Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgr2wwlkeqm Structured version   Visualization version   GIF version

Theorem frgr2wwlkeqm 27311
 Description: If there is a (simple) path of length 2 from one vertex to another vertex and a (simple) path of length 2 from the other vertex back to the first vertex in a friendship graph, then the middle vertex is the same. This is only an observation, which is not required to proof the friendship theorem. (Contributed by Alexander van der Vekens, 20-Feb-2018.) (Revised by AV, 13-May-2021.) (Proof shortened by AV, 7-Jan-2022.)
Assertion
Ref Expression
frgr2wwlkeqm ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃))

Proof of Theorem frgr2wwlkeqm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp3l 1109 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → 𝑃𝑋)
2 eqid 2651 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
32wwlks2onv 26918 . . . 4 ((𝑃𝑋 ∧ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
41, 3sylan 487 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) ∧ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
5 simp3r 1110 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → 𝑄𝑌)
62wwlks2onv 26918 . . . . . . . 8 ((𝑄𝑌 ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
75, 6sylan 487 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
8 frgrusgr 27240 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
9 usgrumgr 26119 . . . . . . . . . . . . . . 15 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
108, 9syl 17 . . . . . . . . . . . . . 14 (𝐺 ∈ FriendGraph → 𝐺 ∈ UMGraph)
11103ad2ant1 1102 . . . . . . . . . . . . 13 ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → 𝐺 ∈ UMGraph)
12 simpr3 1089 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝐵 ∈ (Vtx‘𝐺))
13 simpl 472 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝑄 ∈ (Vtx‘𝐺))
14 simpr1 1087 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝐴 ∈ (Vtx‘𝐺))
1512, 13, 143jca 1261 . . . . . . . . . . . . 13 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
162wwlks2onsym 26924 . . . . . . . . . . . . 13 ((𝐺 ∈ UMGraph ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) ↔ ⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
1711, 15, 16syl2anr 494 . . . . . . . . . . . 12 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) ↔ ⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
18 simpr1 1087 . . . . . . . . . . . . . 14 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → 𝐺 ∈ FriendGraph )
19 3simpb 1079 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
2019ad2antlr 763 . . . . . . . . . . . . . 14 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
21 simpr2 1088 . . . . . . . . . . . . . 14 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → 𝐴𝐵)
222frgr2wwlkeu 27307 . . . . . . . . . . . . . 14 ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ 𝐴𝐵) → ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
2318, 20, 21, 22syl3anc 1366 . . . . . . . . . . . . 13 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
24 s3eq2 13661 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑄 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑄𝐵”⟩)
2524eleq1d 2715 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑄 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
2625riota2 6673 . . . . . . . . . . . . . . 15 ((𝑄 ∈ (Vtx‘𝐺) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄))
2726ad4ant14 1317 . . . . . . . . . . . . . 14 ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄))
28 simplr2 1124 . . . . . . . . . . . . . . . . 17 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → 𝑃 ∈ (Vtx‘𝐺))
29 s3eq2 13661 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑃 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑃𝐵”⟩)
3029eleq1d 2715 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑃 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
3130riota2 6673 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ (Vtx‘𝐺) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑃))
3228, 31sylan 487 . . . . . . . . . . . . . . . 16 ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑃))
33 eqtr2 2671 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄 ∧ (𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑃) → 𝑄 = 𝑃)
3433expcom 450 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑃 → ((𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄𝑄 = 𝑃))
3532, 34syl6bi 243 . . . . . . . . . . . . . . 15 ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → ((𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄𝑄 = 𝑃)))
3635com23 86 . . . . . . . . . . . . . 14 ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄 → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
3727, 36sylbid 230 . . . . . . . . . . . . 13 ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
3823, 37mpdan 703 . . . . . . . . . . . 12 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → (⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
3917, 38sylbid 230 . . . . . . . . . . 11 (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
4039expimpd 628 . . . . . . . . . 10 ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → (((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
4140ex 449 . . . . . . . . 9 (𝑄 ∈ (Vtx‘𝐺) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
4241com23 86 . . . . . . . 8 (𝑄 ∈ (Vtx‘𝐺) → (((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
43423ad2ant2 1103 . . . . . . 7 ((𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) → (((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
447, 43mpcom 38 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
4544ex 449 . . . . 5 ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
4645com24 95 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → 𝑄 = 𝑃))))
4746imp 444 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) ∧ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → 𝑄 = 𝑃)))
484, 47mpd 15 . 2 (((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) ∧ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → 𝑄 = 𝑃))
4948expimpd 628 1 ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∃!wreu 2943  ‘cfv 5926  ℩crio 6650  (class class class)co 6690  2c2 11108  ⟨“cs3 13633  Vtxcvtx 25919  UMGraphcumgr 26021  USGraphcusgr 26089   WWalksNOn cwwlksnon 26775   FriendGraph cfrgr 27236 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-ac2 9323  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-ac 8977  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-edg 25985  df-uhgr 25998  df-upgr 26022  df-umgr 26023  df-usgr 26091  df-wlks 26551  df-wwlks 26778  df-wwlksn 26779  df-wwlksnon 26780  df-frgr 27237 This theorem is referenced by: (None)
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