Theorem usgr2wlkspth 26655

Description: In a simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.)

Assertion

Ref	Expression
usgr2wlkspth	|- ( ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) -> ( F ( A (WalksOn ` G ) B ) P <-> F ( A (SPathsOn ` G ) B ) P ) )

Proof of Theorem usgr2wlkspth

Step	Hyp	Ref	Expression
1		simpl31 1142	. . . . . . . 8 |- ( ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) ) -> F (Walks ` G ) P )
2		simp2 1062	. . . . . . . . . . . . . 14 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( P ` 0 ) = A )
3		simp3 1063	. . . . . . . . . . . . . 14 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( P ` ( # ` F ) ) = B )
4	2, 3	neeq12d 2855	. . . . . . . . . . . . 13 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( ( P ` 0 ) =/= ( P ` ( # ` F ) ) <-> A =/= B ) )
5	4	bicomd 213	. . . . . . . . . . . 12 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( A =/= B <-> ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) )
6	5	3anbi3d 1405	. . . . . . . . . . 11 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) <-> ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) )
7		usgr2wlkspthlem1 26653	. . . . . . . . . . . . 13 |- ( ( F (Walks ` G ) P /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> Fun `' F )
8	7	ex 450	. . . . . . . . . . . 12 |- ( F (Walks ` G ) P -> ( ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> Fun `' F ) )
9	8	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> Fun `' F ) )
10	6, 9	sylbid 230	. . . . . . . . . 10 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) -> Fun `' F ) )
11	10	3ad2ant3 1084	. . . . . . . . 9 |- ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) -> Fun `' F ) )
12	11	imp 445	. . . . . . . 8 |- ( ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) ) -> Fun `' F )
13		istrl 26593	. . . . . . . 8 |- ( F (Trails ` G ) P <-> ( F (Walks ` G ) P /\ Fun `' F ) )
14	1, 12, 13	sylanbrc 698	. . . . . . 7 |- ( ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) ) -> F (Trails ` G ) P )
15		usgr2wlkspthlem2 26654	. . . . . . . . . . . 12 |- ( ( F (Walks ` G ) P /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) ) -> Fun `' P )
16	15	ex 450	. . . . . . . . . . 11 |- ( F (Walks ` G ) P -> ( ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> Fun `' P ) )
17	16	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 /\ ( P ` 0 ) =/= ( P ` ( # ` F ) ) ) -> Fun `' P ) )
18	6, 17	sylbid 230	. . . . . . . . 9 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) -> Fun `' P ) )
19	18	3ad2ant3 1084	. . . . . . . 8 |- ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) -> Fun `' P ) )
20	19	imp 445	. . . . . . 7 |- ( ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) ) -> Fun `' P )
21		isspth 26620	. . . . . . 7 |- ( F (SPaths ` G ) P <-> ( F (Trails ` G ) P /\ Fun `' P ) )
22	14, 20, 21	sylanbrc 698	. . . . . 6 |- ( ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) ) -> F (SPaths ` G ) P )
23		3simpc 1060	. . . . . . . 8 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) )
24	23	3ad2ant3 1084	. . . . . . 7 |- ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) )
25	24	adantr 481	. . . . . 6 |- ( ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) ) -> ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) )
26		3anass 1042	. . . . . 6 |- ( ( F (SPaths ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) <-> ( F (SPaths ` G ) P /\ ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) )
27	22, 25, 26	sylanbrc 698	. . . . 5 |- ( ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) ) -> ( F (SPaths ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) )
28		3simpa 1058	. . . . . . 7 |- ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) )
29	28	adantr 481	. . . . . 6 |- ( ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) ) -> ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) )
30		eqid 2622	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
31	30	isspthonpth 26645	. . . . . 6 |- ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( A (SPathsOn ` G ) B ) P <-> ( F (SPaths ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) )
32	29, 31	syl 17	. . . . 5 |- ( ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) ) -> ( F ( A (SPathsOn ` G ) B ) P <-> ( F (SPaths ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) )
33	27, 32	mpbird 247	. . . 4 |- ( ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) /\ ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) ) -> F ( A (SPathsOn ` G ) B ) P )
34	33	ex 450	. . 3 |- ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) -> F ( A (SPathsOn ` G ) B ) P ) )
35	30	wlkonprop 26554	. . . 4 |- ( F ( A (WalksOn ` G ) B ) P -> ( ( G e. _V /\ A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) )
36		3simpc 1060	. . . . 5 |- ( ( G e. _V /\ A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) -> ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) )
37	36	3anim1i 1248	. . . 4 |- ( ( ( G e. _V /\ A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) )
38	35, 37	syl 17	. . 3 |- ( F ( A (WalksOn ` G ) B ) P -> ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) )
39	34, 38	syl11 33	. 2 |- ( ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) -> ( F ( A (WalksOn ` G ) B ) P -> F ( A (SPathsOn ` G ) B ) P ) )
40		spthonpthon 26647	. . 3 |- ( F ( A (SPathsOn ` G ) B ) P -> F ( A (PathsOn ` G ) B ) P )
41		pthontrlon 26643	. . 3 |- ( F ( A (PathsOn ` G ) B ) P -> F ( A (TrailsOn ` G ) B ) P )
42		trlsonwlkon 26606	. . 3 |- ( F ( A (TrailsOn ` G ) B ) P -> F ( A (WalksOn ` G ) B ) P )
43	40, 41, 42	3syl 18	. 2 |- ( F ( A (SPathsOn ` G ) B ) P -> F ( A (WalksOn ` G ) B ) P )
44	39, 43	impbid1 215	1 |- ( ( G e. USGraph /\ ( # ` F ) = 2 /\ A =/= B ) -> ( F ( A (WalksOn ` G ) B ) P <-> F ( A (SPathsOn ` G ) B ) P ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   class class class wbr 4653   `'ccnv 5113   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   0cc0 9936   2c2 11070   #chash 13117  Vtxcvtx 25874   USGraph cusgr 26044  Walkscwlks 26492  WalksOncwlkson 26493  Trailsctrls 26587  TrailsOnctrlson 26588  SPathscspths 26609  PathsOncpthson 26610  SPathsOncspthson 26611

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-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-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

This theorem is referenced by:  usgr2trlspth  26657  wpthswwlks2on  26854