Theorem umgr2wlk 26845

Description: In a multigraph, there is a walk of length 2 for each pair of adjacent 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
umgr2wlk	|- ( ( 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 ) ) ) )

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

Allowed substitution hints:   E( f, p)

Proof of Theorem umgr2wlk

Dummy variables i j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		umgruhgr 25999	. . . . . 6 |- ( G e. UMGraph -> G e. UHGraph )
2		umgr2wlk.e	. . . . . . . 8 |- E = (Edg ` G )
3	2	eleq2i 2693	. . . . . . 7 |- ( { B , C } e. E <-> { B , C } e. (Edg ` G ) )
4		eqid 2622	. . . . . . . 8 |- (iEdg ` G ) = (iEdg ` G )
5	4	uhgredgiedgb 26021	. . . . . . 7 |- ( G e. UHGraph -> ( { B , C } e. (Edg ` G ) <-> E. i e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` i ) ) )
6	3, 5	syl5bb 272	. . . . . 6 |- ( G e. UHGraph -> ( { B , C } e. E <-> E. i e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` i ) ) )
7	1, 6	syl 17	. . . . 5 |- ( G e. UMGraph -> ( { B , C } e. E <-> E. i e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` i ) ) )
8	7	biimpd 219	. . . 4 |- ( G e. UMGraph -> ( { B , C } e. E -> E. i e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` i ) ) )
9	8	a1d 25	. . 3 |- ( G e. UMGraph -> ( { A , B } e. E -> ( { B , C } e. E -> E. i e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` i ) ) ) )
10	9	3imp 1256	. 2 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> E. i e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` i ) )
11	2	eleq2i 2693	. . . . . . 7 |- ( { A , B } e. E <-> { A , B } e. (Edg ` G ) )
12	4	uhgredgiedgb 26021	. . . . . . 7 |- ( G e. UHGraph -> ( { A , B } e. (Edg ` G ) <-> E. j e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` j ) ) )
13	11, 12	syl5bb 272	. . . . . 6 |- ( G e. UHGraph -> ( { A , B } e. E <-> E. j e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` j ) ) )
14	1, 13	syl 17	. . . . 5 |- ( G e. UMGraph -> ( { A , B } e. E <-> E. j e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` j ) ) )
15	14	biimpd 219	. . . 4 |- ( G e. UMGraph -> ( { A , B } e. E -> E. j e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` j ) ) )
16	15	a1dd 50	. . 3 |- ( G e. UMGraph -> ( { A , B } e. E -> ( { B , C } e. E -> E. j e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` j ) ) ) )
17	16	3imp 1256	. 2 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> E. j e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` j ) )
18		s2cli 13625	. . . . . . . . . 10 |- <" j i "> e. Word _V
19		s3cli 13626	. . . . . . . . . 10 |- <" A B C "> e. Word _V
20	18, 19	pm3.2i 471	. . . . . . . . 9 |- ( <" j i "> e. Word _V /\ <" A B C "> e. Word _V )
21		eqid 2622	. . . . . . . . . 10 |- <" j i "> = <" j i ">
22		eqid 2622	. . . . . . . . . 10 |- <" A B C "> = <" A B C ">
23		simpl1 1064	. . . . . . . . . 10 |- ( ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) /\ ( { A , B } = ( (iEdg ` G ) ` j ) /\ { B , C } = ( (iEdg ` G ) ` i ) ) ) -> G e. UMGraph )
24		3simpc 1060	. . . . . . . . . . 11 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( { A , B } e. E /\ { B , C } e. E ) )
25	24	adantr 481	. . . . . . . . . 10 |- ( ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) /\ ( { A , B } = ( (iEdg ` G ) ` j ) /\ { B , C } = ( (iEdg ` G ) ` i ) ) ) -> ( { A , B } e. E /\ { B , C } e. E ) )
26		simpl 473	. . . . . . . . . . . 12 |- ( ( { A , B } = ( (iEdg ` G ) ` j ) /\ { B , C } = ( (iEdg ` G ) ` i ) ) -> { A , B } = ( (iEdg ` G ) ` j ) )
27	26	eqcomd 2628	. . . . . . . . . . 11 |- ( ( { A , B } = ( (iEdg ` G ) ` j ) /\ { B , C } = ( (iEdg ` G ) ` i ) ) -> ( (iEdg ` G ) ` j ) = { A , B } )
28	27	adantl 482	. . . . . . . . . 10 |- ( ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) /\ ( { A , B } = ( (iEdg ` G ) ` j ) /\ { B , C } = ( (iEdg ` G ) ` i ) ) ) -> ( (iEdg ` G ) ` j ) = { A , B } )
29		simpr 477	. . . . . . . . . . . 12 |- ( ( { A , B } = ( (iEdg ` G ) ` j ) /\ { B , C } = ( (iEdg ` G ) ` i ) ) -> { B , C } = ( (iEdg ` G ) ` i ) )
30	29	eqcomd 2628	. . . . . . . . . . 11 |- ( ( { A , B } = ( (iEdg ` G ) ` j ) /\ { B , C } = ( (iEdg ` G ) ` i ) ) -> ( (iEdg ` G ) ` i ) = { B , C } )
31	30	adantl 482	. . . . . . . . . 10 |- ( ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) /\ ( { A , B } = ( (iEdg ` G ) ` j ) /\ { B , C } = ( (iEdg ` G ) ` i ) ) ) -> ( (iEdg ` G ) ` i ) = { B , C } )
32	2, 4, 21, 22, 23, 25, 28, 31	umgr2adedgwlk 26841	. . . . . . . . 9 |- ( ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) /\ ( { A , B } = ( (iEdg ` G ) ` j ) /\ { B , C } = ( (iEdg ` G ) ` i ) ) ) -> ( <" j i "> (Walks ` G ) <" A B C "> /\ ( # ` <" j i "> ) = 2 /\ ( A = ( <" A B C "> ` 0 ) /\ B = ( <" A B C "> ` 1 ) /\ C = ( <" A B C "> ` 2 ) ) ) )
