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Theorem elwwlks2 29487

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 29436	. . 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 29480	. . . 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 1118	. . 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 3215	. 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 3285	. . . 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 14836	. . . . . . . . . 10 ⊢ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V
9	8	a1i 11	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)
10		simplr 765	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
11		simpr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑝 = ⟨“𝑎𝑏𝑐”⟩)
12	10, 11	eqtr4d 2773	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑊 = 𝑝)
13	12	breq2d 5159	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(Walks‘𝐺)𝑊 ↔ 𝑓(Walks‘𝐺)𝑝))
14	13	biimpd 228	. . . . . . . . . . . . . 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 479	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 769	. . . . . . . . . . 11 ⊢ ((((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ (𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) → (♯‘𝑓) = 2)
19		vex 3476	. . . . . . . . . . . . . . . 16 ⊢ 𝑎 ∈ V
20		s3fv0 14846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎)
21	20	eqcomd 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ V → 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0))
22	19, 21	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0))
23		fveq1 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘0) = (⟨“𝑎𝑏𝑐”⟩‘0))
24	22, 23	eqtr4d 2773	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑎 = (𝑝‘0))
25		vex 3476	. . . . . . . . . . . . . . . 16 ⊢ 𝑏 ∈ V
26		s3fv1 14847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏)
27	26	eqcomd 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ V → 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1))
28	25, 27	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1))
29		fveq1 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘1) = (⟨“𝑎𝑏𝑐”⟩‘1))
30	28, 29	eqtr4d 2773	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑏 = (𝑝‘1))
31		vex 3476	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
32		s3fv2 14848	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)
33	32	eqcomd 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ V → 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))
34	31, 33	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))
35		fveq1 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘2) = (⟨“𝑎𝑏𝑐”⟩‘2))
36	34, 35	eqtr4d 2773	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑐 = (𝑝‘2))
37	24, 30, 36	3jca 1126	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))
38	37	adantl 480	. . . . . . . . . . . 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 479	. . . . . . . . . . 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 1126	. . . . . . . . . 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 411	. . . . . . . . 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 3584	. . . . . . . 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 29142	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑝) = ((♯‘𝑓) + 1))
44		simpl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2) → (♯‘𝑝) = ((♯‘𝑓) + 1))
45		oveq1 7418	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = (2 + 1))
46	45	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2) → ((♯‘𝑓) + 1) = (2 + 1))
47	44, 46	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2) → (♯‘𝑝) = (2 + 1))
48	47	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2)) → (♯‘𝑝) = (2 + 1))
49		2p1e3 12358	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (2 + 1) = 3
50	48, 49	eqtrdi 2786	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2)) → (♯‘𝑝) = 3)
51	50	exp32 419	. . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) → ((♯‘𝑓) = 2 → (♯‘𝑝) = 3))
54	53	imp 405	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (♯‘𝑓) = 2) → (♯‘𝑝) = 3)
55		eqcom 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (𝑝‘0) ↔ (𝑝‘0) = 𝑎)
56	55	biimpi 215	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝑝‘0) → (𝑝‘0) = 𝑎)
57		eqcom 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = (𝑝‘1) ↔ (𝑝‘1) = 𝑏)
58	57	biimpi 215	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = (𝑝‘1) → (𝑝‘1) = 𝑏)
59		eqcom 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = (𝑝‘2) ↔ (𝑝‘2) = 𝑐)
60	59	biimpi 215	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = (𝑝‘2) → (𝑝‘2) = 𝑐)
61	56, 58, 60	3anim123i 1149	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
62	54, 61	anim12i 611	. . . . . . . . . . . . . . . . . . 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 29141	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓(Walks‘𝐺)𝑝 → 𝑝 ∈ Word 𝑉)
64		simpr 483	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉)
65	64	anim1i 613	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
66		3anass 1093	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ↔ (𝑎 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
67	65, 66	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))
68	67	adantr 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))
69	63, 68	anim12i 611	. . . . . . . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . . . . 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 14916	. . . . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) → 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
75	74	ad2antrr 722	. . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . 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 5159	. . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . 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 525	. . . . . . . . . . . . . 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 433	. . . . . . . . . . . . 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 1109	. . . . . . . . . 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 1934	. . . . . . . 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 211	. . . . . . 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 1922	. . . . . 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 577	. . . . 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 3215	. . . 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	bitrid 282	. . 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 3174	. 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 304	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))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∃wex 1779 ∈ wcel 2104 ∃wrex 3068 Vcvv 3472 class class class wbr 5147 ‘cfv 6542 (class class class)co 7411 0cc0 11112 1c1 11113 + caddc 11115 2c2 12271 3c3 12272 ♯chash 14294 Word cword 14468 ⟨“cs3 14797 Vtxcvtx 28523 UPGraphcupgr 28607 Walkscwlks 29120 WWalksN cwwlksn 29347 WWalksNOn cwwlksnon 29348

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-hash 14295 df-word 14469 df-concat 14525 df-s1 14550 df-s2 14803 df-s3 14804 df-edg 28575 df-uhgr 28585 df-upgr 28609 df-wlks 29123 df-wwlks 29351 df-wwlksn 29352 df-wwlksnon 29353

This theorem is referenced by: (None)

W3C validator