Theorem elwwlks2 27302

Description: A walk of length 2 between two vertices as length 3 string in a pseudograph. (Contributed by Alexander van der Vekens, 21-Feb-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 14-Mar-2022.)

Hypothesis

Ref	Expression
elwwlks2.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
elwwlks2	⊢ (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WWalksN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))

Distinct variable groups:   𝐺,𝑎,𝑏,𝑐,𝑓,𝑝   𝑉,𝑎,𝑏,𝑐,𝑓,𝑝   𝑊,𝑎,𝑏,𝑐,𝑓,𝑝

Proof of Theorem elwwlks2

Step	Hyp	Ref	Expression
1		elwwlks2.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	wwlksnwwlksnon 27251	. . 3 ⊢ (𝑊 ∈ (2 WWalksN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 ∈ (𝑎(2 WWalksNOn 𝐺)𝑐))
3	2	a1i 11	. 2 ⊢ (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WWalksN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)))
4	1	elwwlks2on 27295	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝑊 ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2))))
5	4	3expb 1153	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑊 ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2))))
6	5	2rexbidva 3266	. 2 ⊢ (𝐺 ∈ UPGraph → (∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ ∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2))))
7		rexcom 3309	. . . 4 ⊢ (∃𝑐 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) ↔ ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)))
8		s3cli 14009	. . . . . . . . . 10 ⊢ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V
9	8	a1i 11	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)
10		simplr 785	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
11		simpr 479	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑝 = ⟨“𝑎𝑏𝑐”⟩)
12	10, 11	eqtr4d 2864	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑊 = 𝑝)
13	12	breq2d 4887	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(Walks‘𝐺)𝑊 ↔ 𝑓(Walks‘𝐺)𝑝))
14	13	biimpd 221	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(Walks‘𝐺)𝑊 → 𝑓(Walks‘𝐺)𝑝))
15	14	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑓(Walks‘𝐺)𝑊 → (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑓(Walks‘𝐺)𝑝))
16	15	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2) → (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑓(Walks‘𝐺)𝑝))
17	16	impcom 398	. . . . . . . . . . 11 ⊢ ((((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ (𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) → 𝑓(Walks‘𝐺)𝑝)
18		simprr 789	. . . . . . . . . . 11 ⊢ ((((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ (𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) → (♯‘𝑓) = 2)
19		vex 3417	. . . . . . . . . . . . . . . 16 ⊢ 𝑎 ∈ V
20		s3fv0 14019	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎)
21	20	eqcomd 2831	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ V → 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0))
22	19, 21	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0))
23		fveq1 6436	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘0) = (⟨“𝑎𝑏𝑐”⟩‘0))
24	22, 23	eqtr4d 2864	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑎 = (𝑝‘0))
25		vex 3417	. . . . . . . . . . . . . . . 16 ⊢ 𝑏 ∈ V
26		s3fv1 14020	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏)
27	26	eqcomd 2831	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ V → 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1))
28	25, 27	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1))
29		fveq1 6436	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘1) = (⟨“𝑎𝑏𝑐”⟩‘1))
30	28, 29	eqtr4d 2864	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑏 = (𝑝‘1))
31		vex 3417	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
32		s3fv2 14021	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)
33	32	eqcomd 2831	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ V → 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))
34	31, 33	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))
35		fveq1 6436	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘2) = (⟨“𝑎𝑏𝑐”⟩‘2))
36	34, 35	eqtr4d 2864	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑐 = (𝑝‘2))
37	24, 30, 36	3jca 1162	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))
38	37	adantl 475	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))
39	38	adantr 474	. . . . . . . . . . 11 ⊢ ((((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ (𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) → (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))
40	17, 18, 39	3jca 1162	. . . . . . . . . 10 ⊢ ((((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ (𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) → (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))
41	40	ex 403	. . . . . . . . 9 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2) → (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
42	9, 41	spcimedv 3509	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2) → ∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
43		wlklenvp1 26923	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑝) = ((♯‘𝑓) + 1))
44		simpl 476	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2) → (♯‘𝑝) = ((♯‘𝑓) + 1))
45		oveq1 6917	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = (2 + 1))
46	45	adantl 475	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2) → ((♯‘𝑓) + 1) = (2 + 1))
47	44, 46	eqtrd 2861	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2) → (♯‘𝑝) = (2 + 1))
48	47	adantl 475	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2)) → (♯‘𝑝) = (2 + 1))
49		2p1e3 11507	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (2 + 1) = 3
50	48, 49	syl6eq 2877	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2)) → (♯‘𝑝) = 3)
51	50	exp32 413	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓(Walks‘𝐺)𝑝 → ((♯‘𝑝) = ((♯‘𝑓) + 1) → ((♯‘𝑓) = 2 → (♯‘𝑝) = 3)))
52	43, 51	mpd 15	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓(Walks‘𝐺)𝑝 → ((♯‘𝑓) = 2 → (♯‘𝑝) = 3))
53	52	adantr 474	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) → ((♯‘𝑓) = 2 → (♯‘𝑝) = 3))
