MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wlkcompim Structured version   Unicode version

Theorem wlkcompim 24653
Description: Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
Hypotheses
Ref Expression
wlkcomp.1  |-  F  =  ( 1st `  W
)
wlkcomp.2  |-  P  =  ( 2nd `  W
)
Assertion
Ref Expression
wlkcompim  |-  ( W  e.  ( V Walks  E
)  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
Distinct variable groups:    k, E    k, F    P, k    k, V   
k, W

Proof of Theorem wlkcompim
Dummy variables  e 
f  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wlk 24635 . . . 4  |- Walks  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f ) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( e `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) } )
2 vex 3112 . . . . 5  |-  v  e. 
_V
3 vex 3112 . . . . 5  |-  e  e. 
_V
4 wlks 24646 . . . . . 6  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  ( v Walks  e )  =  { <. f ,  p >.  |  (
f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f
) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( e `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) } )
5 ovex 6324 . . . . . 6  |-  ( v Walks 
e )  e.  _V
64, 5syl6eqelr 2554 . . . . 5  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  { <. f ,  p >.  |  ( f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f
) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( e `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) }  e.  _V )
72, 3, 6mp2an 672 . . . 4  |-  { <. f ,  p >.  |  ( f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f
) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( e `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) }  e.  _V
8 dmeq 5213 . . . . . . . . 9  |-  ( e  =  E  ->  dom  e  =  dom  E )
9 wrdeq 12571 . . . . . . . . 9  |-  ( dom  e  =  dom  E  -> Word 
dom  e  = Word  dom  E )
108, 9syl 16 . . . . . . . 8  |-  ( e  =  E  -> Word  dom  e  = Word  dom  E )
1110eleq2d 2527 . . . . . . 7  |-  ( e  =  E  ->  (
f  e. Word  dom  e  <->  f  e. Word  dom 
E ) )
1211adantl 466 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f  e. Word  dom  e 
<->  f  e. Word  dom  E
) )
13 feq3 5721 . . . . . . 7  |-  ( v  =  V  ->  (
p : ( 0 ... ( # `  f
) ) --> v  <->  p :
( 0 ... ( # `
 f ) ) --> V ) )
1413adantr 465 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( p : ( 0 ... ( # `  f ) ) --> v  <-> 
p : ( 0 ... ( # `  f
) ) --> V ) )
15 fveq1 5871 . . . . . . . . 9  |-  ( e  =  E  ->  (
e `  ( f `  k ) )  =  ( E `  (
f `  k )
) )
1615adantl 466 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( e `  (
f `  k )
)  =  ( E `
 ( f `  k ) ) )
1716eqeq1d 2459 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( e `  ( f `  k
) )  =  {
( p `  k
) ,  ( p `
 ( k  +  1 ) ) }  <-> 
( E `  (
f `  k )
)  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) )
1817ralbidv 2896 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( A. k  e.  ( 0..^ ( # `  f ) ) ( e `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) }  <->  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) )
1912, 14, 183anbi123d 1299 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f
) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( e `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } )  <->  ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) ) )
2019opabbidv 4520 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  { <. f ,  p >.  |  ( f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f
) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( e `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) }  =  { <. f ,  p >.  |  ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) } )
211, 7, 20elovmpt2 6519 . . 3  |-  ( W  e.  ( V Walks  E
)  <->  ( V  e. 
_V  /\  E  e.  _V  /\  W  e.  { <. f ,  p >.  |  ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) } ) )
22 elopaelxp 5081 . . . 4  |-  ( W  e.  { <. f ,  p >.  |  (
f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) } ) }  ->  W  e.  ( _V  X.  _V ) )
23 wlkcomp.1 . . . . . 6  |-  F  =  ( 1st `  W
)
24 wlkcomp.2 . . . . . 6  |-  P  =  ( 2nd `  W
)
2523, 24wlkcomp 24652 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  W  e.  ( _V  X.  _V ) )  ->  ( W  e.  ( V Walks  E )  <->  ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) ) )
2625biimpd 207 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  W  e.  ( _V  X.  _V ) )  ->  ( W  e.  ( V Walks  E )  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
2722, 26syl3an3 1263 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  W  e.  { <. f ,  p >.  |  ( f  e. Word  dom  E  /\  p : ( 0 ... ( # `
 f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 k ) )  =  { ( p `
 k ) ,  ( p `  (
k  +  1 ) ) } ) } )  ->  ( W  e.  ( V Walks  E )  ->  ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) ) )
2821, 27sylbi 195 . 2  |-  ( W  e.  ( V Walks  E
)  ->  ( W  e.  ( V Walks  E )  ->  ( F  e. Word  dom  E  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) ) )
2928pm2.43i 47 1  |-  ( W  e.  ( V Walks  E
)  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   {cpr 4034   {copab 4514    X. cxp 5006   dom cdm 5008   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   0cc0 9509   1c1 9510    + caddc 9512   ...cfz 11697  ..^cfzo 11821   #chash 12408  Word cword 12538   Walks cwalk 24625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-wlk 24635
This theorem is referenced by:  usg2wlkeq  24835
  Copyright terms: Public domain W3C validator