Theorem elwwlks2ons3 26848

Description: For each walk of length 2 between two vertices, there is a third vertex in the middle of the walk. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.)

Hypothesis

Ref	Expression
elwwlks2ons3.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
elwwlks2ons3	⊢ ((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))

Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐺,𝑏   𝑈,𝑏   𝑉,𝑏   𝑊,𝑏

Proof of Theorem elwwlks2ons3

Step	Hyp	Ref	Expression
1		simpr 477	. . . . 5 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
2		elwwlks2ons3.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	wwlknon 26742	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ (𝑊 ∈ (2 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶)))
4	3	3adant1 1079	. . . . . . 7 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ (𝑊 ∈ (2 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶)))
5		wwlknbp2 26752	. . . . . . . . . 10 ⊢ (𝑊 ∈ (2 WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (2 + 1)))
6		2p1e3 11151	. . . . . . . . . . . 12 ⊢ (2 + 1) = 3
7	6	eqeq2i 2634	. . . . . . . . . . 11 ⊢ ((#‘𝑊) = (2 + 1) ↔ (#‘𝑊) = 3)
8		1ex 10035	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
9	8	tpid2 4304	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ {0, 1, 2}
10		oveq2 6658	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑊) = 3 → (0..^(#‘𝑊)) = (0..^3))
11		fzo0to3tp 12554	. . . . . . . . . . . . . . . . 17 ⊢ (0..^3) = {0, 1, 2}
12	10, 11	syl6eq 2672	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) = 3 → (0..^(#‘𝑊)) = {0, 1, 2})
13	9, 12	syl5eleqr 2708	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑊) = 3 → 1 ∈ (0..^(#‘𝑊)))
14		wrdsymbcl 13318	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ∈ (0..^(#‘𝑊))) → (𝑊‘1) ∈ (Vtx‘𝐺))
15	13, 14	sylan2 491	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) → (𝑊‘1) ∈ (Vtx‘𝐺))
16	15	3ad2ant1 1082	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑊‘1) ∈ (Vtx‘𝐺))
17		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) → (#‘𝑊) = 3)
18	17	3ad2ant1 1082	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (#‘𝑊) = 3)
19	18	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (#‘𝑊) = 3)
20		simpl 473	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊‘0) = 𝐴)
21		eqidd 2623	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊‘1) = (𝑊‘1))
22		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊‘2) = 𝐶)
23	20, 21, 22	3jca 1242	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))
24	23	3ad2ant2 1083	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))
25	24	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))
26	2	eqcomi 2631	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Vtx‘𝐺) = 𝑉
27	26	wrdeqi 13328	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Word (Vtx‘𝐺) = Word 𝑉
28	27	eleq2i 2693	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) ↔ 𝑊 ∈ Word 𝑉)
29	28	biimpi 206	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → 𝑊 ∈ Word 𝑉)
30	29	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) → 𝑊 ∈ Word 𝑉)
31	30	3ad2ant1 1082	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑊 ∈ Word 𝑉)
32	31	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝑊 ∈ Word 𝑉)
33		simpl32 1143	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝐴 ∈ 𝑉)
34	26	eleq2i 2693	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊‘1) ∈ (Vtx‘𝐺) ↔ (𝑊‘1) ∈ 𝑉)
35	34	biimpi 206	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊‘1) ∈ (Vtx‘𝐺) → (𝑊‘1) ∈ 𝑉)
36	35	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (𝑊‘1) ∈ 𝑉)
37		simpl33 1144	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝐶 ∈ 𝑉)
38		eqwrds3 13704	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ (𝑊‘1) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ↔ ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))))
39	32, 33, 36, 37, 38	syl13anc 1328	. . . . . . . . . . . . . . 15 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ↔ ((#‘𝑊) = 3 ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))))
40	19, 25, 39	mpbir2and 957	. . . . . . . . . . . . . 14 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩)
41	40, 36	jca 554	. . . . . . . . . . . . 13 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
42	16, 41	mpdan 702	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
43	42	3exp 1264	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 3) → (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))))
44	7, 43	sylan2b 492	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (2 + 1)) → (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))))
45	5, 44	syl 17	. . . . . . . . 9 ⊢ (𝑊 ∈ (2 WWalksN 𝐺) → (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))))
46	45	3impib 1262	. . . . . . . 8 ⊢ ((𝑊 ∈ (2 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉)))
47	46	com12 32	. . . . . . 7 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝑊 ∈ (2 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉)))
48	4, 47	sylbid 230	. . . . . 6 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉)))
49	48	imp 445	. . . . 5 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
50		anass 681	. . . . 5 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) ↔ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉)))
51	1, 49, 50	sylanbrc 698	. . . 4 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉))
52		simpr 477	. . . . 5 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊‘1) ∈ 𝑉)
53		eqidd 2623	. . . . . . . 8 ⊢ (𝑏 = (𝑊‘1) → 𝐴 = 𝐴)
54		id 22	. . . . . . . 8 ⊢ (𝑏 = (𝑊‘1) → 𝑏 = (𝑊‘1))
55		eqidd 2623	. . . . . . . 8 ⊢ (𝑏 = (𝑊‘1) → 𝐶 = 𝐶)
56	53, 54, 55	s3eqd 13609	. . . . . . 7 ⊢ (𝑏 = (𝑊‘1) → ⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩)
57		eqeq2 2633	. . . . . . . 8 ⊢ (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ↔ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩))
58		eleq1 2689	. . . . . . . 8 ⊢ (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
59	57, 58	anbi12d 747	. . . . . . 7 ⊢ (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
60	56, 59	syl 17	. . . . . 6 ⊢ (𝑏 = (𝑊‘1) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
61	60	adantl 482	. . . . 5 ⊢ ((((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) ∧ 𝑏 = (𝑊‘1)) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
62		simpr 477	. . . . . . 7 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩)
63		eleq1 2689	. . . . . . . 8 ⊢ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
64	63	biimpac 503	. . . . . . 7 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
65	62, 64	jca 554	. . . . . 6 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
66	65	adantr 481	. . . . 5 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
67	52, 61, 66	rspcedvd 3317	. . . 4 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
68	51, 67	syl 17	. . 3 ⊢ (((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
69	68	ex 450	. 2 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
70		eleq1 2689	. . . . . 6 ⊢ (⟨“𝐴𝑏𝐶”⟩ = 𝑊 → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
71	70	eqcoms 2630	. . . . 5 ⊢ (𝑊 = ⟨“𝐴𝑏𝐶”⟩ → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
72	71	biimpa 501	. . . 4 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
73	72	a1i 11	. . 3 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
74	73	rexlimdvw 3034	. 2 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
75	69, 74	impbid 202	1 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  {ctp 4181  ‘cfv 5888  (class class class)co 6650  0cc0 9936  1c1 9937   + caddc 9939  2c2 11070  3c3 11071  ..^cfzo 12465  #chash 13117  Word cword 13291  ⟨“cs3 13587  Vtxcvtx 25874   WWalksN cwwlksn 26718   WWalksNOn cwwlksnon 26719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724

This theorem is referenced by:  elwwlks2on  26852  frgr2wwlk1  27193