Theorem usgr2wspthons3 26857

Description: A simple path of length 2 between two vertices represented as length 3 string corresponds to two adjacent edges in a simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.) (Revised by AV, 17-May-2021.)

Hypotheses

Ref	Expression
usgr2wspthon0.v	|- V = (Vtx ` G )
usgr2wspthon0.e	|- E = (Edg ` G )

Assertion

Ref	Expression
usgr2wspthons3	|- ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) <-> ( A =/= C /\ { A , B } e. E /\ { B , C } e. E ) ) )

Proof of Theorem usgr2wspthons3

Step	Hyp	Ref	Expression
1		2nn 11185	. . . . . 6 |- 2 e. NN
2		ne0i 3921	. . . . . 6 |- ( <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) -> ( A ( 2 WSPathsNOn G ) C ) =/= (/) )
3		wspthsnonn0vne 26813	. . . . . 6 |- ( ( 2 e. NN /\ ( A ( 2 WSPathsNOn G ) C ) =/= (/) ) -> A =/= C )
4	1, 2, 3	sylancr 695	. . . . 5 |- ( <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) -> A =/= C )
5		simplr 792	. . . . . . . 8 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) /\ <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) ) -> A =/= C )
6		simpll 790	. . . . . . . . . . 11 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> G e. USGraph )
7		3simpb 1059	. . . . . . . . . . . 12 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( A e. V /\ C e. V ) )
8	7	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> ( A e. V /\ C e. V ) )
9		simpr 477	. . . . . . . . . . 11 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> A =/= C )
10	6, 8, 9	3jca 1242	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> ( G e. USGraph /\ ( A e. V /\ C e. V ) /\ A =/= C ) )
11		usgr2wspthon0.v	. . . . . . . . . . . 12 |- V = (Vtx ` G )
12	11	wpthswwlks2on 26854	. . . . . . . . . . 11 |- ( ( G e. USGraph /\ ( A e. V /\ C e. V ) /\ A =/= C ) -> ( A ( 2 WSPathsNOn G ) C ) = ( A ( 2 WWalksNOn G ) C ) )
13	12	eleq2d 2687	. . . . . . . . . 10 |- ( ( G e. USGraph /\ ( A e. V /\ C e. V ) /\ A =/= C ) -> ( <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) <-> <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) )
14	10, 13	syl 17	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> ( <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) <-> <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) )
15	14	biimpa 501	. . . . . . . 8 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) /\ <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) ) -> <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) )
16	5, 15	jca 554	. . . . . . 7 |- ( ( ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) /\ <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) ) -> ( A =/= C /\ <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) )
17	16	exp31 630	. . . . . 6 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A =/= C -> ( <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) -> ( A =/= C /\ <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) ) )
18	17	com13 88	. . . . 5 |- ( <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) -> ( A =/= C -> ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A =/= C /\ <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) ) )
19	4, 18	mpd 15	. . . 4 |- ( <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) -> ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A =/= C /\ <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) )
20	19	com12 32	. . 3 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) -> ( A =/= C /\ <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) )
21	13	bicomd 213	. . . . . 6 |- ( ( G e. USGraph /\ ( A e. V /\ C e. V ) /\ A =/= C ) -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) <-> <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) ) )
22	10, 21	syl 17	. . . . 5 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) <-> <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) ) )
23	22	biimpd 219	. . . 4 |- ( ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ A =/= C ) -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) -> <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) ) )
24	23	expimpd 629	. . 3 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A =/= C /\ <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) -> <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) ) )
25	20, 24	impbid 202	. 2 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) <-> ( A =/= C /\ <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) ) )
26		usgrumgr 26074	. . . . 5 |- ( G e. USGraph -> G e. UMGraph )
27		usgr2wspthon0.e	. . . . . 6 |- E = (Edg ` G )
28	11, 27	umgrwwlks2on 26850	. . . . 5 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) <-> ( { A , B } e. E /\ { B , C } e. E ) ) )
29	26, 28	sylan 488	. . . 4 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) <-> ( { A , B } e. E /\ { B , C } e. E ) ) )
30	29	anbi2d 740	. . 3 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A =/= C /\ <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) <-> ( A =/= C /\ ( { A , B } e. E /\ { B , C } e. E ) ) ) )
31		3anass 1042	. . 3 |- ( ( A =/= C /\ { A , B } e. E /\ { B , C } e. E ) <-> ( A =/= C /\ ( { A , B } e. E /\ { B , C } e. E ) ) )
32	30, 31	syl6bbr 278	. 2 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A =/= C /\ <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) ) <-> ( A =/= C /\ { A , B } e. E /\ { B , C } e. E ) ) )
33	25, 32	bitrd 268	1 |- ( ( G e. USGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <" A B C "> e. ( A ( 2 WSPathsNOn G ) C ) <-> ( A =/= C /\ { A , B } e. E /\ { B , C } e. E ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   {cpr 4179   ` cfv 5888  (class class class)co 6650   NNcn 11020   2c2 11070   <"cs3 13587  Vtxcvtx 25874  Edgcedg 25939   UMGraph cumgr 25976   USGraph cusgr 26044   WWalksNOn cwwlksnon 26719   WSPathsNOn cwwspthsnon 26721

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-fal 1489  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-uspgr 26045  df-usgr 26046  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-pthson 26614  df-spthson 26615  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724  df-wspthsnon 26726

This theorem is referenced by:  usgr2wspthon  26858