clwwlkinwwlk - Metamath Proof Explorer

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Theorem clwwlkinwwlk 30069

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 2735	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2735	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	wwlknp 29873	. 2 ⊢ (𝑊 ∈ (𝑀 WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ ∀𝑖 ∈ (0..^𝑀){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
4		pfxcl 14712	. . . . . . . . . . . 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 12882	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ_≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ≤ 𝑀))
10		zre 12615	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
11		zre 12615	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
12		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ≤ 𝑀 → 𝑁 ≤ 𝑀)
13	10, 11, 12	3anim123i 1150	. . . . . . . . . . . . . . . 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 12109	. . . . . . . . . . . . . . 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 5152	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (𝑁 ≤ (♯‘𝑊) ↔ 𝑁 ≤ (𝑀 + 1)))
20	19	ad2antlr 727	. . . . . . . . . . . 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 14721	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ ∧ 𝑁 ≤ (♯‘𝑊)) → (𝑊 prefix 𝑁) ≠ ∅)
23	7, 8, 21, 22	syl3anc 1370	. . . . . . . . . 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 1167	. . . . . . . 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 12632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
28		1nn0 12540	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ₀
29		eluzmn 12883	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
30	27, 28, 29	sylancl 586	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
31		uzss 12899	. . . . . . . . . . . . . . . . 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 3995	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑀 ∈ (ℤ_≥‘(𝑁 − 1)))
34		fzoss2 13724	. . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (0..^(𝑁 − 1)) ⊆ (0..^𝑀))
37		ssralv 4064	. . . . . . . . . . . . 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 1118	. . . . . . . . . . 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 1346	. . . . . . . . 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 12531	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
44		elnn0uz 12921	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (ℤ_≥‘0))
45	43, 44	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘0))
46		eluzfz 13556	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (ℤ_≥‘0) ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ∈ (0...𝑀))
47	45, 46	sylan 580	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ∈ (0...𝑀))
48		fzelp1 13613	. . . . . . . . . . . . . . . . . 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 7439	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (0...(♯‘𝑊)) = (0...(𝑀 + 1)))
52	51	eleq2d 2825	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (𝑁 ∈ (0...(♯‘𝑊)) ↔ 𝑁 ∈ (0...(𝑀 + 1))))
53	52	ad2antlr 727	. . . . . . . . . . . . . . . 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 14718	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝑁)) = 𝑁)
56	7, 54, 55	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (♯‘(𝑊 prefix 𝑁)) = 𝑁)
57	56	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → ((♯‘(𝑊 prefix 𝑁)) − 1) = (𝑁 − 1))
58	57	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → (0..^((♯‘(𝑊 prefix 𝑁)) − 1)) = (0..^(𝑁 − 1)))
59	58	raleqdv 3324	. . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
63		fzoss2 13724	. . . . . . . . . . . . . . . . 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 3995	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑖 ∈ (0..^𝑁))
66		pfxfv 14717	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑊 prefix 𝑁)‘𝑖) = (𝑊‘𝑖))
67	60, 61, 65, 66	syl3anc 1370	. . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) → 𝑁 ∈ ℤ)
69		elfzom1elp1fzo 13768	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑖 + 1) ∈ (0..^𝑁))
70	68, 69	sylan 580	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑀 + 1)) ∧ (𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁))) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑖 + 1) ∈ (0..^𝑁))
71		pfxfv 14717	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 2824	. . . . . . . . . . . 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 3174	. . . . . . . . . . 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 1167	. . . . . . . . 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 13631	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_≥‘𝑁)) → 𝑁 ∈ (1...𝑀))
81		fzelp1 13613	. . . . . . . . . . . . . 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 7439	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (1...(♯‘𝑊)) = (1...(𝑀 + 1)))
85	84	eleq2d 2825	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) = (𝑀 + 1) → (𝑁 ∈ (1...(♯‘𝑊)) ↔ 𝑁 ∈ (1...(𝑀 + 1))))
86	85	ad2antlr 727	. . . . . . . . . . . 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 14730	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑊))) → (lastS‘(𝑊 prefix 𝑁)) = (𝑊‘(𝑁 − 1)))
89		pfxfv0 14727	. . . . . . . . . . . 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 584	. . . . . . . . . 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 1167	. . . . . . . . 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 13747	. . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . 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 6911	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → (𝑊‘𝑖) = (𝑊‘(𝑁 − 1)))
99		oveq1 7438	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = (𝑁 − 1) → (𝑖 + 1) = ((𝑁 − 1) + 1))
100		nncn 12272	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
101		npcan1 11686	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
102	100, 101	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁)
103	99, 102	sylan9eqr 2797	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ 𝑖 = (𝑁 − 1)) → (𝑖 + 1) = 𝑁)
104	103	fveq2d 6911	. . . . . . . . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . 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 3614	. . . . . . . . . . . . 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 1118	. . . . . . . . . . . 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 1346	. . . . . . . . . 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 4739	. . . . . . . . . . 11 ⊢ ((𝑊‘𝑁) = (𝑊‘0) → {(𝑊‘(𝑁 − 1)), (𝑊‘𝑁)} = {(𝑊‘(𝑁 − 1)), (𝑊‘0)})
117	116	eleq1d 2824	. . . . . . . . . 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 2839	. . . . . . 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 1127	. . . . . 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 1113	. . . 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 30013	. . . 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 1131	. . . . . 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 1131	. . . 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 30056	. . 3 ⊢ ((𝑊 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 prefix 𝑁) ∈ (ClWWalks‘𝐺) ∧ (♯‘(𝑊 prefix 𝑁)) = 𝑁))
136	125, 134, 135	sylanbrc 583	. 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 1163	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 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ⊆ wss 3963 ∅c0 4339 {cpr 4633 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 ≤ cle 11294 − cmin 11490 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ...cfz 13544 ..^cfzo 13691 ♯chash 14366 Word cword 14549 lastSclsw 14597 prefix cpfx 14705 Vtxcvtx 29028 Edgcedg 29079 WWalksN cwwlksn 29856 ClWWalkscclwwlk 30010 ClWWalksN cclwwlkn 30053

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-lsw 14598 df-substr 14676 df-pfx 14706 df-wwlks 29860 df-wwlksn 29861 df-clwwlk 30011 df-clwwlkn 30054

This theorem is referenced by: clwwnrepclwwn 30373

W3C validator