Theorem elwwlks2 30573

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 30519	. . 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 30565	. . . 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 1138	. . 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 3229	. 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 3295	. . . 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 15046	. . . . . . . . . 10 ⊢ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V
9	8	a1i 11	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)
10		simplr 781	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
11		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑝 = ⟨“𝑎𝑏𝑐”⟩)
12	10, 11	eqtr4d 2801	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑊 = 𝑝)
13	12	breq2d 5118	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(Walks‘𝐺)𝑊 ↔ 𝑓(Walks‘𝐺)𝑝))
14	13	biimpd 232	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(Walks‘𝐺)𝑊 → 𝑓(Walks‘𝐺)𝑝))
15	14	com12 33	. . . . . . . . . . . . 13 ⊢ (𝑓(Walks‘𝐺)𝑊 → (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑓(Walks‘𝐺)𝑝))
16	15	adantr 486	. . . . . . . . . . . 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 413	. . . . . . . . . . 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 785	. . . . . . . . . . 11 ⊢ ((((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ (𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) → (♯‘𝑓) = 2)
19		vex 3458	. . . . . . . . . . . . . . . 16 ⊢ 𝑎 ∈ V
20		s3fv0 15057	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎)
21	20	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ V → 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0))
22	19, 21	mp1i 14	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0))
23		fveq1 6895	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘0) = (⟨“𝑎𝑏𝑐”⟩‘0))
24	22, 23	eqtr4d 2801	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑎 = (𝑝‘0))
25		vex 3458	. . . . . . . . . . . . . . . 16 ⊢ 𝑏 ∈ V
26		s3fv1 15058	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏)
27	26	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ V → 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1))
28	25, 27	mp1i 14	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1))
29		fveq1 6895	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘1) = (⟨“𝑎𝑏𝑐”⟩‘1))
30	28, 29	eqtr4d 2801	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑏 = (𝑝‘1))
31		vex 3458	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
32		s3fv2 15059	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)
33	32	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ V → 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))
34	31, 33	mp1i 14	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))
35		fveq1 6895	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘2) = (⟨“𝑎𝑏𝑐”⟩‘2))
36	34, 35	eqtr4d 2801	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑐 = (𝑝‘2))
37	24, 30, 36	3jca 1146	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))
38	37	adantl 487	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 1146	. . . . . . . . . 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 418	. . . . . . . . 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 3553	. . . . . . . 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 30214	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑝) = ((♯‘𝑓) + 1))
44		simpl 488	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2) → (♯‘𝑝) = ((♯‘𝑓) + 1))
45		oveq1 7437	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = (2 + 1))
46	45	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2) → ((♯‘𝑓) + 1) = (2 + 1))
47	44, 46	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2) → (♯‘𝑝) = (2 + 1))
48	47	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2)) → (♯‘𝑝) = (2 + 1))
49		2p1e3 12497	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (2 + 1) = 3
50	48, 49	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑝) = ((♯‘𝑓) + 1) ∧ (♯‘𝑓) = 2)) → (♯‘𝑝) = 3)
51	50	exp32 426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓(Walks‘𝐺)𝑝 → ((♯‘𝑝) = ((♯‘𝑓) + 1) → ((♯‘𝑓) = 2 → (♯‘𝑝) = 3)))
52	43, 51	mpd 16	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓(Walks‘𝐺)𝑝 → ((♯‘𝑓) = 2 → (♯‘𝑝) = 3))
53	52	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) → ((♯‘𝑓) = 2 → (♯‘𝑝) = 3))
54	53	imp 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (♯‘𝑓) = 2) → (♯‘𝑝) = 3)
55		eqcom 2770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (𝑝‘0) ↔ (𝑝‘0) = 𝑎)
56	55	biimpi 219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝑝‘0) → (𝑝‘0) = 𝑎)
57		eqcom 2770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = (𝑝‘1) ↔ (𝑝‘1) = 𝑏)
58	57	biimpi 219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = (𝑝‘1) → (𝑝‘1) = 𝑏)
59		eqcom 2770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = (𝑝‘2) ↔ (𝑝‘2) = 𝑐)
60	59	biimpi 219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = (𝑝‘2) → (𝑝‘2) = 𝑐)
61	56, 58, 60	3anim123i 1169	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
62	54, 61	anim12i 625	. . . . . . . . . . . . . . . . . . 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 30213	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓(Walks‘𝐺)𝑝 → 𝑝 ∈ Word 𝑉)
64		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉)
65	64	anim1i 627	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
66		3anass 1111	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ↔ (𝑎 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
67	65, 66	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))
68	67	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))
69	63, 68	anim12i 625	. . . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . 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 15129	. . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) → 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
75	74	ad2antrr 739	. . . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . . 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 5118	. . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . 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 536	. . . . . . . . . . . . . 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 440	. . . . . . . . . . . . 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 100	. . . . . . . . . . . 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 53	. . . . . . . . . . 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 1128	. . . . . . . . . 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 33	. . . . . . . . 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 1967	. . . . . . . 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 215	. . . . . . 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 1955	. . . . . 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 590	. . . . 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 3229	. . . 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 286	. . 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 3188	. 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 308	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))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2146  ∃wrex 3090  Vcvv 3454   class class class wbr 5106  ‘cfv 6547  (class class class)co 7430  0cc0 11213  1c1 11214   + caddc 11216  2c2 12410  3c3 12411  ♯chash 14487  Word cword 14671  ⟨“cs3 15007  Vtxcvtx 29589  UPGraphcupgr 29673  Walkscwlks 30192   WWalksN cwwlksn 30430   WWalksNOn cwwlksnon 30431

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-ac2 10554  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-isom 6556  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-2o 8491  df-oadd 8494  df-er 8731  df-map 8863  df-pm 8864  df-en 8988  df-dom 8989  df-sdom 8990  df-fin 8991  df-dju 9995  df-card 10033  df-ac 10208  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-nn 12349  df-2 12418  df-3 12419  df-n0 12620  df-xnn0 12693  df-z 12707  df-uz 12979  df-fz 13653  df-fzo 13802  df-hash 14488  df-word 14672  df-concat 14729  df-s1 14756  df-s2 15013  df-s3 15014  df-edg 29641  df-uhgr 29651  df-upgr 29675  df-wlks 30195  df-wwlks 30434  df-wwlksn 30435  df-wwlksnon 30436

This theorem is used by: (None)