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Theorem clwwlkinwwlk 27821

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 2824	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2824	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	wwlknp 27624	. 2 ⊢ (𝑊 ∈ (𝑀 WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
4		pfxcl 14042	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺))
5	4	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) → (𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺))
6	5	adantr 483	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺))
7		simpll 765	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑊 ∈ Word (Vtx‘𝐺))
8		simprl 769	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ ℕ)
9		eluz2 12252	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ_≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ≤ 𝑀))
10		zre 11988	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
11		zre 11988	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
12		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ≤ 𝑀 → 𝑁 ≤ 𝑀)
13	10, 11, 12	3anim123i 1147	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ≤ 𝑀) → (𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ≤ 𝑀))
14	9, 13	sylbi 219	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ_≥‘𝑁) → (𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ≤ 𝑀))
15		letrp1 11487	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ≤ (𝑀 + 1))
18	17	adantl 484	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ≤ (𝑀 + 1))
19		breq2 5073	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (𝑁 ≤ (♯‘𝑊) ↔ 𝑁 ≤ (𝑀 + 1)))
20	19	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑁 ≤ (♯‘𝑊) ↔ 𝑁 ≤ (𝑀 + 1)))
21	18, 20	mpbird 259	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ≤ (♯‘𝑊))
22		pfxn0 14051	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ ∧ 𝑁 ≤ (♯‘𝑊)) → (𝑊 prefix 𝑁) ≠ ∅)
23	7, 8, 21, 22	syl3anc 1367	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑊 prefix 𝑁) ≠ ∅)
24	6, 23	jca 514	. . . . . . . . 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 1164	. . . . . . . 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 483	. . . . . . 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 12007	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
28		1nn0 11916	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ₀
29		eluzmn 12253	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
30	27, 28, 29	sylancl 588	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
31		uzss 12268	. . . . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑀 ∈ (ℤ_≥‘(𝑁 − 1)))
34		fzoss2 13068	. . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (0..^(𝑁 − 1)) ⊆ (0..^𝑀))
37		ssralv 4036	. . . . . . . . . . . . 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 1115	. . . . . . . . . . 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 1343	. . . . . . . . 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 483	. . . . . . . 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 11907	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
44		elnn0uz 12286	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (ℤ_≥‘0))
45	43, 44	sylib 220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘0))
46		eluzfz 12906	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (ℤ_≥‘0) ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ∈ (0...𝑀))
47	45, 46	sylan 582	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ∈ (0...𝑀))
48		fzelp1 12962	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (0...(𝑀 + 1)))
51		oveq2 7167	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (0...(♯‘𝑊)) = (0...(𝑀 + 1)))
52	51	eleq2d 2901	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (𝑁 ∈ (0...(♯‘𝑊)) ↔ 𝑁 ∈ (0...(𝑀 + 1))))
53	52	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑁 ∈ (0...(♯‘𝑊)) ↔ 𝑁 ∈ (0...(𝑀 + 1))))
54	50, 53	mpbird 259	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (0...(♯‘𝑊)))
55		pfxlen 14048	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝑁)) = 𝑁)
56	7, 54, 55	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (♯‘(𝑊 prefix 𝑁)) = 𝑁)
57	56	oveq1d 7174	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → ((♯‘(𝑊 prefix 𝑁)) − 1) = (𝑁 − 1))
58	57	oveq2d 7175	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (0..^((♯‘(𝑊 prefix 𝑁)) − 1)) = (0..^(𝑁 − 1)))
59	58	raleqdv 3418	. . . . . . . . . . 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 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑊 ∈ Word (Vtx‘𝐺))
61	54	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑁 ∈ (0...(♯‘𝑊)))
62	30	ad2antrl 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
63		fzoss2 13068	. . . . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑖 ∈ (0..^𝑁))
66		pfxfv 14047	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑊 prefix 𝑁)‘𝑖) = (𝑊‘𝑖))
67	60, 61, 65, 66	syl3anc 1367	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ ℤ)
69		elfzom1elp1fzo 13107	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑖 + 1) ∈ (0..^𝑁))
70	68, 69	sylan 582	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑖 + 1) ∈ (0..^𝑁))
