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

Theorem clwwlkfo 24470
Description: Lemma 4 for clwwlkbij 24472: F is an onto function. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
Hypotheses
Ref Expression
clwwlkbij.d  |-  D  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) }
clwwlkbij.f  |-  F  =  ( t  e.  D  |->  ( t substr  <. 0 ,  N >. ) )
Assertion
Ref Expression
clwwlkfo  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  F : D -onto-> (
( V ClWWalksN  E ) `  N ) )
Distinct variable groups:    w, E    w, N    w, V    t, D    t, E, w    t, N    t, V    t, X    t, Y
Allowed substitution hints:    D( w)    F( w, t)    X( w)    Y( w)

Proof of Theorem clwwlkfo
Dummy variables  i  x  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwwlkbij.d . . 3  |-  D  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) }
2 clwwlkbij.f . . 3  |-  F  =  ( t  e.  D  |->  ( t substr  <. 0 ,  N >. ) )
31, 2clwwlkf 24467 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  F : D --> ( ( V ClWWalksN  E ) `  N
) )
4 clwwlknimp 24449 . . . . . . 7  |-  ( p  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( (
p  e. Word  V  /\  ( # `  p )  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p ) ,  ( p `  0 ) }  e.  ran  E
) )
5 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  /\  ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN ) )  -> 
( V  e.  X  /\  E  e.  Y  /\  N  e.  NN ) )
6 simpl1 999 . . . . . . . . . 10  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  /\  ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN ) )  -> 
( p  e. Word  V  /\  ( # `  p
)  =  N ) )
7 3simpc 995 . . . . . . . . . . 11  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  ->  ( A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E ) )
87adantr 465 . . . . . . . . . 10  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  /\  ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN ) )  -> 
( A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E ) )
91clwwlkel 24466 . . . . . . . . . 10  |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  /\  ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E ) )  ->  ( p concat  <" ( p ` 
0 ) "> )  e.  D )
105, 6, 8, 9syl3anc 1228 . . . . . . . . 9  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  /\  ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN ) )  -> 
( p concat  <" (
p `  0 ) "> )  e.  D
)
11 opeq2 4214 . . . . . . . . . . . . . . 15  |-  ( N  =  ( # `  p
)  ->  <. 0 ,  N >.  =  <. 0 ,  ( # `  p
) >. )
1211eqcoms 2479 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  <. 0 ,  N >.  =  <. 0 ,  ( # `  p
) >. )
1312oveq2d 6298 . . . . . . . . . . . . 13  |-  ( (
# `  p )  =  N  ->  ( ( p concat  <" ( p `
 0 ) "> ) substr  <. 0 ,  N >. )  =  ( ( p concat  <" (
p `  0 ) "> ) substr  <. 0 ,  ( # `  p
) >. ) )
1413adantl 466 . . . . . . . . . . . 12  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( ( p concat  <" (
p `  0 ) "> ) substr  <. 0 ,  N >. )  =  ( ( p concat  <" (
p `  0 ) "> ) substr  <. 0 ,  ( # `  p
) >. ) )
15143ad2ant1 1017 . . . . . . . . . . 11  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  ->  ( ( p concat  <" ( p ` 
0 ) "> ) substr  <. 0 ,  N >. )  =  ( ( p concat  <" ( p `
 0 ) "> ) substr  <. 0 ,  ( # `  p
) >. ) )
1615adantr 465 . . . . . . . . . 10  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  /\  ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN ) )  -> 
( ( p concat  <" (
p `  0 ) "> ) substr  <. 0 ,  N >. )  =  ( ( p concat  <" (
p `  0 ) "> ) substr  <. 0 ,  ( # `  p
) >. ) )
17 simpll 753 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  N  e.  NN )  ->  p  e. Word  V
)
18 fstwrdne0 12540 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN  /\  ( p  e. Word  V  /\  ( # `  p )  =  N ) )  ->  ( p ` 
0 )  e.  V
)
1918ancoms 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  N  e.  NN )  ->  ( p ` 
0 )  e.  V
)
2019s1cld 12572 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  N  e.  NN )  ->  <" ( p `
 0 ) ">  e. Word  V )
2117, 20jca 532 . . . . . . . . . . . . . . . 16  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  N  e.  NN )  ->  ( p  e. Word  V  /\  <" ( p `
 0 ) ">  e. Word  V )
)
2221ex 434 . . . . . . . . . . . . . . 15  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( N  e.  NN  ->  ( p  e. Word  V  /\  <" ( p `
 0 ) ">  e. Word  V )
) )
23223ad2ant1 1017 . . . . . . . . . . . . . 14  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  ->  ( N  e.  NN  ->  ( p  e. Word  V  /\  <" (
p `  0 ) ">  e. Word  V )
) )
2423com12 31 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  (
( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  ->  ( p  e. Word  V  /\  <" ( p `
