wwlksnred - Mathbox for Alexander van der Vekens

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Theorem wwlksnred 41098

Description: Reduction of a walk (as word) by removing the trailing edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.)

**Assertion**
Ref	Expression
wwlksnred	⊢ (𝑁 ∈ ℕ₀ → (𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalkSN 𝐺)))

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

Step	Hyp	Ref	Expression
1		peano2nn0 11210	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
2		iswwlksn 41041	. . 3 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺) ↔ (𝑊 ∈ (WWalkS‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1))))
3	1, 2	syl 17	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺) ↔ (𝑊 ∈ (WWalkS‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1))))
4		eqid 2610	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2610	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
6	4, 5	iswwlks 41039	. . . 4 ⊢ (𝑊 ∈ (WWalkS‘𝐺) ↔ (𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
7		simp1 1054	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → 𝑊 ∈ Word (Vtx‘𝐺))
8		nn0p1nn 11209	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
9	8	3ad2ant3 1077	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ ℕ)
10	1	nn0red 11229	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
11	10	lep1d 10834	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
12	11	3ad2ant3 1077	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
13		breq2 4587	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ≤ (#‘𝑊) ↔ (𝑁 + 1) ≤ ((𝑁 + 1) + 1)))
14	13	3ad2ant2 1076	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → ((𝑁 + 1) ≤ (#‘𝑊) ↔ (𝑁 + 1) ≤ ((𝑁 + 1) + 1)))
15	12, 14	mpbird 246	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ≤ (#‘𝑊))
16		swrdn0 13282	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ ℕ ∧ (𝑁 + 1) ≤ (#‘𝑊)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ≠ ∅)
17	7, 9, 15, 16	syl3anc 1318	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ≠ ∅)
18	17	3exp 1256	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ₀ → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ≠ ∅)))
19	18	3ad2ant2 1076	. . . . . . . . 9 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ₀ → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ≠ ∅)))
20	19	imp 444	. . . . . . . 8 ⊢ (((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ₀ → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ≠ ∅))
21	20	impcom 445	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1))) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ≠ ∅)
22		swrdcl 13271	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ Word (Vtx‘𝐺))
23	22	3ad2ant2 1076	. . . . . . . . 9 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ Word (Vtx‘𝐺))
24	23	adantr 480	. . . . . . . 8 ⊢ (((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ Word (Vtx‘𝐺))
25	24	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1))) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ Word (Vtx‘𝐺))
26		oveq1 6556	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → ((#‘𝑊) − 1) = (((𝑁 + 1) + 1) − 1))
27	1	nn0cnd 11230	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℂ)
28		1cnd 9935	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
29	27, 28	pncand 10272	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
30	26, 29	sylan9eq 2664	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → ((#‘𝑊) − 1) = (𝑁 + 1))
31	30	oveq2d 6565	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → (0..^((#‘𝑊) − 1)) = (0..^(𝑁 + 1)))
32	31	raleqdv 3121	. . . . . . . . . . . . . . . . . 18 ⊢ (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
33	32	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
34		nn0z 11277	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
35		nn0z 11277	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (𝑁 + 1) ∈ ℤ)
36	1, 35	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℤ)
37		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
38	37	lep1d 10834	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ≤ (𝑁 + 1))
39	34, 36, 38	3jca 1235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ 𝑁 ≤ (𝑁 + 1)))
40	39	ad2antll 761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ 𝑁 ≤ (𝑁 + 1)))
41		eluz2 11569	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ ∧ 𝑁 ≤ (𝑁 + 1)))
42	40, 41	sylibr 223	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (𝑁 + 1) ∈ (ℤ_≥‘𝑁))
43		fzoss2 12365	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘𝑁) → (0..^𝑁) ⊆ (0..^(𝑁 + 1)))
44	42, 43	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (0..^𝑁) ⊆ (0..^(𝑁 + 1)))
45		ssralv 3629	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0..^𝑁) ⊆ (0..^(𝑁 + 1)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
46	44, 45	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
47		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → 𝑊 ∈ Word (Vtx‘𝐺))
48	47	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑊 ∈ Word (Vtx‘𝐺))
49		nn0fz0 12306	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 + 1) ∈ ℕ₀ ↔ (𝑁 + 1) ∈ (0...(𝑁 + 1)))
50	1, 49	sylib 207	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ (0...(𝑁 + 1)))
51	50	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (𝑁 + 1) ∈ (0...(𝑁 + 1)))
52		fzelp1 12263	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑁 + 1) ∈ (0...(𝑁 + 1)) → (𝑁 + 1) ∈ (0...((𝑁 + 1) + 1)))