33		breq12 4658	. . . . . . . . . . 11 |- ( ( f = <" j i "> /\ p = <" A B C "> ) -> ( f (Walks ` G ) p <-> <" j i "> (Walks ` G ) <" A B C "> ) )
34		fveq2 6191	. . . . . . . . . . . . 13 |- ( f = <" j i "> -> ( # ` f ) = ( # ` <" j i "> ) )
35	34	eqeq1d 2624	. . . . . . . . . . . 12 |- ( f = <" j i "> -> ( ( # ` f ) = 2 <-> ( # ` <" j i "> ) = 2 ) )
36	35	adantr 481	. . . . . . . . . . 11 |- ( ( f = <" j i "> /\ p = <" A B C "> ) -> ( ( # ` f ) = 2 <-> ( # ` <" j i "> ) = 2 ) )
37		fveq1 6190	. . . . . . . . . . . . . 14 |- ( p = <" A B C "> -> ( p ` 0 ) = ( <" A B C "> ` 0 ) )
38	37	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( p = <" A B C "> -> ( A = ( p ` 0 ) <-> A = ( <" A B C "> ` 0 ) ) )
39		fveq1 6190	. . . . . . . . . . . . . 14 |- ( p = <" A B C "> -> ( p ` 1 ) = ( <" A B C "> ` 1 ) )
40	39	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( p = <" A B C "> -> ( B = ( p ` 1 ) <-> B = ( <" A B C "> ` 1 ) ) )
41		fveq1 6190	. . . . . . . . . . . . . 14 |- ( p = <" A B C "> -> ( p ` 2 ) = ( <" A B C "> ` 2 ) )
42	41	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( p = <" A B C "> -> ( C = ( p ` 2 ) <-> C = ( <" A B C "> ` 2 ) ) )
43	38, 40, 42	3anbi123d 1399	. . . . . . . . . . . 12 |- ( p = <" A B C "> -> ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) <-> ( A = ( <" A B C "> ` 0 ) /\ B = ( <" A B C "> ` 1 ) /\ C = ( <" A B C "> ` 2 ) ) ) )
44	43	adantl 482	. . . . . . . . . . 11 |- ( ( f = <" j i "> /\ p = <" A B C "> ) -> ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) <-> ( A = ( <" A B C "> ` 0 ) /\ B = ( <" A B C "> ` 1 ) /\ C = ( <" A B C "> ` 2 ) ) ) )
45	33, 36, 44	3anbi123d 1399	. . . . . . . . . 10 |- ( ( f = <" j i "> /\ p = <" A B C "> ) -> ( ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) <-> ( <" j i "> (Walks ` G ) <" A B C "> /\ ( # ` <" j i "> ) = 2 /\ ( A = ( <" A B C "> ` 0 ) /\ B = ( <" A B C "> ` 1 ) /\ C = ( <" A B C "> ` 2 ) ) ) ) )
46	45	spc2egv 3295	. . . . . . . . 9 |- ( ( <" j i "> e. Word _V /\ <" A B C "> e. Word _V ) -> ( ( <" j i "> (Walks ` G ) <" A B C "> /\ ( # ` <" j i "> ) = 2 /\ ( A = ( <" A B C "> ` 0 ) /\ B = ( <" A B C "> ` 1 ) /\ C = ( <" A B C "> ` 2 ) ) ) -> E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) )
47	20, 32, 46	mpsyl 68	. . . . . . . 8 |- ( ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) /\ ( { A , B } = ( (iEdg ` G ) ` j ) /\ { B , C } = ( (iEdg ` G ) ` i ) ) ) -> E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) )
48	47	exp32 631	. . . . . . 7 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( { A , B } = ( (iEdg ` G ) ` j ) -> ( { B , C } = ( (iEdg ` G ) ` i ) -> E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
49	48	com12 32	. . . . . 6 |- ( { A , B } = ( (iEdg ` G ) ` j ) -> ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( { B , C } = ( (iEdg ` G ) ` i ) -> E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
50	49	rexlimivw 3029	. . . . 5 |- ( E. j e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` j ) -> ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( { B , C } = ( (iEdg ` G ) ` i ) -> E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
51	50	com13 88	. . . 4 |- ( { B , C } = ( (iEdg ` G ) ` i ) -> ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( E. j e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` j ) -> E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
52	51	rexlimivw 3029	. . 3 |- ( E. i e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` i ) -> ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( E. j e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` j ) -> E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
53	52	com12 32	. 2 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( E. i e. dom (iEdg ` G ) { B , C } = ( (iEdg ` G ) ` i ) -> ( E. j e. dom (iEdg ` G ) { A , B } = ( (iEdg ` G ) ` j ) -> E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) ) )
54	10, 17, 53	mp2d 49	1 |- ( ( 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 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   {cpr 4179   class class class wbr 4653   dom cdm 5114   ` cfv 5888   0cc0 9936   1c1 9937   2c2 11070   #chash 13117  Word cword 13291   <"cs2 13586   <"cs3 13587  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   UMGraph cumgr 25976  Walkscwlks 26492

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

This theorem is referenced by:  umgr2wlkon  26846  umgrwwlks2on  26850