54	53	imp 397	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (♯‘𝑓) = 2) → (♯‘𝑝) = 3)
55		eqcom 2832	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (𝑝‘0) ↔ (𝑝‘0) = 𝑎)
56	55	biimpi 208	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝑝‘0) → (𝑝‘0) = 𝑎)
57		eqcom 2832	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = (𝑝‘1) ↔ (𝑝‘1) = 𝑏)
58	57	biimpi 208	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = (𝑝‘1) → (𝑝‘1) = 𝑏)
59		eqcom 2832	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = (𝑝‘2) ↔ (𝑝‘2) = 𝑐)
60	59	biimpi 208	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = (𝑝‘2) → (𝑝‘2) = 𝑐)
61	56, 58, 60	3anim123i 1194	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
62	54, 61	anim12i 606	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (♯‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐)))
63	1	wlkpwrd 26922	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓(Walks‘𝐺)𝑝 → 𝑝 ∈ Word 𝑉)
64		simpr 479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉)
65	64	anim1i 608	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
66		3anass 1120	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ↔ (𝑎 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
67	65, 66	sylibr 226	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))
68	67	adantr 474	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))
69	63, 68	anim12i 606	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) → (𝑝 ∈ Word 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
70	69	ad2antrr 717	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (♯‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑝 ∈ Word 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
71		eqwrds3 14090	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 ∈ Word 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))))
72	70, 71	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (♯‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))))
73	62, 72	mpbird 249	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (♯‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → 𝑝 = ⟨“𝑎𝑏𝑐”⟩)
74		simprr 789	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) → 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
75	74	ad2antrr 717	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (♯‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
76	73, 75	eqtr4d 2864	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (♯‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → 𝑝 = 𝑊)
77	76	breq2d 4887	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (♯‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑓(Walks‘𝐺)𝑝 ↔ 𝑓(Walks‘𝐺)𝑊))
78	77	biimpd 221	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (♯‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑓(Walks‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑊))
79		simplr 785	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (♯‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (♯‘𝑓) = 2)
80	78, 79	jctird 522	. . . . . . . . . . . . . 14 ⊢ ((((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (♯‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑓(Walks‘𝐺)𝑝 → (𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)))
81	80	exp41 427	. . . . . . . . . . . . 13 ⊢ (𝑓(Walks‘𝐺)𝑝 → ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((♯‘𝑓) = 2 → ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → (𝑓(Walks‘𝐺)𝑝 → (𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2))))))
82	81	com25 99	. . . . . . . . . . . 12 ⊢ (𝑓(Walks‘𝐺)𝑝 → (𝑓(Walks‘𝐺)𝑝 → ((♯‘𝑓) = 2 → ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2))))))
83	82	pm2.43i 52	. . . . . . . . . . 11 ⊢ (𝑓(Walks‘𝐺)𝑝 → ((♯‘𝑓) = 2 → ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)))))
84	83	3imp 1141	. . . . . . . . . 10 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)))
85	84	com12 32	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)))
86	85	exlimdv 2032	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)))
87	42, 86	impbid 204	. . . . . . 7 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2) ↔ ∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
88	87	exbidv 2020	. . . . . 6 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2) ↔ ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
89	88	pm5.32da 574	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → ((𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) ↔ (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
90	89	2rexbidva 3266	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) → (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) ↔ ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
91	7, 90	syl5bb 275	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) → (∃𝑐 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) ↔ ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
92	91	rexbidva 3259	. 2 ⊢ (𝐺 ∈ UPGraph → (∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
93	3, 6, 92	3bitrd 297	1 ⊢ (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WWalksN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656  ∃wex 1878   ∈ wcel 2164  ∃wrex 3118  Vcvv 3414   class class class wbr 4875  ‘cfv 6127  (class class class)co 6910  0cc0 10259  1c1 10260   + caddc 10262  2c2 11413  3c3 11414  ♯chash 13417  Word cword 13581  ⟨“cs3 13970  Vtxcvtx 26301  UPGraphcupgr 26385  Walkscwlks 26901   WWalksN cwwlksn 27132   WWalksNOn cwwlksnon 27133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-ac2 9607  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ifp 1090  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-ac 9259  df-cda 9312  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-3 11422  df-n0 11626  df-xnn0 11698  df-z 11712  df-uz 11976  df-fz 12627  df-fzo 12768  df-hash 13418  df-word 13582  df-concat 13638  df-s1 13663  df-s2 13976  df-s3 13977  df-edg 26353  df-uhgr 26363  df-upgr 26387  df-wlks 26904  df-wwlks 27136  df-wwlksn 27137  df-wwlksnon 27138

This theorem is referenced by: (None)