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Theorem umgr2adedgspth 40070
Description: In a multigraph, two adjacent edges with different endvertices form a simple path of length 2. (Contributed by Alexander van der Vekens, 1-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 } )
umgr2adedgspth.n  |-  ( ph  ->  A  =/=  C )
Assertion
Ref Expression
umgr2adedgspth  |-  ( ph  ->  F (SPathS `  G
) P )

Proof of Theorem umgr2adedgspth
StepHypRef Expression
1 umgr2adedgwlk.p . 2  |-  P  = 
<" A B C ">
2 umgr2adedgwlk.f . 2  |-  F  = 
<" J K ">
3 umgr2adedgwlk.g . . . . 5  |-  ( ph  ->  G  e. UMGraph  )
4 umgr2adedgwlk.a . . . . 5  |-  ( ph  ->  ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E )
)
5 3anass 1011 . . . . 5  |-  ( ( G  e. UMGraph  /\  { A ,  B }  e.  E  /\  { B ,  C }  e.  E )  <->  ( G  e. UMGraph  /\  ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E
) ) )
63, 4, 5sylanbrc 677 . . . 4  |-  ( ph  ->  ( G  e. UMGraph  /\  { A ,  B }  e.  E  /\  { B ,  C }  e.  E
) )
7 umgr2adedgwlk.e . . . . 5  |-  E  =  (Edg `  G )
87umgr2adedgwlklem 40066 . . . 4  |-  ( ( 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
) ) ) )
96, 8syl 17 . . 3  |-  ( ph  ->  ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( A  e.  (Vtx `  G
)  /\  B  e.  (Vtx `  G )  /\  C  e.  (Vtx `  G
) ) ) )
109simprd 470 . 2  |-  ( ph  ->  ( A  e.  (Vtx
`  G )  /\  B  e.  (Vtx `  G
)  /\  C  e.  (Vtx `  G ) ) )
119simpld 466 . 2  |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C
) )
12 ssid 3437 . . . 4  |-  { A ,  B }  C_  { A ,  B }
13 umgr2adedgwlk.j . . . 4  |-  ( ph  ->  ( I `  J
)  =  { A ,  B } )
1412, 13syl5sseqr 3467 . . 3  |-  ( ph  ->  { A ,  B }  C_  ( I `  J ) )
15 ssid 3437 . . . 4  |-  { B ,  C }  C_  { B ,  C }
16 umgr2adedgwlk.k . . . 4  |-  ( ph  ->  ( I `  K
)  =  { B ,  C } )
1715, 16syl5sseqr 3467 . . 3  |-  ( ph  ->  { B ,  C }  C_  ( I `  K ) )
1814, 17jca 541 . 2  |-  ( ph  ->  ( { A ,  B }  C_  ( I `
 J )  /\  { B ,  C }  C_  ( I `  K
) ) )
19 eqid 2471 . 2  |-  (Vtx `  G )  =  (Vtx
`  G )
20 umgr2adedgwlk.i . 2  |-  I  =  (iEdg `  G )
21 fveq2 5879 . . . . . . . . 9  |-  ( K  =  J  ->  (
I `  K )  =  ( I `  J ) )
2221eqcoms 2479 . . . . . . . 8  |-  ( J  =  K  ->  (
I `  K )  =  ( I `  J ) )
2322eqeq1d 2473 . . . . . . 7  |-  ( J  =  K  ->  (
( I `  K
)  =  { B ,  C }  <->  ( I `  J )  =  { B ,  C }
) )
24 eqtr2 2491 . . . . . . . 8  |-  ( ( ( I `  J
)  =  { B ,  C }  /\  (
I `  J )  =  { A ,  B } )  ->  { B ,  C }  =  { A ,  B }
)
2524ex 441 . . . . . . 7  |-  ( ( I `  J )  =  { B ,  C }  ->  ( ( I `  J )  =  { A ,  B }  ->  { B ,  C }  =  { A ,  B }
) )
2623, 25syl6bi 236 . . . . . 6  |-  ( J  =  K  ->  (
( I `  K
)  =  { B ,  C }  ->  (
( I `  J
)  =  { A ,  B }  ->  { B ,  C }  =  { A ,  B }
) ) )
2726com13 82 . . . . 5  |-  ( ( I `  J )  =  { A ,  B }  ->  ( ( I `  K )  =  { B ,  C }  ->  ( J  =  K  ->  { B ,  C }  =  { A ,  B }
) ) )
2813, 16, 27sylc 61 . . . 4  |-  ( ph  ->  ( J  =  K  ->  { B ,  C }  =  { A ,  B }
