Theorem frgr2wwlkeqm 27195

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 e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) -> ( ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) ) -> Q = P ) )

Proof of Theorem frgr2wwlkeqm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3l 1089	. . . 4 |- ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) -> P e. X )
2		eqid 2622	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
3	2	wwlks2onv 26847	. . . 4 |- ( ( P e. X /\ <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) )
4	1, 3	sylan 488	. . 3 |- ( ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) /\ <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) )
5		simp3r 1090	. . . . . . . 8 |- ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) -> Q e. Y )
6	2	wwlks2onv 26847	. . . . . . . 8 |- ( ( Q e. Y /\ <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) ) -> ( B e. (Vtx ` G ) /\ Q e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) )
7	5, 6	sylan 488	. . . . . . 7 |- ( ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) /\ <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) ) -> ( B e. (Vtx ` G ) /\ Q e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) )
8		frgrusgr 27124	. . . . . . . . . . . . . . 15 |- ( G e. FriendGraph -> G e. USGraph )
9		usgrumgr 26074	. . . . . . . . . . . . . . 15 |- ( G e. USGraph -> G e. UMGraph )
10	8, 9	syl 17	. . . . . . . . . . . . . 14 |- ( G e. FriendGraph -> G e. UMGraph )
11	10	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) -> G e. UMGraph )
12		simpr3 1069	. . . . . . . . . . . . . 14 |- ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) -> B e. (Vtx ` G ) )
13		simpl 473	. . . . . . . . . . . . . 14 |- ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) -> Q e. (Vtx ` G ) )
14		simpr1 1067	. . . . . . . . . . . . . 14 |- ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) -> A e. (Vtx ` G ) )
15	12, 13, 14	3jca 1242	. . . . . . . . . . . . 13 |- ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) -> ( B e. (Vtx ` G ) /\ Q e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) )
16	2	wwlks2onsym 26851	. . . . . . . . . . . . 13 |- ( ( G e. UMGraph /\ ( B e. (Vtx ` G ) /\ Q e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) ) -> ( <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) <-> <" A Q B "> e. ( A ( 2 WWalksNOn G ) B ) ) )
17	11, 15, 16	syl2anr 495	. . . . . . . . . . . 12 |- ( ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) /\ ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) ) -> ( <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) <-> <" A Q B "> e. ( A ( 2 WWalksNOn G ) B ) ) )
18		simpr1 1067	. . . . . . . . . . . . . 14 |- ( ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) /\ ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) ) -> G e. FriendGraph )
19		3simpb 1059	. . . . . . . . . . . . . . 15 |- ( ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) -> ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) )
20	19	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) /\ ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) ) -> ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) )
21		simpr2 1068	. . . . . . . . . . . . . 14 |- ( ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) /\ ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) ) -> A =/= B )
22	2	frgr2wwlkeu 27191	. . . . . . . . . . . . . 14 |- ( ( G e. FriendGraph /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ A =/= B ) -> E! x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) )
23	18, 20, 21, 22	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) /\ ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) ) -> E! x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) )
24		s3eq2 13615	. . . . . . . . . . . . . . . . 17 |- ( x = Q -> <" A x B "> = <" A Q B "> )
25	24	eleq1d 2686	. . . . . . . . . . . . . . . 16 |- ( x = Q -> ( <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) <-> <" A Q B "> e. ( A ( 2 WWalksNOn G ) B ) ) )
26	25	riota2 6633	. . . . . . . . . . . . . . 15 |- ( ( Q e. (Vtx ` G ) /\ E! x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( <" A Q B "> e. ( A ( 2 WWalksNOn G ) B ) <-> ( iota_ x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) = Q ) )
27	26	ad4ant14 1293	. . . . . . . . . . . . . 14 |- ( ( ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) /\ ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) ) /\ E! x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( <" A Q B "> e. ( A ( 2 WWalksNOn G ) B ) <-> ( iota_ x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) = Q ) )
28		simplr2 1104	. . . . . . . . . . . . . . . . 17 |- ( ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) /\ ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) ) -> P e. (Vtx ` G ) )
29		s3eq2 13615	. . . . . . . . . . . . . . . . . . 19 |- ( x = P -> <" A x B "> = <" A P B "> )
30	29	eleq1d 2686	. . . . . . . . . . . . . . . . . 18 |- ( x = P -> ( <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) <-> <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) ) )
31	30	riota2 6633	. . . . . . . . . . . . . . . . 17 |- ( ( P e. (Vtx ` G ) /\ E! x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) <-> ( iota_ x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) = P ) )