 0 ) ">  e. Word  V )
) )
25243ad2ant3 1019 . . . . . . . . . . . 12  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p ) ,  ( p `  0 ) }  e.  ran  E
)  ->  ( p  e. Word  V  /\  <" (
p `  0 ) ">  e. Word  V )
) )
2625impcom 430 . . . . . . . . . . 11  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  /\  ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN ) )  -> 
( p  e. Word  V  /\  <" ( p `
 0 ) ">  e. Word  V )
)
27 swrdccat1 12639 . . . . . . . . . . 11  |-  ( ( p  e. Word  V  /\  <" ( p ` 
0 ) ">  e. Word  V )  ->  (
( p concat  <" (
p `  0 ) "> ) substr  <. 0 ,  ( # `  p
) >. )  =  p )
2826, 27syl 16 . . . . . . . . . 10  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  /\  ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN ) )  -> 
( ( p concat  <" (
p `  0 ) "> ) substr  <. 0 ,  ( # `  p
) >. )  =  p )
2916, 28eqtr2d 2509 . . . . . . . . 9  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  /\  ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN ) )  ->  p  =  ( (
p concat  <" ( p `
 0 ) "> ) substr  <. 0 ,  N >. ) )
3010, 29jca 532 . . . . . . . 8  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  /\  ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN ) )  -> 
( ( p concat  <" (
p `  0 ) "> )  e.  D  /\  p  =  (
( p concat  <" (
p `  0 ) "> ) substr  <. 0 ,  N >. ) ) )
3130ex 434 . . . . . . 7  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  ->  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  (
( p concat  <" (
p `  0 ) "> )  e.  D  /\  p  =  (
( p concat  <" (
p `  0 ) "> ) substr  <. 0 ,  N >. ) ) ) )
324, 31syl 16 . . . . . 6  |-  ( p  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  ( ( p concat  <" (
p `  0 ) "> )  e.  D  /\  p  =  (
( p concat  <" (
p `  0 ) "> ) substr  <. 0 ,  N >. ) ) ) )
3332impcom 430 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  /\  p  e.  ( ( V ClWWalksN  E ) `  N ) )  -> 
( ( p concat  <" (
p `  0 ) "> )  e.  D  /\  p  =  (
( p concat  <" (
p `  0 ) "> ) substr  <. 0 ,  N >. ) ) )
34 oveq1 6289 . . . . . . 7  |-  ( x  =  ( p concat  <" (
p `  0 ) "> )  ->  (
x substr  <. 0 ,  N >. )  =  ( ( p concat  <" ( p `
 0 ) "> ) substr  <. 0 ,  N >. ) )
3534eqeq2d 2481 . . . . . 6  |-  ( x  =  ( p concat  <" (
p `  0 ) "> )  ->  (
p  =  ( x substr  <. 0 ,  N >. )  <-> 
p  =  ( ( p concat  <" ( p `
 0 ) "> ) substr  <. 0 ,  N >. ) ) )
3635rspcev 3214 . . . . 5  |-  ( ( ( p concat  <" (
p `  0 ) "> )  e.  D  /\  p  =  (
( p concat  <" (
p `  0 ) "> ) substr  <. 0 ,  N >. ) )  ->  E. x  e.  D  p  =  ( x substr  <.
0 ,  N >. ) )
3733, 36syl 16 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  /\  p  e.  ( ( V ClWWalksN  E ) `  N ) )  ->  E. x  e.  D  p  =  ( x substr  <.
0 ,  N >. ) )
381, 2clwwlkfv 24468 . . . . . . 7  |-  ( x  e.  D  ->  ( F `  x )  =  ( x substr  <. 0 ,  N >. ) )
3938eqeq2d 2481 . . . . . 6  |-  ( x  e.  D  ->  (
p  =  ( F `
 x )  <->  p  =  ( x substr  <. 0 ,  N >. ) ) )
4039adantl 466 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  /\  p  e.  ( ( V ClWWalksN  E ) `
 N ) )  /\  x  e.  D
)  ->  ( p  =  ( F `  x )  <->  p  =  ( x substr  <. 0 ,  N >. ) ) )
4140rexbidva 2970 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  /\  p  e.  ( ( V ClWWalksN  E ) `  N ) )  -> 
( E. x  e.  D  p  =  ( F `  x )  <->  E. x  e.  D  p  =  ( x substr  <.
0 ,  N >. ) ) )
4237, 41mpbird 232 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  /\  p  e.  ( ( V ClWWalksN  E ) `  N ) )  ->  E. x  e.  D  p  =  ( F `  x ) )
4342ralrimiva 2878 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  A. p  e.  ( ( V ClWWalksN  E ) `  N ) E. x  e.  D  p  =  ( F `  x ) )
44 dffo3 6034 . 2  |-  ( F : D -onto-> ( ( V ClWWalksN  E ) `  N
)  <->  ( F : D
--> ( ( V ClWWalksN  E ) `
 N )  /\  A. p  e.  ( ( V ClWWalksN  E ) `  N
) E. x  e.  D  p  =  ( F `  x ) ) )
453, 43, 44sylanbrc 664 1  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  F : D -onto-> (
( V ClWWalksN  E ) `  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   {cpr 4029   <.cop 4033    |-> cmpt 4505   ran crn 5000   -->wf 5582   -onto->wfo 5584   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489    + caddc 9491    - cmin 9801   NNcn 10532  ..^cfzo 11788   #chash 12367  Word cword 12494   lastS clsw 12495   concat cconcat 12496   <"cs1 12497   substr csubstr 12498   WWalksN cwwlkn 24351   ClWWalksN cclwwlkn 24422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12368  df-word 12502  df-lsw 12503  df-concat 12504  df-s1 12505  df-substr 12506  df-wwlk 24352  df-wwlkn 24353  df-clwwlk 24424  df-clwwlkn 24425
This theorem is referenced by:  clwwlkf1o  24471
  Copyright terms: Public domain W3C validator