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Theorem clwlkcompim 24629
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 24615 . . . 4  |- ClWalks  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
2 vex 3096 . . . . 5  |-  v  e. 
_V
3 vex 3096 . . . . 5  |-  e  e. 
_V
4 clwlk 24618 . . . . . 6  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  ( v ClWalks  e )  =  { <. f ,  p >.  |  (
f ( v Walks  e
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) } )
5 ovex 6305 . . . . . 6  |-  ( v ClWalks 
e )  e.  _V
64, 5syl6eqelr 2538 . . . . 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 6286 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Walks  e )  =  ( V Walks  E
) )
98breqd 4444 . . . . . 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 4496 . . . 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 6501 . . 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 5058 . . . 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 24628 . . . . 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 1262 . . 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 972    = wceq 1381    e. wcel 1802   A.wral 2791   _Vcvv 3093   {cpr 4012   class class class wbr 4433   {copab 4490    X. cxp 4983   dom cdm 4985   -->wf 5570   ` cfv 5574  (class class class)co 6277   1stc1st 6779   2ndc2nd 6780   0cc0 9490   1c1 9491    + caddc 9493   ...cfz 11676  ..^cfzo 11798   #chash 12379  Word cword 12508   Walks cwalk 24363   ClWalks cclwlk 24612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-fzo 11799  df-hash 12380  df-word 12516  df-wlk 24373  df-clwlk 24615
This theorem is referenced by:  clwlkfclwwlk2wrd  24705  clwlkfclwwlk1hash  24707  clwlkfclwwlk  24709  clwlkf1clwwlklem1  24711  clwlkf1clwwlklem2  24712  clwlkf1clwwlklem3  24713
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