32	28, 31	sylan 488	. . . . . . . . . . . . . . . 16 |- ( ( ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) /\ ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) ) /\ E! x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) <-> ( iota_ x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) = P ) )
33		eqtr2 2642	. . . . . . . . . . . . . . . . 17 |- ( ( ( iota_ x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) = Q /\ ( iota_ x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) = P ) -> Q = P )
34	33	expcom 451	. . . . . . . . . . . . . . . 16 |- ( ( iota_ x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) = P -> ( ( iota_ x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) = Q -> Q = P ) )
35	32, 34	syl6bi 243	. . . . . . . . . . . . . . 15 |- ( ( ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) /\ ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) ) /\ E! x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) -> ( ( iota_ x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) = Q -> Q = P ) ) )
36	35	com23 86	. . . . . . . . . . . . . 14 |- ( ( ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) /\ ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) ) /\ E! x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( ( iota_ x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) = Q -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) -> Q = P ) ) )
37	27, 36	sylbid 230	. . . . . . . . . . . . 13 |- ( ( ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) /\ ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) ) /\ E! x e. (Vtx ` G ) <" A x B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( <" A Q B "> e. ( A ( 2 WWalksNOn G ) B ) -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) -> Q = P ) ) )
38	23, 37	mpdan 702	. . . . . . . . . . . 12 |- ( ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) /\ ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) ) -> ( <" A Q B "> e. ( A ( 2 WWalksNOn G ) B ) -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) -> Q = P ) ) )
39	17, 38	sylbid 230	. . . . . . . . . . 11 |- ( ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) /\ ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) ) -> ( <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) -> Q = P ) ) )
40	39	expimpd 629	. . . . . . . . . 10 |- ( ( Q e. (Vtx ` G ) /\ ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) -> ( ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) /\ <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) ) -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) -> Q = P ) ) )
41	40	ex 450	. . . . . . . . 9 |- ( Q e. (Vtx ` G ) -> ( ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) -> ( ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) /\ <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) ) -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) -> Q = P ) ) ) )
42	41	com23 86	. . . . . . . 8 |- ( Q e. (Vtx ` G ) -> ( ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) /\ <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) ) -> ( ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) -> Q = P ) ) ) )
43	42	3ad2ant2 1083	. . . . . . 7 |- ( ( B e. (Vtx ` G ) /\ Q e. (Vtx ` G ) /\ A e. (Vtx ` G ) ) -> ( ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) /\ <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) ) -> ( ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) -> Q = P ) ) ) )
44	7, 43	mpcom 38	. . . . . 6 |- ( ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) /\ <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) ) -> ( ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) -> Q = P ) ) )
45	44	ex 450	. . . . 5 |- ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) -> ( <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) -> ( ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) -> Q = P ) ) ) )
46	45	com24 95	. . . 4 |- ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) -> ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) -> ( ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) -> ( <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) -> Q = P ) ) ) )
47	46	imp 445	. . 3 |- ( ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) /\ <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( ( A e. (Vtx ` G ) /\ P e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) -> ( <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) -> Q = P ) ) )
48	4, 47	mpd 15	. 2 |- ( ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) /\ <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) ) -> ( <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) -> Q = P ) )
49	48	expimpd 629	1 |- ( ( G e. FriendGraph /\ A =/= B /\ ( P e. X /\ Q e. Y ) ) -> ( ( <" A P B "> e. ( A ( 2 WWalksNOn G ) B ) /\ <" B Q A "> e. ( B ( 2 WWalksNOn G ) A ) ) -> Q = P ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E!wreu 2914   ` cfv 5888   iota_crio 6610  (class class class)co 6650   2c2 11070   <"cs3 13587  Vtxcvtx 25874   UMGraph cumgr 25976   USGraph cusgr 26044   WWalksNOn cwwlksnon 26719   FriendGraph cfrgr 27120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-usgr 26046  df-wlks 26495  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724  df-frgr 27121

This theorem is referenced by: (None)