Theorem wwlknextbi 26253

Description: Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.)

Assertion

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

Proof of Theorem wwlknextbi

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlknimp 26215	. . . 4 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
2		wwlknred 26251	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
3	2	ad2antrr 758	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
4		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (#‘𝑊) = (#‘(𝑇 ++ ⟨“𝑆”⟩)))
5	4	eqeq1d 2612	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ((#‘𝑊) = ((𝑁 + 1) + 1) ↔ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
6	5	3ad2ant2 1076	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → ((#‘𝑊) = ((𝑁 + 1) + 1) ↔ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
7	6	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ((#‘𝑊) = ((𝑁 + 1) + 1) ↔ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
8		s1cl 13235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑆 ∈ 𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉)
9	8	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) → ⟨“𝑆”⟩ ∈ Word 𝑉)
10	9	anim2i 591	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑇 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉)) → (𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉))
11	10	ancoms 468	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉))
12		ccatlen 13213	. . . . . . . . . . . . . . . . . . 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 2612	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((#‘𝑇) + (#‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
15		s1len 13238	. . . . . . . . . . . . . . . . . . . 20 ⊢ (#‘⟨“𝑆”⟩) = 1
16	15	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (#‘⟨“𝑆”⟩) = 1)
17	16	oveq2d 6565	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘𝑇) + (#‘⟨“𝑆”⟩)) = ((#‘𝑇) + 1))
18	17	eqeq1d 2612	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((#‘𝑇) + (#‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((#‘𝑇) + 1) = ((𝑁 + 1) + 1)))
19		lencl 13179	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ Word 𝑉 → (#‘𝑇) ∈ ℕ0)
20	19	nn0cnd 11230	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇 ∈ Word 𝑉 → (#‘𝑇) ∈ ℂ)
21	20	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (#‘𝑇) ∈ ℂ)
22		peano2nn0 11210	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
23	22	nn0cnd 11230	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
24	23	ad2antrr 758	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑁 + 1) ∈ ℂ)
25		1cnd 9935	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → 1 ∈ ℂ)
26	21, 24, 25	addcan2d 10119	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((#‘𝑇) + 1) = ((𝑁 + 1) + 1) ↔ (#‘𝑇) = (𝑁 + 1)))
27	14, 18, 26	3bitrd 293	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ (#‘𝑇) = (𝑁 + 1)))
28		opeq2 4341	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 + 1) = (#‘𝑇) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (#‘𝑇)⟩)
29	28	eqcoms 2618	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑇) = (𝑁 + 1) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (#‘𝑇)⟩)
30	29	oveq2d 6565	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑇) = (𝑁 + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (#‘𝑇)⟩))
31		swrdccat1 13309	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (#‘𝑇)⟩) = 𝑇)
32	11, 31	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (#‘𝑇)⟩) = 𝑇)
33	30, 32	sylan9eqr 2666	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) ∧ (#‘𝑇) = (𝑁 + 1)) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇)
34	33	ex 449	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘𝑇) = (𝑁 + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
35	27, 34	sylbid 229	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
36	35	3ad2antr1 1219	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
37	7, 36	sylbid 229	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
38	37	imp 444	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇)
39		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩))
40	39	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
41	40	3ad2ant2 1076	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
42	41	ad2antlr 759	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
43	38, 42	mpbird 246	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇)
44	43	eleq1d 2672	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
45	44	biimpd 218	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
46	45	ex 449	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))))
47	46	com23 84	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((#‘𝑊) = ((𝑁 + 1) + 1) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))))
48	3, 47	syld 46	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ((#‘𝑊) = ((𝑁 + 1) + 1) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))))
49	48	com13 86	. . . . 5 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))))
50	49	3ad2ant2 1076	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))))
51	1, 50	mpcom 37	. . 3 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
52	51	com12 32	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
53		wwlknext 26252	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))
54		eleq1 2676	. . . . . . . . . . 11 ⊢ (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
55	53, 54	syl5ibrcom 236	. . . . . . . . . 10 ⊢ ((𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑆 ∈ 𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
56	55	3exp 1256	. . . . . . . . 9 ⊢ (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑆 ∈ 𝑉 → ({( lastS ‘𝑇), 𝑆} ∈ ran 𝐸 → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))))
57	56	com23 84	. . . . . . . 8 ⊢ (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ({( lastS ‘𝑇), 𝑆} ∈ ran 𝐸 → (𝑆 ∈ 𝑉 → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))))
58	57	com14 94	. . . . . . 7 ⊢ (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ({( lastS ‘𝑇), 𝑆} ∈ ran 𝐸 → (𝑆 ∈ 𝑉 → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))))
59	58	imp 444	. . . . . 6 ⊢ ((𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑆 ∈ 𝑉 → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
60	59	3adant1 1072	. . . . 5 ⊢ ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑆 ∈ 𝑉 → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
61	60	com12 32	. . . 4 ⊢ (𝑆 ∈ 𝑉 → ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
62	61	adantl 481	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) → ((𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
63	62	imp 444	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
64	52, 63	impbid 201	1 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {cpr 4127  ⟨cop 4131  ran crn 5039  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818  ℕ₀cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   substr csubstr 13150   WWalksN cwwlkn 26206

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-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-wwlk 26207  df-wwlkn 26208

This theorem is referenced by:  wwlkextwrd  26256