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Theorem clwlkcompim 24885
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 24871 . . . 4  |- ClWalks  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
2 vex 3037 . . . . 5  |-  v  e. 
_V
3 vex 3037 . . . . 5  |-  e  e. 
_V
4 clwlk 24874 . . . . . 6  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  ( v ClWalks  e )  =  { <. f ,  p >.  |  (
f ( v Walks  e
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) } )
5 ovex 6224 . . . . . 6  |-  ( v ClWalks 
e )  e.  _V
64, 5syl6eqelr 2479 . . . . 5  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  { <. f ,  p >.  |  ( f ( v Walks  e ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) ) }  e.  _V )
72, 3, 6mp2an 670 . . . 4  |-  { <. f ,  p >.  |  ( f ( v Walks  e
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) }  e.  _V
8 oveq12 6205 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Walks  e )  =  ( V Walks  E
) )
98breqd 4378 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Walks 
e ) p  <->  f ( V Walks  E ) p ) )
109anbi1d 702 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f ( v Walks  e ) p  /\  ( p ` 
0 )  =  ( p `  ( # `  f ) ) )  <-> 
( f ( V Walks 
E ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) ) )
1110opabbidv 4430 . . . 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 6419 . . 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 4986 . . . 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 24884 . . . . 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 1261 . . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034   {cpr 3946   class class class wbr 4367   {copab 4424    X. cxp 4911   dom cdm 4913   -->wf 5492   ` cfv 5496  (class class class)co 6196   1stc1st 6697   2ndc2nd 6698   0cc0 9403   1c1 9404    + caddc 9406   ...cfz 11593  ..^cfzo 11717   #chash 12307  Word cword 12438   Walks cwalk 24619   ClWalks cclwlk 24868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-hash 12308  df-word 12446  df-wlk 24629  df-clwlk 24871
This theorem is referenced by:  clwlkfclwwlk2wrd  24961  clwlkfclwwlk1hash  24963  clwlkfclwwlk  24965  clwlkf1clwwlklem1  24967  clwlkf1clwwlklem2  24968  clwlkf1clwwlklem3  24969
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