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Theorem clwlkcompim 24426
Description: Implications for the properties of the components of a closed walk. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
Hypotheses
Ref Expression
clwlkcomp.1  |-  F  =  ( 1st `  W
)
clwlkcomp.2  |-  P  =  ( 2nd `  W
)
Assertion
Ref Expression
clwlkcompim  |-  ( W  e.  ( V ClWalks  E
)  ->  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  ( A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
Distinct variable groups:    k, E    k, F    P, k    k, V   
k, W

Proof of Theorem clwlkcompim
Dummy variables  e 
f  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-clwlk 24412 . . . 4  |- ClWalks  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
2 vex 3109 . . . . 5  |-  v  e. 
_V
3 vex 3109 . . . . 5  |-  e  e. 
_V
4 clwlk 24415 . . . . . 6  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  ( v ClWalks  e )  =  { <. f ,  p >.  |  (
f ( v Walks  e
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) } )
5 ovex 6300 . . . . . 6  |-  ( v ClWalks 
e )  e.  _V
64, 5syl6eqelr 2557 . . . . 5  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  { <. f ,  p >.  |  ( f ( v Walks  e ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) }  e.  _V )
72, 3, 6mp2an 672 . . . 4  |-  { <. f ,  p >.  |  ( f ( v Walks  e
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) }  e.  _V
8 oveq12 6284 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Walks  e )  =  ( V Walks  E
) )
98breqd 4451 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Walks 
e ) p  <->  f ( V Walks  E ) p ) )
109anbi1d 704 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f ( v Walks  e ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) )  <-> 
( f ( V Walks 
E ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) ) )
1110opabbidv 4503 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  { <. f ,  p >.  |  ( f ( v Walks  e ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) }  =  { <. f ,  p >.  |  ( f ( V Walks  E
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) } )
121, 7, 11elovmpt2 6495 . . 3  |-  ( W  e.  ( V ClWalks  E
)  <->  ( V  e. 
_V  /\  E  e.  _V  /\  W  e.  { <. f ,  p >.  |  ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } ) )
13 elopaelxp 5064 . . . 4  |-  ( W  e.  { <. f ,  p >.  |  (
f ( V Walks  E
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) }  ->  W  e.  ( _V  X.  _V ) )
14 clwlkcomp.1 . . . . . 6  |-  F  =  ( 1st `  W
)
15 clwlkcomp.2 . . . . . 6  |-  P  =  ( 2nd `  W
)
1614, 15clwlkcomp 24425 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  W  e.  ( _V  X.  _V ) )  ->  ( W  e.  ( V ClWalks  E )  <->  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  ( A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) ) )
1716biimpd 207 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  W  e.  ( _V  X.  _V ) )  ->  ( W  e.  ( V ClWalks  E )  ->  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  ( A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) ) )
1813, 17syl3an3 1258 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  W  e.  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) } )  ->  ( W  e.  ( V ClWalks  E )  ->  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  ( A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) ) )
1912, 18sylbi 195 . 2  |-  ( W  e.  ( V ClWalks  E
)  ->  ( W  e.  ( V ClWalks  E )  ->  ( ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  ( A. k  e.  (
0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  /\  ( P `
 0 )  =  ( P `  ( # `
 F ) ) ) ) ) )
2019pm2.43i 47 1  |-  ( W  e.  ( V ClWalks  E
)  ->  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  ( A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   _Vcvv 3106   {cpr 4022   class class class wbr 4440   {copab 4497    X. cxp 4990   dom cdm 4992   -->wf 5575   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   0cc0 9481   1c1 9482    + caddc 9484   ...cfz 11661  ..^cfzo 11781   #chash 12360  Word cword 12487   Walks cwalk 24160   ClWalks cclwlk 24409
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-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-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-wlk 24170  df-clwlk 24412
This theorem is referenced by:  clwlkfclwwlk2wrd  24502  clwlkfclwwlk1hash  24504  clwlkfclwwlk  24506  clwlkf1clwwlklem1  24508  clwlkf1clwwlklem2  24509  clwlkf1clwwlklem3  24510
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