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Theorem usgra2adedgwlkon 30333
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 23292 . . . . . . 7  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
21adantr 465 . . . . . 6  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( V  e. 
_V  /\  E  e.  _V ) )
3 usgraedgrnv 23318 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  ( A  e.  V  /\  B  e.  V ) )
43ancomd 451 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  ( B  e.  V  /\  A  e.  V ) )
54adantrr 716 . . . . . . . 8  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( B  e.  V  /\  A  e.  V ) )
65simprd 463 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  A  e.  V
)
73adantrr 716 . . . . . . . 8  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( A  e.  V  /\  B  e.  V ) )
87simprd 463 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  B  e.  V
)
9 usgraedgrnv 23318 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  ( B  e.  V  /\  C  e.  V ) )
109adantrl 715 . . . . . . . 8  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( B  e.  V  /\  C  e.  V ) )
1110simprd 463 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  C  e.  V
)
126, 8, 113jca 1168 . . . . . 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 532 . . . . 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 30329 . . . . 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 5722 . . . . . . 7  |-  ( `' E `  { A ,  B } )  e. 
_V
16 fvex 5722 . . . . . . 7  |-  ( `' E `  { B ,  C } )  e. 
_V
1715, 16pm3.2i 455 . . . . . 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 23519 . . . . 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 60 . . . 4  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  F ( V Walks 
E ) P )
223ex 434 . . . . . . 7  |-  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  ->  ( A  e.  V  /\  B  e.  V ) ) )
239ex 434 . . . . . . 7  |-  ( V USGrph  E  ->  ( { B ,  C }  e.  ran  E  ->  ( B  e.  V  /\  C  e.  V ) ) )
2422, 23anim12d 563 . . . . . 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 23475 . . . . . . . 8  |-  ( A  e.  V  ->  ( P `  0 )  =  A )
2625adantr 465 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( P `  0
)  =  A )
2717, 182trllemA 23471 . . . . . . . . . 10  |-  ( # `  F )  =  2
2827fveq2i 5715 . . . . . . . . 9  |-  ( P `
 ( # `  F
) )  =  ( P `  2 )
29192wlklemC 23477 . . . . . . . . 9  |-  ( C  e.  V  ->  ( P `  2 )  =  C )
3028, 29syl5eq 2487 . . . . . . . 8  |-  ( C  e.  V  ->  ( P `  ( # `  F
) )  =  C )
3130adantl 466 . . . . . . 7  |-  ( ( B  e.  V  /\  C  e.  V )  ->  ( P `  ( # `
 F ) )  =  C )
3226, 31anim12i 566 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  -> 
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  C ) )
3324, 32syl6 33 . . . . 5  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  -> 
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  C ) ) )
3433imp 429 . . . 4  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  C ) )
35 3anass 969 . . . 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 664 . . 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 4555 . . . . . . 7  |-  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. }  e.  _V
3818, 37eqeltri 2513 . . . . . 6  |-  F  e. 
_V
39 tpex 6400 . . . . . . 7  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V
4019, 39eqeltri 2513 . . . . . 6  |-  P  e. 
_V
4138, 40pm3.2i 455 . . . . 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 459 . . . . 5  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  A  e.  V )
449simprd 463 . . . . 5  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  C  e.  V )
4543, 44anim12dan 833 . . . 4  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( A  e.  V  /\  C  e.  V ) )
46 iswlkon 23452 . . . 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 1218 . . 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 232 . 2  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  F ( A ( V WalkOn  E ) C ) P )
4948ex 434 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2993   {cpr 3900   {ctp 3902   <.cop 3904   class class class wbr 4313   `'ccnv 4860   ran crn 4862   ` cfv 5439  (class class class)co 6112   0cc0 9303   1c1 9304   2c2 10392   #chash 12124   USGrph cusg 23286   Walks cwalk 23427   WalkOn cwlkon 23431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-fzo 11570  df-hash 12125  df-word 12250  df-usgra 23288  df-wlk 23437  df-wlkon 23443
This theorem is referenced by:  usg2wlkon  30336
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