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Theorem usgra2adedgwlkon 25343
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
usgra2adedgwlkon  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  F ( A ( V WalkOn  E ) C ) P ) )

Proof of Theorem usgra2adedgwlkon
StepHypRef Expression
1 usgrav 25065 . . . . . . 7  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
21adantr 467 . . . . . 6  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( V  e. 
_V  /\  E  e.  _V ) )
3 usgraedgrnv 25104 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  ( A  e.  V  /\  B  e.  V ) )
43ancomd 453 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  ( B  e.  V  /\  A  e.  V ) )
54adantrr 723 . . . . . . . 8  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( B  e.  V  /\  A  e.  V ) )
65simprd 465 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  A  e.  V
)
73adantrr 723 . . . . . . . 8  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( A  e.  V  /\  B  e.  V ) )
87simprd 465 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  B  e.  V
)
9 usgraedgrnv 25104 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  ( B  e.  V  /\  C  e.  V ) )
109adantrl 722 . . . . . . . 8  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( B  e.  V  /\  C  e.  V ) )
1110simprd 465 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  C  e.  V
)
126, 8, 113jca 1188 . . . . . 6  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )
132, 12jca 535 . . . . 5  |-  ( ( 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 usgra2adedgspthlem1 25339 . . . . 5  |-  ( ( 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 } ) )
15 fvex 5875 . . . . . . 7  |-  ( `' E `  { A ,  B } )  e. 
_V
16 fvex 5875 . . . . . . 7  |-  ( `' E `  { B ,  C } )  e. 
_V
1715, 16pm3.2i 457 . . . . . 6  |-  ( ( `' E `  { A ,  B } )  e. 
_V  /\  ( `' E `  { B ,  C } )  e. 
_V )
18 usgra2adedgspth.f . . . . . 6  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. }
19 usgra2adedgspth.p . . . . . 6  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
2017, 18, 19constr2wlk 25328 . . . . 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 ) )
2113, 14, 20sylc 62 . . . 4  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  F ( V Walks 
E ) P )
223ex 436 . . . . . . 7  |-  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  ->  ( A  e.  V  /\  B  e.  V ) ) )
239ex 436 . . . . . . 7  |-  ( V USGrph  E  ->  ( { B ,  C }  e.  ran  E  ->  ( B  e.  V  /\  C  e.  V ) ) )
2422, 23anim12d 566 . . . . . 6  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  -> 
( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) ) )
25192wlklemA 25284 . . . . . . . 8  |-  ( A  e.  V  ->  ( P `  0 )  =  A )
2625adantr 467 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( P `  0
)  =  A )
2717, 182trllemA 25280 . . . . . . . . . 10  |-  ( # `  F )  =  2
2827fveq2i 5868 . . . . . . . . 9  |-  ( P `
 ( # `  F
) )  =  ( P `  2 )
29192wlklemC 25286 . . . . . . . . 9  |-  ( C  e.  V  ->  ( P `  2 )  =  C )
3028, 29syl5eq 2497 . . . . . . . 8  |-  ( C  e.  V  ->  ( P `  ( # `  F
) )  =  C )
3130adantl 468 . . . . . . 7  |-  ( ( B  e.  V  /\  C  e.  V )  ->  ( P `  ( # `
 F ) )  =  C )
3226, 31anim12i 570 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  -> 
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  C ) )
3324, 32syl6 34 . . . . 5  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  -> 
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  C ) ) )
3433imp 431 . . . 4  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  C ) )
35 3anass 989 . . . 4  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  C )  <->  ( F ( V Walks  E ) P  /\  ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  C ) ) )
3621, 34, 35sylanbrc 670 . . 3  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( F ( V Walks  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  C ) )
37 prex 4642 . . . . . . 7  |-  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. }  e.  _V
3818, 37eqeltri 2525 . . . . . 6  |-  F  e. 
_V
39 tpex 6590 . . . . . . 7  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V
4019, 39eqeltri 2525 . . . . . 6  |-  P  e. 
_V
4138, 40pm3.2i 457 . . . . 5  |-  ( F  e.  _V  /\  P  e.  _V )
4241a1i 11 . . . 4  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( F  e. 
_V  /\  P  e.  _V ) )
433simpld 461 . . . . 5  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  A  e.  V )
449simprd 465 . . . . 5  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  C  e.  V )
4543, 44anim12dan 848 . . . 4  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( A  e.  V  /\  C  e.  V ) )
46 iswlkon 25262 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( F ( A ( V WalkOn  E ) C ) P  <->  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  C ) ) )
472, 42, 45, 46syl3anc 1268 . . 3  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( F ( A ( V WalkOn  E
) C ) P  <-> 
( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  C ) ) )
4836, 47mpbird 236 . 2  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  F ( A ( V WalkOn  E ) C ) P )
4948ex 436 1  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  F ( A ( V WalkOn  E ) C ) P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   _Vcvv 3045   {cpr 3970   {ctp 3972   <.cop 3974   class class class wbr 4402   `'ccnv 4833   ran crn 4835   ` cfv 5582  (class class class)co 6290   0cc0 9539   1c1 9540   2c2 10659   #chash 12515   USGrph cusg 25057   Walks cwalk 25226   WalkOn cwlkon 25230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-usgra 25060  df-wlk 25236  df-wlkon 25242
This theorem is referenced by:  usg2wlkon  25346
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