Theorem wwlksnextbi 28259

Description: Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 16-Apr-2021.) (Proof shortened by AV, 27-Oct-2022.)

Hypotheses

Ref	Expression
wwlksnext.v	⊢ 𝑉 = (Vtx‘𝐺)
wwlksnext.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
wwlksnextbi	⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))

Proof of Theorem wwlksnextbi

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlksnext.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		wwlksnext.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	wwlknp 28208	. . . 4 ⊢ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
4		wwlksnred 28257	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
5	4	ad2antrr 723	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
6		fveqeq2 6783	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
7	6	3ad2ant2 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
8	7	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
9		s1cl 14307	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ 𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉)
10	9	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) → ⟨“𝑆”⟩ ∈ Word 𝑉)
11	10	anim1ci 616	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉))
12		ccatlen 14278	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)))
13	11, 12	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)))
14	13	eqeq1d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
15		s1len 14311	. . . . . . . . . . . . . . . . . . . 20 ⊢ (♯‘⟨“𝑆”⟩) = 1
16	15	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘⟨“𝑆”⟩) = 1)
17	16	oveq2d 7291	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) = ((♯‘𝑇) + 1))
18	17	eqeq1d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((♯‘𝑇) + 1) = ((𝑁 + 1) + 1)))
19		lencl 14236	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ Word 𝑉 → (♯‘𝑇) ∈ ℕ0)
20	19	nn0cnd 12295	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇 ∈ Word 𝑉 → (♯‘𝑇) ∈ ℂ)
21	20	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘𝑇) ∈ ℂ)
22		peano2nn0 12273	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
23	22	nn0cnd 12295	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
24	23	ad2antrr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑁 + 1) ∈ ℂ)
25		1cnd 10970	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → 1 ∈ ℂ)
26	21, 24, 25	addcan2d 11179	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((♯‘𝑇) + 1) = ((𝑁 + 1) + 1) ↔ (♯‘𝑇) = (𝑁 + 1)))
27	14, 18, 26	3bitrd 305	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ (♯‘𝑇) = (𝑁 + 1)))
28		oveq2 7283	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 + 1) = (♯‘𝑇) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩) prefix (♯‘𝑇)))
29	28	eqcoms 2746	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑇) = (𝑁 + 1) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩) prefix (♯‘𝑇)))
30		pfxccat1 14415	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (♯‘𝑇)) = 𝑇)
31	11, 30	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (♯‘𝑇)) = 𝑇)
32	29, 31	sylan9eqr 2800	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) ∧ (♯‘𝑇) = (𝑁 + 1)) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇)
33	32	ex 413	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘𝑇) = (𝑁 + 1) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
34	27, 33	sylbid 239	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
35	34	3ad2antr1 1187	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
36	8, 35	sylbid 239	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
37	36	imp 407	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇)
38		oveq1 7282	. . . . . . . . . . . . . . 15 ⊢ (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (𝑊 prefix (𝑁 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)))
39	38	eqeq1d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ((𝑊 prefix (𝑁 + 1)) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
40	39	3ad2ant2 1133	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → ((𝑊 prefix (𝑁 + 1)) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
41	40	ad2antlr 724	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 prefix (𝑁 + 1)) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
42	37, 41	mpbird 256	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊 prefix (𝑁 + 1)) = 𝑇)
43	42	eleq1d 2823	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))
44	43	biimpd 228	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))
45	44	ex 413	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺))))
46	45	com23 86	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → 𝑇 ∈ (𝑁 WWalksN 𝐺))))
47	5, 46	syld 47	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → 𝑇 ∈ (𝑁 WWalksN 𝐺))))
48	47	com13 88	. . . . 5 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → 𝑇 ∈ (𝑁 WWalksN 𝐺))))
49	48	3ad2ant2 1133	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → 𝑇 ∈ (𝑁 WWalksN 𝐺))))
50	3, 49	mpcom 38	. . 3 ⊢ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))
51	50	com12 32	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))
52	1, 2	wwlksnext 28258	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
53		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
54	52, 53	syl5ibrcom 246	. . . . . . . . . 10 ⊢ ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))
55	54	3exp 1118	. . . . . . . . 9 ⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝑆 ∈ 𝑉 → ({(lastS‘𝑇), 𝑆} ∈ 𝐸 → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))))
56	55	com23 86	. . . . . . . 8 ⊢ (𝑇 ∈ (𝑁 WWalksN 𝐺) → ({(lastS‘𝑇), 𝑆} ∈ 𝐸 → (𝑆 ∈ 𝑉 → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))))
57	56	com14 96	. . . . . . 7 ⊢ (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ({(lastS‘𝑇), 𝑆} ∈ 𝐸 → (𝑆 ∈ 𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))))
58	57	imp 407	. . . . . 6 ⊢ ((𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑆 ∈ 𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
59	58	3adant1 1129	. . . . 5 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑆 ∈ 𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
60	59	com12 32	. . . 4 ⊢ (𝑆 ∈ 𝑉 → ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
61	60	adantl 482	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) → ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
62	61	imp 407	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))
63	51, 62	impbid 211	1 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {cpr 4563  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  0cc0 10871  1c1 10872   + caddc 10874  ℕ₀cn0 12233  ..^cfzo 13382  ♯chash 14044  Word cword 14217  lastSclsw 14265   ++ cconcat 14273  ⟨“cs1 14300   prefix cpfx 14383  Vtxcvtx 27366  Edgcedg 27417   WWalksN cwwlksn 28191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-lsw 14266  df-concat 14274  df-s1 14301  df-substr 14354  df-pfx 14384  df-wwlks 28195  df-wwlksn 28196

This theorem is referenced by:  wwlksnextwrd  28262