) )
29 eqcom 2478 . . . . . 6  |-  ( { B ,  C }  =  { A ,  B } 
<->  { A ,  B }  =  { B ,  C } )
30 prcom 4041 . . . . . . 7  |-  { B ,  C }  =  { C ,  B }
3130eqeq2i 2483 . . . . . 6  |-  ( { A ,  B }  =  { B ,  C } 
<->  { A ,  B }  =  { C ,  B } )
3229, 31bitri 257 . . . . 5  |-  ( { B ,  C }  =  { A ,  B } 
<->  { A ,  B }  =  { C ,  B } )
3319, 7umgrpredgav 39391 . . . . . . . . . . 11  |-  ( ( G  e. UMGraph  /\  { A ,  B }  e.  E
)  ->  ( A  e.  (Vtx `  G )  /\  B  e.  (Vtx `  G ) ) )
34 elex 3040 . . . . . . . . . . . 12  |-  ( A  e.  (Vtx `  G
)  ->  A  e.  _V )
3534adantr 472 . . . . . . . . . . 11  |-  ( ( A  e.  (Vtx `  G )  /\  B  e.  (Vtx `  G )
)  ->  A  e.  _V )
3633, 35syl 17 . . . . . . . . . 10  |-  ( ( G  e. UMGraph  /\  { A ,  B }  e.  E
)  ->  A  e.  _V )
3736ex 441 . . . . . . . . 9  |-  ( G  e. UMGraph  ->  ( { A ,  B }  e.  E  ->  A  e.  _V )
)
3819, 7umgrpredgav 39391 . . . . . . . . . . 11  |-  ( ( G  e. UMGraph  /\  { B ,  C }  e.  E
)  ->  ( B  e.  (Vtx `  G )  /\  C  e.  (Vtx `  G ) ) )
39 elex 3040 . . . . . . . . . . . 12  |-  ( C  e.  (Vtx `  G
)  ->  C  e.  _V )
4039adantl 473 . . . . . . . . . . 11  |-  ( ( B  e.  (Vtx `  G )  /\  C  e.  (Vtx `  G )
)  ->  C  e.  _V )
4138, 40syl 17 . . . . . . . . . 10  |-  ( ( G  e. UMGraph  /\  { B ,  C }  e.  E
)  ->  C  e.  _V )
4241ex 441 . . . . . . . . 9  |-  ( G  e. UMGraph  ->  ( { B ,  C }  e.  E  ->  C  e.  _V )
)
4337, 42anim12d 572 . . . . . . . 8  |-  ( G  e. UMGraph  ->  ( ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E
)  ->  ( A  e.  _V  /\  C  e. 
_V ) ) )
443, 4, 43sylc 61 . . . . . . 7  |-  ( ph  ->  ( A  e.  _V  /\  C  e.  _V )
)
45 preqr1g 4145 . . . . . . 7  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( { A ,  B }  =  { C ,  B }  ->  A  =  C ) )
4644, 45syl 17 . . . . . 6  |-  ( ph  ->  ( { A ,  B }  =  { C ,  B }  ->  A  =  C ) )
47 umgr2adedgspth.n . . . . . 6  |-  ( ph  ->  A  =/=  C )
48 eqneqall 2654 . . . . . 6  |-  ( A  =  C  ->  ( A  =/=  C  ->  J  =/=  K ) )
4946, 47, 48syl6ci 66 . . . . 5  |-  ( ph  ->  ( { A ,  B }  =  { C ,  B }  ->  J  =/=  K ) )
5032, 49syl5bi 225 . . . 4  |-  ( ph  ->  ( { B ,  C }  =  { A ,  B }  ->  J  =/=  K ) )
5128, 50syld 44 . . 3  |-  ( ph  ->  ( J  =  K  ->  J  =/=  K
) )
52 neqne 2651 . . 3  |-  ( -.  J  =  K  ->  J  =/=  K )
5351, 52pm2.61d1 164 . 2  |-  ( ph  ->  J  =/=  K )
541, 2, 10, 11, 18, 19, 20, 53, 472spthd 40063 1  |-  ( ph  ->  F (SPathS `  G
) P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    C_ wss 3390   {cpr 3961   class class class wbr 4395   ` cfv 5589   <"cs2 12996   <"cs3 12997  Vtxcvtx 39251  iEdgciedg 39252   UMGraph cumgr 39327  Edgcedga 39371  SPathScspths 39908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ifp 984  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-s2 13003  df-s3 13004  df-umgr 39329  df-edga 39372  df-1wlks 39804  df-trls 39889  df-spths 39912
This theorem is referenced by: (None)
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