)){((𝑊 prefix 𝑁)‘𝑖), ((𝑊 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘(𝑊 prefix 𝑁)), ((𝑊 prefix 𝑁)‘0)} ∈ (Edg‘𝐺)))
124	1, 2	isclwwlk 29237	. . . 4 ⊢ ((𝑊 prefix 𝑁) ∈ (ClWWalks‘𝐺) ↔ (((𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ (𝑊 prefix 𝑁) ≠ ∅) ∧ ∀𝑖 ∈ (0..^((♯‘(𝑊 prefix 𝑁)) − 1)){((𝑊 prefix 𝑁)‘𝑖), ((𝑊 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘(𝑊 prefix 𝑁)), ((𝑊 prefix 𝑁)‘0)} ∈ (Edg‘𝐺)))
125	123, 124	sylibr 233	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑊‘𝑁) = (𝑊‘0)) → (𝑊 prefix 𝑁) ∈ (ClWWalks‘𝐺))
126	47	adantl 483	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (0...𝑀))
127	126, 48	syl 17	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (0...(𝑀 + 1)))
128	127, 53	mpbird 257	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (0...(♯‘𝑊)))
129	7, 128	jca 513	. . . . . . . 8 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))))
130	129	ex 414	. . . . . . 7 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) → ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))))
131	130	3adant3 1133	. . . . . 6 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))))
132	131	impcom 409	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))))
133	132	3adant3 1133	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑊‘𝑁) = (𝑊‘0)) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))))
134	133, 55	syl 17	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑊‘𝑁) = (𝑊‘0)) → (♯‘(𝑊 prefix 𝑁)) = 𝑁)
135		isclwwlkn 29280	. . 3 ⊢ ((𝑊 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 prefix 𝑁) ∈ (ClWWalks‘𝐺) ∧ (♯‘(𝑊 prefix 𝑁)) = 𝑁))
136	125, 134, 135	sylanbrc 584	. 2 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (𝑊‘𝑁) = (𝑊‘0)) → (𝑊 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺))
137	3, 136	syl3an2 1165	1 ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) ∧ 𝑊 ∈ (𝑀 WWalksN 𝐺) ∧ (𝑊‘𝑁) = (𝑊‘0)) → (𝑊 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ⊆ wss 3949 ∅c0 4323 {cpr 4631 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 ≤ cle 11249 − cmin 11444 ℕcn 12212 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ...cfz 13484 ..^cfzo 13627 ♯chash 14290 Word cword 14464 lastSclsw 14512 prefix cpfx 14620 Vtxcvtx 28256 Edgcedg 28307 WWalksN cwwlksn 29080 ClWWalkscclwwlk 29234 ClWWalksN cclwwlkn 29277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-lsw 14513 df-substr 14591 df-pfx 14621 df-wwlks 29084 df-wwlksn 29085 df-clwwlk 29235 df-clwwlkn 29278

This theorem is referenced by: clwwnrepclwwn 29597

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