Theorem frg2woteq 26587

Description: There is a (simple) path of length 2 from one vertex to another vertex in a friendship graph if and only if there is a (simple) path of length 2 from the other vertex back to the first vertex. (Contributed by Alexander van der Vekens, 14-Feb-2018.)

Assertion

Ref	Expression
frg2woteq	⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))

Proof of Theorem frg2woteq

Dummy variables 𝑐 𝑑 𝑓 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2wlkonot3v 26402	. . . 4 ⊢ (𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)))
2	1	adantr 480	. . 3 ⊢ ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)))
3		el2wlkonot 26396	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ∃𝑐 ∈ 𝑉 (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))))))
4		pm3.22 464	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
5	4	anim2i 591	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
6		el2wlkonot 26396	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)) → (𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) ↔ ∃𝑑 ∈ 𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2))))))
7	5, 6	syl 17	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) ↔ ∃𝑑 ∈ 𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2))))))
8	3, 7	anbi12d 743	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) ↔ (∃𝑐 ∈ 𝑉 (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) ∧ ∃𝑑 ∈ 𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))))))
9	8	3adant3 1074	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) ↔ (∃𝑐 ∈ 𝑉 (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) ∧ ∃𝑑 ∈ 𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))))))
10	5	3adant3 1074	. . . . . . . . . . . . 13 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
11	10	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
12	11	ad2antrr 758	. . . . . . . . . . 11 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
13		el2wlkonotot 26400	. . . . . . . . . . . 12 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) ↔ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))))
14	13	bicomd 212	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)) → (∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2))) ↔ ⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)))
15	12, 14	syl 17	. . . . . . . . . 10 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) → (∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2))) ↔ ⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)))
16		3simpa 1051	. . . . . . . . . . . . . . . 16 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))
17	16	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))
18	17	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))
19		el2wlkonotot 26400	. . . . . . . . . . . . . . 15 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))))
20	19	bicomd 212	. . . . . . . . . . . . . 14 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))) ↔ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
21	18, 20	syl 17	. . . . . . . . . . . . 13 ⊢ (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))) ↔ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
22		frg2woteqm 26586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → 𝑑 = 𝑐))
23		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ → (1st ‘𝑃) = (1st ‘⟨𝐴, 𝑐, 𝐵⟩))
24	23	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ → (1st ‘(1st ‘𝑃)) = (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)))
25	24	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (1st ‘(1st ‘𝑃)) = (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)))
26	25	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (1st ‘(1st ‘𝑃)) = (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)))
27		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑐 ∈ V
28		ot1stg 7073	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑐 ∈ V ∧ 𝐵 ∈ 𝑉) → (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)) = 𝐴)
29	27, 28	mp3an2 1404	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)) = 𝐴)
30	29	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)) = 𝐴)
31	30	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (1st ‘(1st ‘⟨𝐴, 𝑐, 𝐵⟩)) = 𝐴)
32		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ → (2nd ‘𝑄) = (2nd ‘⟨𝐵, 𝑑, 𝐴⟩))
33	32	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (2nd ‘𝑄) = (2nd ‘⟨𝐵, 𝑑, 𝐴⟩))
34	33	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (2nd ‘𝑄) = (2nd ‘⟨𝐵, 𝑑, 𝐴⟩))
35		ot3rdg 7075	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐴 ∈ 𝑉 → (2nd ‘⟨𝐵, 𝑑, 𝐴⟩) = 𝐴)
36	35	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (2nd ‘⟨𝐵, 𝑑, 𝐴⟩) = 𝐴)
37	36	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (2nd ‘⟨𝐵, 𝑑, 𝐴⟩) = 𝐴)
38	37	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (2nd ‘⟨𝐵, 𝑑, 𝐴⟩) = 𝐴)
39	34, 38	eqtr2d 2645	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → 𝐴 = (2nd ‘𝑄))
40	26, 31, 39	3eqtrd 2648	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (1st ‘(1st ‘𝑃)) = (2nd ‘𝑄))
41		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)))
42		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ → (1st ‘𝑄) = (1st ‘⟨𝐵, 𝑑, 𝐴⟩))
43	42	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ → (1st ‘(1st ‘𝑄)) = (1st ‘(1st ‘⟨𝐵, 𝑑, 𝐴⟩)))
44	43	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (1st ‘(1st ‘𝑄)) = (1st ‘(1st ‘⟨𝐵, 𝑑, 𝐴⟩)))
45	44	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (1st ‘(1st ‘𝑄)) = (1st ‘(1st ‘⟨𝐵, 𝑑, 𝐴⟩)))
46		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
47		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑑 ∈ V
