clwwlkinwwlk - Metamath Proof Explorer

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Theorem clwwlkinwwlk 30110

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 2736	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2736	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	wwlknp 29911	. 2 ⊢ (𝑊 ∈ (𝑀 WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
4		pfxcl 14640	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺))
5	4	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) → (𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺))
6	5	adantr 480	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑊 prefix 𝑁) ∈ Word (Vtx‘𝐺))
7		simpll 767	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑊 ∈ Word (Vtx‘𝐺))
8		simprl 771	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ ℕ)
9		eluz2 12794	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ_≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ≤ 𝑀))
10		zre 12528	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
11		zre 12528	. . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ_≥‘𝑁) → (𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ≤ 𝑀))
15		letrp1 11999	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ≤ (𝑀 + 1))
18	17	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ≤ (𝑀 + 1))
19		breq2 5089	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (𝑁 ≤ (♯‘𝑊) ↔ 𝑁 ≤ (𝑀 + 1)))
20	19	ad2antlr 728	. . . . . . . . . . . 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 14649	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ ∧ 𝑁 ≤ (♯‘𝑊)) → (𝑊 prefix 𝑁) ≠ ∅)
23	7, 8, 21, 22	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑊 prefix 𝑁) ≠ ∅)
24	6, 23	jca 511	. . . . . . . . 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 1170	. . . . . . . 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 480	. . . . . . 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 12545	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
28		1nn0 12453	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ₀
29		eluzmn 12795	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
30	27, 28, 29	sylancl 587	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
31		uzss 12811	. . . . . . . . . . . . . . . . 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 3921	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑀 ∈ (ℤ_≥‘(𝑁 − 1)))
34		fzoss2 13642	. . . . . . . . . . . . . . 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 3990	. . . . . . . . . . . . 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 1349	. . . . . . . . 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 480	. . . . . . . 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 12444	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
44		elnn0uz 12829	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (ℤ_≥‘0))
45	43, 44	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘0))
46		eluzfz 13473	. . . . . . . . . . . . . . . . . . 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 13530	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (0...(𝑀 + 1)))
51		oveq2 7375	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (0...(♯‘𝑊)) = (0...(𝑀 + 1)))
52	51	eleq2d 2822	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (𝑁 ∈ (0...(♯‘𝑊)) ↔ 𝑁 ∈ (0...(𝑀 + 1))))
53	52	ad2antlr 728	. . . . . . . . . . . . . . . 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 14646	. . . . . . . . . . . . . . 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 7382	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → ((♯‘(𝑊 prefix 𝑁)) − 1) = (𝑁 − 1))
58	57	oveq2d 7383	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (0..^((♯‘(𝑊 prefix 𝑁)) − 1)) = (0..^(𝑁 − 1)))
59	58	raleqdv 3295	. . . . . . . . . . 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 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑊 ∈ Word (Vtx‘𝐺))
61	54	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑁 ∈ (0...(♯‘𝑊)))
62	30	ad2antrl 729	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
63		fzoss2 13642	. . . . . . . . . . . . . . . . 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 3921	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑖 ∈ (0..^𝑁))
66		pfxfv 14645	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑊 prefix 𝑁)‘𝑖) = (𝑊‘𝑖))
67	60, 61, 65, 66	syl3anc 1374	. . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ ℤ)
69		elfzom1elp1fzo 13687	. . . . . . . . . . . . . . . 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 14645	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 4685	. . . . . . . . . . . . 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 2821	. . . . . . . . . . . 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 3158	. . . . . . . . . . 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 1170	. . . . . . . . 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 480	. . . . . . . 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 13548	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ∈ (1...𝑀))
81		fzelp1 13530	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (1...(𝑀 + 1)))
84		oveq2 7375	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (1...(♯‘𝑊)) = (1...(𝑀 + 1)))
85	84	eleq2d 2822	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (𝑁 ∈ (1...(♯‘𝑊)) ↔ 𝑁 ∈ (1...(𝑀 + 1))))
86	85	ad2antlr 728	. . . . . . . . . . . 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 14657	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑊))) → (lastS‘(𝑊 prefix 𝑁)) = (𝑊‘(𝑁 − 1)))
89		pfxfv0 14654	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((𝑊 prefix 𝑁)‘0) = (𝑊‘0))
90	88, 89	preq12d 4685	. . . . . . . . . . 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 1170	. . . . . . . . 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 480	. . . . . . . 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 13665	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → 𝑖 = (𝑁 − 1))
98	97	fveq2d 6844	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → (𝑊‘𝑖) = (𝑊‘(𝑁 − 1)))
99		oveq1 7374	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = (𝑁 − 1) → (𝑖 + 1) = ((𝑁 − 1) + 1))
100		nncn 12182	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
101		npcan1 11575	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
102	100, 101	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁)
103	99, 102	sylan9eqr 2793	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → (𝑖 + 1) = 𝑁)
104	103	fveq2d 6844	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → (𝑊‘(𝑖 + 1)) = (𝑊‘𝑁))
105	98, 104	preq12d 4685	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)})
106	105	eleq1d 2821	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺)))
107	106	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑖 = (𝑁 − 1) → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺))))
108	107	adantr 480	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . 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 3556	. . . . . . . . . . . . 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 1349	. . . . . . . . . 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 480	. . . . . . . . 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 4678	. . . . . . . . . . 11 ⊢ ((𝑊‘𝑁) = (𝑊‘0) → {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} = {(𝑊‘(𝑁 − 1)), (𝑊‘0)})
117	116	eleq1d 2821	. . . . . . . . . 10 ⊢ ((𝑊‘𝑁) = (𝑊‘0) → ({(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} ∈ (Edg‘𝐺) ↔ {(𝑊‘(𝑁 − 1)), (𝑊‘0)} ∈ (Edg‘𝐺)))
118	117	adantl 481	. . . . . . . . 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 232	. . . . . . . 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 2836	. . . . . . 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 419	. . . . 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 1114	. . . 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 30054	. . . 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 234	. . 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 481	. . . . . . . . . . 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 511	. . . . . . . 8 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))))
130	129	ex 412	. . . . . . 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 407	. . . . 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 30097	. . 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ⊆ wss 3889 ∅c0 4273 {cpr 4569 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 ≤ cle 11180 − cmin 11377 ℕcn 12174 ℕ₀cn0 12437 ℤcz 12524 ℤ_≥cuz 12788 ...cfz 13461 ..^cfzo 13608 ♯chash 14292 Word cword 14475 lastSclsw 14524 prefix cpfx 14633 Vtxcvtx 29065 Edgcedg 29116 WWalksN cwwlksn 29894 ClWWalkscclwwlk 30051 ClWWalksN cclwwlkn 30094

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-hash 14293 df-word 14476 df-lsw 14525 df-substr 14604 df-pfx 14634 df-wwlks 29898 df-wwlksn 29899 df-clwwlk 30052 df-clwwlkn 30095

This theorem is referenced by: clwwnrepclwwn 30414

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