71		pfxfv 14047	. . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . 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 4680	. . . . . . . . . . . . 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 2900	. . . . . . . . . . . 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 3199	. . . . . . . . . . 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 281	. . . . . . . . . 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 1164	. . . . . . . . 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 483	. . . . . . . 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 259	. . . . . . 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 12980	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ∈ (1...𝑀))
81		fzelp1 12962	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (1...(𝑀 + 1)))
84		oveq2 7167	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (1...(♯‘𝑊)) = (1...(𝑀 + 1)))
85	84	eleq2d 2901	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (𝑁 ∈ (1...(♯‘𝑊)) ↔ 𝑁 ∈ (1...(𝑀 + 1))))
86	85	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑁 ∈ (1...(♯‘𝑊)) ↔ 𝑁 ∈ (1...(𝑀 + 1))))
87	83, 86	mpbird 259	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (1...(♯‘𝑊)))
88		pfxfvlsw 14060	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑊))) → (lastS‘(𝑊 prefix 𝑁)) = (𝑊‘(𝑁 − 1)))
89		pfxfv0 14057	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((𝑊 prefix 𝑁)‘0) = (𝑊‘0))
90	88, 89	preq12d 4680	. . . . . . . . . . 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 586	. . . . . . . . . 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 1164	. . . . . . . . 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 483	. . . . . . . 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 13088	. . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑁 − 1) ∈ (0..^𝑀))
97		simpr 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → 𝑖 = (𝑁 − 1))
98	97	fveq2d 6677	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → (𝑊‘𝑖) = (𝑊‘(𝑁 − 1)))
99		oveq1 7166	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = (𝑁 − 1) → (𝑖 + 1) = ((𝑁 − 1) + 1))
100		nncn 11649	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
101		npcan1 11068	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
102	100, 101	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁)
103	99, 102	sylan9eqr 2881	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → (𝑖 + 1) = 𝑁)
104	103	fveq2d 6677	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → (𝑊‘(𝑖 + 1)) = (𝑊‘𝑁))
105	98, 104	preq12d 4680	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)})
106	105	eleq1d 2900	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺)))
107	106	ex 415	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑖 = (𝑁 − 1) → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺))))
108	107	adantr 483	. . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . 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 3618	. . . . . . . . . . . . 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 1115	. . . . . . . . . . . 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 1343	. . . . . . . . . 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 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))
116		preq2 4673	. . . . . . . . . . 11 ⊢ ((𝑊‘𝑁) = (𝑊‘0) → {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} = {(𝑊‘(𝑁 − 1)), (𝑊‘0)})
117	116	eleq1d 2900	. . . . . . . . . 10 ⊢ ((𝑊‘𝑁) = (𝑊‘0) → ({(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘0)} ∈ (Edg‘𝐺)))
118	117	adantl 484	. . . . . . . . 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 234	. . . . . . . 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 2916	. . . . . . 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 1124	. . . . . 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 422	. . . . 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 1110	. . . 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 27765	. . . 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 236	. . 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 484	. . . . . . . . . . 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 259	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (0...(♯‘𝑊)))
129	7, 128	jca 514	. . . . . . . 8 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))))
130	129	ex 415	. . . . . . 7 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) → ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)))))
131	130	3adant3 1128	. . . . . 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 410	. . . . 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 1128	. . . 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 27808	. . 3 ⊢ ((𝑊 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 prefix 𝑁) ∈ (ClWWalks‘𝐺) ∧ (♯‘(𝑊 prefix 𝑁)) = 𝑁))
136	125, 134, 135	sylanbrc 585	. 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 1160	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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∀wral 3141 ⊆ wss 3939 ∅c0 4294 {cpr 4572 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 1c1 10541 + caddc 10543 ≤ cle 10679 − cmin 10873 ℕcn 11641 ℕ₀cn0 11900 ℤcz 11984 ℤ_≥cuz 12246 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Word cword 13864 lastSclsw 13917 prefix cpfx 14035 Vtxcvtx 26784 Edgcedg 26835 WWalksN cwwlksn 27607 ClWWalkscclwwlk 27762 ClWWalksN cclwwlkn 27805

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-lsw 13918 df-substr 14006 df-pfx 14036 df-wwlks 27611 df-wwlksn 27612 df-clwwlk 27763 df-clwwlkn 27806

This theorem is referenced by: clwwnrepclwwn 28126

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