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Theorem usgra2adedgwlk 24387
Description: In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
Hypotheses
Ref Expression
usgra2adedgspth.f  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. }
usgra2adedgspth.p  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
Assertion
Ref Expression
usgra2adedgwlk  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  -> 
( F ( V Walks 
E ) P  /\  ( # `  F )  =  2  /\  ( A  =  ( P `  0 )  /\  B  =  ( P `  1 )  /\  C  =  ( P `  2 ) ) ) ) )

Proof of Theorem usgra2adedgwlk
StepHypRef Expression
1 usgrav 24111 . . . . 5  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
21adantr 465 . . . 4  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( V  e. 
_V  /\  E  e.  _V ) )
3 usgraedgrnv 24150 . . . . . . . 8  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  ( A  e.  V  /\  B  e.  V ) )
43ancomd 451 . . . . . . 7  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  ( B  e.  V  /\  A  e.  V ) )
54adantrr 716 . . . . . 6  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( B  e.  V  /\  A  e.  V ) )
65simprd 463 . . . . 5  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  A  e.  V
)
73adantrr 716 . . . . . 6  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( A  e.  V  /\  B  e.  V ) )
87simprd 463 . . . . 5  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  B  e.  V
)
9 usgraedgrnv 24150 . . . . . . 7  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  ( B  e.  V  /\  C  e.  V ) )
109adantrl 715 . . . . . 6  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( B  e.  V  /\  C  e.  V ) )
1110simprd 463 . . . . 5  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  C  e.  V
)
126, 8, 113jca 1176 . . . 4  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )
132, 12jca 532 . . 3  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )
14 simpl 457 . . . . 5  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )  -> 
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )
15 usgra2adedgspthlem1 24384 . . . . . 6  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( ( E `
 ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) )
1615adantl 466 . . . . 5  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )  -> 
( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) )
17 fvex 5876 . . . . . . 7  |-  ( `' E `  { A ,  B } )  e. 
_V
18 fvex 5876 . . . . . . 7  |-  ( `' E `  { B ,  C } )  e. 
_V
1917, 18pm3.2i 455 . . . . . 6  |-  ( ( `' E `  { A ,  B } )  e. 
_V  /\  ( `' E `  { B ,  C } )  e. 
_V )
20 usgra2adedgspth.f . . . . . 6  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. }
21 usgra2adedgspth.p . . . . . 6  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
2219, 20, 21constr2wlk 24373 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  ->  (
( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } )  ->  F
( V Walks  E ) P ) )
2314, 16, 22sylc 60 . . . 4  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )  ->  F ( V Walks  E
) P )
2419, 202trllemA 24325 . . . . 5  |-  ( # `  F )  =  2
2524a1i 11 . . . 4  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )  -> 
( # `  F )  =  2 )
26212wlklemA 24329 . . . . . . . 8  |-  ( A  e.  V  ->  ( P `  0 )  =  A )
2726eqcomd 2475 . . . . . . 7  |-  ( A  e.  V  ->  A  =  ( P ` 
0 ) )
28212wlklemB 24330 . . . . . . . 8  |-  ( B  e.  V  ->  ( P `  1 )  =  B )
2928eqcomd 2475 . . . . . . 7  |-  ( B  e.  V  ->  B  =  ( P ` 
1 ) )
30212wlklemC 24331 . . . . . . . 8  |-  ( C  e.  V  ->  ( P `  2 )  =  C )
3130eqcomd 2475 . . . . . . 7  |-  ( C  e.  V  ->  C  =  ( P ` 
2 ) )
3227, 29, 313anim123i 1181 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( A  =  ( P `  0 )  /\  B  =  ( P `  1 )  /\  C  =  ( P `  2 ) ) )
3332adantl 466 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  ->  ( A  =  ( P `  0 )  /\  B  =  ( P `  1 )  /\  C  =  ( P `  2 ) ) )
3433adantr 465 . . . 4  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )  -> 
( A  =  ( P `  0 )  /\  B  =  ( P `  1 )  /\  C  =  ( P `  2 ) ) )
3523, 25, 343jca 1176 . . 3  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )  -> 
( F ( V Walks 
E ) P  /\  ( # `  F )  =  2  /\  ( A  =  ( P `  0 )  /\  B  =  ( P `  1 )  /\  C  =  ( P `  2 ) ) ) )
3613, 35mpancom 669 . 2  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( F ( V Walks  E ) P  /\  ( # `  F
)  =  2  /\  ( A  =  ( P `  0 )  /\  B  =  ( P `  1 )  /\  C  =  ( P `  2 ) ) ) )
3736ex 434 1  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  -> 
( F ( V Walks 
E ) P  /\  ( # `  F )  =  2  /\  ( A  =  ( P `  0 )  /\  B  =  ( P `  1 )  /\  C  =  ( P `  2 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   {cpr 4029   {ctp 4031   <.cop 4033   class class class wbr 4447   `'ccnv 4998   ran crn 5000   ` cfv 5588  (class class class)co 6285   0cc0 9493   1c1 9494   2c2 10586   #chash 12374   USGrph cusg 24103   Walks cwalk 24271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-hash 12375  df-word 12509  df-usgra 24106  df-wlk 24281
This theorem is referenced by:  usgra2adedgwlkonALT  24389  usg2wlk  24390
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