53	51, 52	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (𝑁 + 1) ∈ (0...((𝑁 + 1) + 1)))
54		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → (0...(#‘𝑊)) = (0...((𝑁 + 1) + 1)))
55	54	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ∈ (0...(#‘𝑊)) ↔ (𝑁 + 1) ∈ (0...((𝑁 + 1) + 1))))
56	55	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → ((𝑁 + 1) ∈ (0...(#‘𝑊)) ↔ (𝑁 + 1) ∈ (0...((𝑁 + 1) + 1))))
57	56	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → ((𝑁 + 1) ∈ (0...(#‘𝑊)) ↔ (𝑁 + 1) ∈ (0...((𝑁 + 1) + 1))))
58	53, 57	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (𝑁 + 1) ∈ (0...(#‘𝑊)))
59	58	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑁 + 1) ∈ (0...(#‘𝑊)))
60		fzossfzop1 12412	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ₀ → (0..^𝑁) ⊆ (0..^(𝑁 + 1)))
61	60	sseld 3567	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ₀ → (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^(𝑁 + 1))))
62	61	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0..^(𝑁 + 1))))
63	62	imp 444	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^(𝑁 + 1)))
64		swrd0fv 13291	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (0...(#‘𝑊)) ∧ 𝑖 ∈ (0..^(𝑁 + 1))) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖) = (𝑊‘𝑖))
65	48, 59, 63, 64	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖) = (𝑊‘𝑖))
66	65	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑊‘𝑖) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖))
67		fzofzp1 12431	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁))
68	67	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁))
69		fzval3 12404	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑁 ∈ ℤ → (0...𝑁) = (0..^(𝑁 + 1)))
70	69	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ ℤ → (0..^(𝑁 + 1)) = (0...𝑁))
71	34, 70	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → (0..^(𝑁 + 1)) = (0...𝑁))
72	71	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ₀ → ((𝑖 + 1) ∈ (0..^(𝑁 + 1)) ↔ (𝑖 + 1) ∈ (0...𝑁)))
73	72	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → ((𝑖 + 1) ∈ (0..^(𝑁 + 1)) ↔ (𝑖 + 1) ∈ (0...𝑁)))
74	73	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑖 + 1) ∈ (0..^(𝑁 + 1)) ↔ (𝑖 + 1) ∈ (0...𝑁)))
75	68, 74	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0..^(𝑁 + 1)))
76		swrd0fv 13291	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (0...(#‘𝑊)) ∧ (𝑖 + 1) ∈ (0..^(𝑁 + 1))) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1)) = (𝑊‘(𝑖 + 1)))
77	48, 59, 75, 76	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1)) = (𝑊‘(𝑖 + 1)))
78	77	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑊‘(𝑖 + 1)) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1)))
79	66, 78	preq12d 4220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑖 ∈ (0..^𝑁)) → {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} = {((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))})
80	79	eleq1d 2672	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑖 ∈ (0..^𝑁)) → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
81	80	biimpd 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑖 ∈ (0..^𝑁)) → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → {((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
82	81	ralimdva 2945	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^𝑁){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
83	46, 82	syld 46	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^𝑁){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
84	33, 83	sylbid 229	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^𝑁){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
85	84	imp 444	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ∀𝑖 ∈ (0..^𝑁){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
86		nn0cn 11179	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
87	86, 28	pncand 10272	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) − 1) = 𝑁)
88	87	oveq2d 6565	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (0..^((𝑁 + 1) − 1)) = (0..^𝑁))
89	88	ad2antll 761	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (0..^((𝑁 + 1) − 1)) = (0..^𝑁))
90	89	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (0..^((𝑁 + 1) − 1)) = (0..^𝑁))
91	90	raleqdv 3121	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (∀𝑖 ∈ (0..^((𝑁 + 1) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^𝑁){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
92	85, 91	mpbird 246	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ∀𝑖 ∈ (0..^((𝑁 + 1) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
93	1	ad2antll 761	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (𝑁 + 1) ∈ ℕ₀)
94		simpl 472	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → (#‘𝑊) = ((𝑁 + 1) + 1))
95	94	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (#‘𝑊) = ((𝑁 + 1) + 1))
96		swrd0len0 13288	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ ℕ₀ ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1))
97	47, 93, 95, 96	syl3anc 1318	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1))
98	97	oveq1d 6564	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → ((#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) − 1) = ((𝑁 + 1) − 1))
99	98	oveq2d 6565	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (0..^((#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) − 1)) = (0..^((𝑁 + 1) − 1)))
