Theorem umgr2wlkon 26846

Description: For each pair of adjacent edges in a multigraph, there is a walk of length 2 between the not common vertices of the edges. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)

Hypothesis

Ref	Expression
umgr2wlk.e	|- E = (Edg ` G )

Assertion

Ref	Expression
umgr2wlkon	|- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> E. f E. p f ( A (WalksOn ` G ) C ) p )

Distinct variable groups:   A, f, p   B, f, p   C, f, p   f, G, p   f, E, p

Proof of Theorem umgr2wlkon

Step	Hyp	Ref	Expression
1		umgr2wlk.e	. . 3 |- E = (Edg ` G )
2	1	umgr2wlk 26845	. 2 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) )
3		simp1 1061	. . . . . . 7 |- ( ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> f (Walks ` G ) p )
4		eqcom 2629	. . . . . . . . . 10 |- ( A = ( p ` 0 ) <-> ( p ` 0 ) = A )
5	4	biimpi 206	. . . . . . . . 9 |- ( A = ( p ` 0 ) -> ( p ` 0 ) = A )
6	5	3ad2ant1 1082	. . . . . . . 8 |- ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( p ` 0 ) = A )
7	6	3ad2ant3 1084	. . . . . . 7 |- ( ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( p ` 0 ) = A )
8		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( 2 = ( # ` f ) -> ( p ` 2 ) = ( p ` ( # ` f ) ) )
9	8	eqcoms 2630	. . . . . . . . . . . . . 14 |- ( ( # ` f ) = 2 -> ( p ` 2 ) = ( p ` ( # ` f ) ) )
10	9	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( ( # ` f ) = 2 -> ( ( p ` 2 ) = C <-> ( p ` ( # ` f ) ) = C ) )
11	10	biimpcd 239	. . . . . . . . . . . 12 |- ( ( p ` 2 ) = C -> ( ( # ` f ) = 2 -> ( p ` ( # ` f ) ) = C ) )
12	11	eqcoms 2630	. . . . . . . . . . 11 |- ( C = ( p ` 2 ) -> ( ( # ` f ) = 2 -> ( p ` ( # ` f ) ) = C ) )
13	12	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( ( # ` f ) = 2 -> ( p ` ( # ` f ) ) = C ) )
14	13	com12 32	. . . . . . . . 9 |- ( ( # ` f ) = 2 -> ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( p ` ( # ` f ) ) = C ) )
15	14	a1i 11	. . . . . . . 8 |- ( f (Walks ` G ) p -> ( ( # ` f ) = 2 -> ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( p ` ( # ` f ) ) = C ) ) )
16	15	3imp 1256	. . . . . . 7 |- ( ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( p ` ( # ` f ) ) = C )
17	3, 7, 16	3jca 1242	. . . . . 6 |- ( ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( f (Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) )
18	17	adantl 482	. . . . 5 |- ( ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) /\ ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( f (Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) )
19	1	umgr2adedgwlklem 26840	. . . . . . 7 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( ( A =/= B /\ B =/= C ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) /\ C e. (Vtx ` G ) ) ) )
20		simprr1 1109	. . . . . . . 8 |- ( ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) /\ ( ( A =/= B /\ B =/= C ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) /\ C e. (Vtx ` G ) ) ) ) -> A e. (Vtx ` G ) )
21		simprr3 1111	. . . . . . . 8 |- ( ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) /\ ( ( A =/= B /\ B =/= C ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) /\ C e. (Vtx ` G ) ) ) ) -> C e. (Vtx ` G ) )
22	20, 21	jca 554	. . . . . . 7 |- ( ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) /\ ( ( A =/= B /\ B =/= C ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) /\ C e. (Vtx ` G ) ) ) ) -> ( A e. (Vtx ` G ) /\ C e. (Vtx ` G ) ) )
23	19, 22	mpdan 702	. . . . . 6 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( A e. (Vtx ` G ) /\ C e. (Vtx ` G ) ) )
24		vex 3203	. . . . . . . 8 |- f e. _V
25		vex 3203	. . . . . . . 8 |- p e. _V
26	24, 25	pm3.2i 471	. . . . . . 7 |- ( f e. _V /\ p e. _V )
27	26	a1i 11	. . . . . 6 |- ( ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) /\ ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( f e. _V /\ p e. _V ) )
28		eqid 2622	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
29	28	iswlkon 26553	. . . . . 6 |- ( ( ( A e. (Vtx ` G ) /\ C e. (Vtx ` G ) ) /\ ( f e. _V /\ p e. _V ) ) -> ( f ( A (WalksOn ` G ) C ) p <-> ( f (Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) ) )
30	23, 27, 29	syl2an2r 876	. . . . 5 |- ( ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) /\ ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( f ( A (WalksOn ` G ) C ) p <-> ( f (Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = C ) ) )
31	18, 30	mpbird 247	. . . 4 |- ( ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) /\ ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> f ( A (WalksOn ` G ) C ) p )
32	31	ex 450	. . 3 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> f ( A (WalksOn ` G ) C ) p ) )
33	32	2eximdv 1848	. 2 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> E. f E. p f ( A (WalksOn ` G ) C ) p ) )
34	2, 33	mpd 15	1 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> E. f E. p f ( A (WalksOn ` G ) C ) p )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   {cpr 4179   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   2c2 11070   #chash 13117  Vtxcvtx 25874  Edgcedg 25939   UMGraph cumgr 25976  Walkscwlks 26492  WalksOncwlkson 26493

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-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-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-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-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-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-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-wlks 26495  df-wlkson 26496

This theorem is referenced by: (None)