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Theorem usgra2adedgwlkon 24277
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 24001 . . . . . . 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 24039 . . . . . . . . . 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 24039 . . . . . . . . 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 1171 . . . . . 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 24273 . . . . 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 5867 . . . . . . 7  |-  ( `' E `  { A ,  B } )  e. 
_V
16 fvex 5867 . . . . . . 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 24262 . . . . 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 24218 . . . . . . . 8  |-  ( A  e.  V  ->  ( P `  0 )  =  A )
2625adantr 465 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( P `  0
)  =  A )
2717, 182trllemA 24214 . . . . . . . . . 10  |-  ( # `  F )  =  2
2827fveq2i 5860 . . . . . . . . 9  |-  ( P `
 ( # `  F
) )  =  ( P `  2 )
29192wlklemC 24220 . . . . . . . . 9  |-  ( C  e.  V  ->  ( P `  2 )  =  C )
3028, 29syl5eq 2513 . . . . . . . 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 972 . . . 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 4682 . . . . . . 7  |-  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. }  e.  _V
3818, 37eqeltri 2544 . . . . . 6  |-  F  e. 
_V
39 tpex 6574 . . . . . . 7  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V
4019, 39eqeltri 2544 . . . . . 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 834 . . . 4  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( A  e.  V  /\  C  e.  V ) )
46 iswlkon 24196 . . . 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 1223 . . 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 968    = wceq 1374    e. wcel 1762   _Vcvv 3106   {cpr 4022   {ctp 4024   <.cop 4026   class class class wbr 4440   `'ccnv 4991   ran crn 4993   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482   2c2 10574   #chash 12360   USGrph cusg 23993   Walks cwalk 24160   WalkOn cwlkon 24164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-usgra 23996  df-wlk 24170  df-wlkon 24176
This theorem is referenced by:  usg2wlkon  24280
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