100	99	raleqdv 3121	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (∀𝑖 ∈ (0..^((#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((𝑁 + 1) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
101	100	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (∀𝑖 ∈ (0..^((#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((𝑁 + 1) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
102	92, 101	mpbird 246	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ∀𝑖 ∈ (0..^((#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
103	102	exp31 628	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^((#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
104	103	com23 84	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) → (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → ∀𝑖 ∈ (0..^((#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
105	104	imp 444	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → ∀𝑖 ∈ (0..^((#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
106	105	3adant1 1072	. . . . . . . . 9 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → ∀𝑖 ∈ (0..^((#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
107	106	expdimp 452	. . . . . . . 8 ⊢ (((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ₀ → ∀𝑖 ∈ (0..^((#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
108	107	impcom 445	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1))) → ∀𝑖 ∈ (0..^((#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
109	4, 5	iswwlks 41039	. . . . . . 7 ⊢ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (WWalkS‘𝐺) ↔ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ≠ ∅ ∧ (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) − 1)){((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘𝑖), ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
110	21, 25, 108, 109	syl3anbrc 1239	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1))) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (WWalkS‘𝐺))
111		peano2nn0 11210	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 + 1) ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ₀)
112	1, 111	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ₀)
113		elfz2nn0 12300	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 + 1) ∈ (0...((𝑁 + 1) + 1)) ↔ ((𝑁 + 1) ∈ ℕ₀ ∧ ((𝑁 + 1) + 1) ∈ ℕ₀ ∧ (𝑁 + 1) ≤ ((𝑁 + 1) + 1)))
114	1, 112, 11, 113	syl3anbrc 1239	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ (0...((𝑁 + 1) + 1)))
115	114	adantl 481	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ (0...((𝑁 + 1) + 1)))
116	115, 56	mpbird 246	. . . . . . . . . . . 12 ⊢ (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ (0...(#‘𝑊)))
117	116	anim2i 591	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀)) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (0...(#‘𝑊))))
118	117	exp32 629	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ₀ → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (0...(#‘𝑊))))))
119	118	3ad2ant2 1076	. . . . . . . . 9 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ₀ → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (0...(#‘𝑊))))))
120	119	imp 444	. . . . . . . 8 ⊢ (((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ₀ → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (0...(#‘𝑊)))))
121	120	impcom 445	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1))) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (0...(#‘𝑊))))
122		swrd0len 13274	. . . . . . 7 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (0...(#‘𝑊))) → (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1))
123	121, 122	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1))) → (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1))
124		iswwlksn 41041	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalkSN 𝐺) ↔ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (WWalkS‘𝐺) ∧ (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1))))
125	124	adantr 480	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1))) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalkSN 𝐺) ↔ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (WWalkS‘𝐺) ∧ (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1))))
126	110, 123, 125	mpbir2and 959	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1))) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalkSN 𝐺))
127	126	expcom 450	. . . 4 ⊢ (((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ₀ → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalkSN 𝐺)))
128	6, 127	sylanb 488	. . 3 ⊢ ((𝑊 ∈ (WWalkS‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ₀ → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalkSN 𝐺)))
129	128	com12 32	. 2 ⊢ (𝑁 ∈ ℕ₀ → ((𝑊 ∈ (WWalkS‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalkSN 𝐺)))
130	3, 129	sylbid 229	1 ⊢ (𝑁 ∈ ℕ₀ → (𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalkSN 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ⊆ wss 3540 ∅c0 3874 {cpr 4127 ⟨cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 substr csubstr 13150 Vtxcvtx 25673 Edgcedga 25792 WWalkScwwlks 41028 WWalkSN cwwlksn 41029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-substr 13158 df-wwlks 41033 df-wwlksn 41034

This theorem is referenced by: wwlksnextbi 41100 wwlksnredwwlkn 41101

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