Theorem umgr2adedgwlk 26841

Description: In a multigraph, two adjacent edges form a walk of length 2. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 29-Jan-2021.)

Hypotheses

Ref	Expression
umgr2adedgwlk.e	|- E = (Edg ` G )
umgr2adedgwlk.i	|- I = (iEdg ` G )
umgr2adedgwlk.f	|- F = <" J K ">
umgr2adedgwlk.p	|- P = <" A B C ">
umgr2adedgwlk.g	|- ( ph -> G e. UMGraph )
umgr2adedgwlk.a	|- ( ph -> ( { A , B } e. E /\ { B , C } e. E ) )
umgr2adedgwlk.j	|- ( ph -> ( I ` J ) = { A , B } )
umgr2adedgwlk.k	|- ( ph -> ( I ` K ) = { B , C } )

Assertion

Ref	Expression
umgr2adedgwlk	|- ( ph -> ( F (Walks ` G ) P /\ ( # ` F ) = 2 /\ ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) ) )

Proof of Theorem umgr2adedgwlk

Step	Hyp	Ref	Expression
1		umgr2adedgwlk.p	. . 3 |- P = <" A B C ">
2		umgr2adedgwlk.f	. . 3 |- F = <" J K ">
3		umgr2adedgwlk.g	. . . . . 6 |- ( ph -> G e. UMGraph )
4		umgr2adedgwlk.a	. . . . . 6 |- ( ph -> ( { A , B } e. E /\ { B , C } e. E ) )
5		3anass 1042	. . . . . 6 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) <-> ( G e. UMGraph /\ ( { A , B } e. E /\ { B , C } e. E ) ) )
6	3, 4, 5	sylanbrc 698	. . . . 5 |- ( ph -> ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) )
7		umgr2adedgwlk.e	. . . . . 6 |- E = (Edg ` G )
8	7	umgr2adedgwlklem 26840	. . . . 5 |- ( ( 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 ) ) ) )
9	6, 8	syl 17	. . . 4 |- ( ph -> ( ( A =/= B /\ B =/= C ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) /\ C e. (Vtx ` G ) ) ) )
10	9	simprd 479	. . 3 |- ( ph -> ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) /\ C e. (Vtx ` G ) ) )
11	9	simpld 475	. . 3 |- ( ph -> ( A =/= B /\ B =/= C ) )
12		ssid 3624	. . . . 5 |- { A , B } C_ { A , B }
13		umgr2adedgwlk.j	. . . . 5 |- ( ph -> ( I ` J ) = { A , B } )
14	12, 13	syl5sseqr 3654	. . . 4 |- ( ph -> { A , B } C_ ( I ` J ) )
15		ssid 3624	. . . . 5 |- { B , C } C_ { B , C }
16		umgr2adedgwlk.k	. . . . 5 |- ( ph -> ( I ` K ) = { B , C } )
17	15, 16	syl5sseqr 3654	. . . 4 |- ( ph -> { B , C } C_ ( I ` K ) )
18	14, 17	jca 554	. . 3 |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
19		eqid 2622	. . 3 |- (Vtx ` G ) = (Vtx ` G )
20		umgr2adedgwlk.i	. . 3 |- I = (iEdg ` G )
21	1, 2, 10, 11, 18, 19, 20	2wlkd 26832	. 2 |- ( ph -> F (Walks ` G ) P )
22	2	fveq2i 6194	. . . 4 |- ( # ` F ) = ( # ` <" J K "> )
23		s2len 13634	. . . 4 |- ( # ` <" J K "> ) = 2
24	22, 23	eqtri 2644	. . 3 |- ( # ` F ) = 2
25	24	a1i 11	. 2 |- ( ph -> ( # ` F ) = 2 )
26		s3fv0 13636	. . . . 5 |- ( A e. (Vtx ` G ) -> ( <" A B C "> ` 0 ) = A )
27		s3fv1 13637	. . . . 5 |- ( B e. (Vtx ` G ) -> ( <" A B C "> ` 1 ) = B )
28		s3fv2 13638	. . . . 5 |- ( C e. (Vtx ` G ) -> ( <" A B C "> ` 2 ) = C )
29	26, 27, 28	3anim123i 1247	. . . 4 |- ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) /\ C e. (Vtx ` G ) ) -> ( ( <" A B C "> ` 0 ) = A /\ ( <" A B C "> ` 1 ) = B /\ ( <" A B C "> ` 2 ) = C ) )
30	10, 29	syl 17	. . 3 |- ( ph -> ( ( <" A B C "> ` 0 ) = A /\ ( <" A B C "> ` 1 ) = B /\ ( <" A B C "> ` 2 ) = C ) )
31	1	fveq1i 6192	. . . . . 6 |- ( P ` 0 ) = ( <" A B C "> ` 0 )
32	31	eqeq2i 2634	. . . . 5 |- ( A = ( P ` 0 ) <-> A = ( <" A B C "> ` 0 ) )
33		eqcom 2629	. . . . 5 |- ( A = ( <" A B C "> ` 0 ) <-> ( <" A B C "> ` 0 ) = A )
34	32, 33	bitri 264	. . . 4 |- ( A = ( P ` 0 ) <-> ( <" A B C "> ` 0 ) = A )
35	1	fveq1i 6192	. . . . . 6 |- ( P ` 1 ) = ( <" A B C "> ` 1 )
36	35	eqeq2i 2634	. . . . 5 |- ( B = ( P ` 1 ) <-> B = ( <" A B C "> ` 1 ) )
37		eqcom 2629	. . . . 5 |- ( B = ( <" A B C "> ` 1 ) <-> ( <" A B C "> ` 1 ) = B )
38	36, 37	bitri 264	. . . 4 |- ( B = ( P ` 1 ) <-> ( <" A B C "> ` 1 ) = B )
39	1	fveq1i 6192	. . . . . 6 |- ( P ` 2 ) = ( <" A B C "> ` 2 )
40	39	eqeq2i 2634	. . . . 5 |- ( C = ( P ` 2 ) <-> C = ( <" A B C "> ` 2 ) )
41		eqcom 2629	. . . . 5 |- ( C = ( <" A B C "> ` 2 ) <-> ( <" A B C "> ` 2 ) = C )
42	40, 41	bitri 264	. . . 4 |- ( C = ( P ` 2 ) <-> ( <" A B C "> ` 2 ) = C )
43	34, 38, 42	3anbi123i 1251	. . 3 |- ( ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) <-> ( ( <" A B C "> ` 0 ) = A /\ ( <" A B C "> ` 1 ) = B /\ ( <" A B C "> ` 2 ) = C ) )
44	30, 43	sylibr 224	. 2 |- ( ph -> ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) )
45	21, 25, 44	3jca 1242	1 |- ( ph -> ( F (Walks ` G ) P /\ ( # ` F ) = 2 /\ ( A = ( P ` 0 ) /\ B = ( P ` 1 ) /\ C = ( P ` 2 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   {cpr 4179   class class class wbr 4653   ` cfv 5888   0cc0 9936   1c1 9937   2c2 11070   #chash 13117   <"cs2 13586   <"cs3 13587  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   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-umgr 25978  df-wlks 26495

This theorem is referenced by:  umgr2adedgwlkonALT  26843  umgr2wlk  26845