Theorem wwlksnextbi 29924

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 29873	. . . 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 29922	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
5	4	ad2antrr 726	. . . . . . 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 6916	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 14637	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ 𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉)
10	9	adantl 481	. . . . . . . . . . . . . . . . . . . 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 14610	. . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
15		s1len 14641	. . . . . . . . . . . . . . . . . . . 20 ⊢ (♯‘⟨“𝑆”⟩) = 1
16	15	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘⟨“𝑆”⟩) = 1)
17	16	oveq2d 7447	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) = ((♯‘𝑇) + 1))
18	17	eqeq1d 2737	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((♯‘𝑇) + 1) = ((𝑁 + 1) + 1)))
19		lencl 14568	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ Word 𝑉 → (♯‘𝑇) ∈ ℕ0)
20	19	nn0cnd 12587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇 ∈ Word 𝑉 → (♯‘𝑇) ∈ ℂ)
21	20	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘𝑇) ∈ ℂ)
22		peano2nn0 12564	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
23	22	nn0cnd 12587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
24	23	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑁 + 1) ∈ ℂ)
25		1cnd 11254	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → 1 ∈ ℂ)
26	21, 24, 25	addcan2d 11463	. . . . . . . . . . . . . . . . 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 7439	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 + 1) = (♯‘𝑇) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩) prefix (♯‘𝑇)))
29	28	eqcoms 2743	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑇) = (𝑁 + 1) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩) prefix (♯‘𝑇)))
30		pfxccat1 14737	. . . . . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) ∧ (♯‘𝑇) = (𝑁 + 1)) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇)
33	32	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘𝑇) = (𝑁 + 1) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
34	27, 33	sylbid 240	. . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . 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 406	. . . . . . . . . . . 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 7438	. . . . . . . . . . . . . . 15 ⊢ (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (𝑊 prefix (𝑁 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)))
39	38	eqeq1d 2737	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . 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 257	. . . . . . . . . . 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 2824	. . . . . . . . . 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 229	. . . . . . . . 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 412	. . . . . . . 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 29923	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
53		eleq1 2827	. . . . . . . . . . 11 ⊢ (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
54	52, 53	syl5ibrcom 247	. . . . . . . . . 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 406	. . . . . 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 481	. . 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 406	. 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 212	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {cpr 4633  ‘cfv 6563  (class class class)co 7431  ℂcc 11151  0cc0 11153  1c1 11154   + caddc 11156  ℕ₀cn0 12524  ..^cfzo 13691  ♯chash 14366  Word cword 14549  lastSclsw 14597   ++ cconcat 14605  ⟨“cs1 14630   prefix cpfx 14705  Vtxcvtx 29028  Edgcedg 29079   WWalksN cwwlksn 29856

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-concat 14606  df-s1 14631  df-substr 14676  df-pfx 14706  df-wwlks 29860  df-wwlksn 29861

This theorem is referenced by:  wwlksnextwrd  29927