48	47	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑑 ∈ V)
49		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
50	46, 48, 49	3jca 1235	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ 𝑉 ∧ 𝑑 ∈ V ∧ 𝐴 ∈ 𝑉))
51	50	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (𝐵 ∈ 𝑉 ∧ 𝑑 ∈ V ∧ 𝐴 ∈ 𝑉))
52	51	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (𝐵 ∈ 𝑉 ∧ 𝑑 ∈ V ∧ 𝐴 ∈ 𝑉))
53		ot1stg 7073	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑑 ∈ V ∧ 𝐴 ∈ 𝑉) → (1st ‘(1st ‘⟨𝐵, 𝑑, 𝐴⟩)) = 𝐵)
54	52, 53	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (1st ‘(1st ‘⟨𝐵, 𝑑, 𝐴⟩)) = 𝐵)
55		ot3rdg 7075	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐵 ∈ 𝑉 → (2nd ‘⟨𝐴, 𝑐, 𝐵⟩) = 𝐵)
56	55	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (2nd ‘⟨𝐴, 𝑐, 𝐵⟩) = 𝐵)
57	56	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (2nd ‘⟨𝐴, 𝑐, 𝐵⟩) = 𝐵)
58	57	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (2nd ‘⟨𝐴, 𝑐, 𝐵⟩) = 𝐵)
59		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)
60	59	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)
61	60	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → ⟨𝐴, 𝑐, 𝐵⟩ = 𝑃)
62	61	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (2nd ‘⟨𝐴, 𝑐, 𝐵⟩) = (2nd ‘𝑃))
63	58, 62	eqtr3d 2646	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → 𝐵 = (2nd ‘𝑃))
64	45, 54, 63	3eqtrd 2648	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))
65	40, 41, 64	3jca 1235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑑 = 𝑐 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩)) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃)))
66	65	ex 449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 = 𝑐 → ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))
67	22, 66	syl6 34	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃)))))
68	67	expd 451	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))
69	68	com14 94	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))
70	69	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) → (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃)))))))
71	70	ex 449	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ → (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))))
72	71	3ad2ant2 1076	. . . . . . . . . . . . . . 15 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ → (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))))
73	72	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ → (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))))
74	73	imp31 447	. . . . . . . . . . . . 13 ⊢ (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))
75	21, 74	sylbid 229	. . . . . . . . . . . 12 ⊢ (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) ∧ 𝑃 = ⟨𝐴, 𝑐, 𝐵⟩) → (∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))
76	75	expimpd 627	. . . . . . . . . . 11 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) → ((𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))
77	76	com23 84	. . . . . . . . . 10 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) → (⟨𝐵, 𝑑, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴) → ((𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))
78	15, 77	sylbid 229	. . . . . . . . 9 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑄 = ⟨𝐵, 𝑑, 𝐴⟩) → (∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2))) → ((𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))
79	78	expimpd 627	. . . . . . . 8 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → ((𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))) → ((𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))
80	79	rexlimdva 3013	. . . . . . 7 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) → (∃𝑑 ∈ 𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))) → ((𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))
81	80	com23 84	. . . . . 6 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → (∃𝑑 ∈ 𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))
82	81	rexlimdva 3013	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → (∃𝑐 ∈ 𝑉 (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → (∃𝑑 ∈ 𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2)))) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))))
83	82	impd 446	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((∃𝑐 ∈ 𝑉 (𝑃 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) ∧ ∃𝑑 ∈ 𝑉 (𝑄 = ⟨𝐵, 𝑑, 𝐴⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐵 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐴 = (𝑝‘2))))) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃)))))
84	9, 83	sylbid 229	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑃 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃)))))
85	2, 84	mpcom 37	. 2 ⊢ ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))
86	85	com12 32	1 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝐴 ≠ 𝐵) → ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((1st ‘(1st ‘𝑃)) = (2nd ‘𝑄) ∧ (2nd ‘(1st ‘𝑃)) = (2nd ‘(1st ‘𝑃)) ∧ (1st ‘(1st ‘𝑄)) = (2nd ‘𝑃))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173  ⟨cotp 4133   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   Walks cwalk 26026   2WalksOnOt c2wlkonot 26382   FriendGrph cfrgra 26515

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-rmo 2904  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-ot 4134  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-cda 8873  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-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wlkon 26042  df-2wlkonot 26385  df-frgra 26516

This theorem is referenced by: (None)