Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clwwlkfo Structured version   Unicode version

Theorem clwwlkfo 30459
Description: Lemma 4 for clwwlkbij 30461: 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 30456 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  F : D --> ( ( V ClWWalksN  E ) `  N
) )
4 clwwlknimp 30439 . . . . . . 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 991 . . . . . . . . . 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 987 . . . . . . . . . . 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 30455 . . . . . . . . . 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 1218 . . . . . . . . 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 4060 . . . . . . . . . . . . . . 15  |-  ( N  =  ( # `  p
)  ->  <. 0 ,  N >.  =  <. 0 ,  ( # `  p
) >. )
1211eqcoms 2446 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  <. 0 ,  N >.  =  <. 0 ,  ( # `  p
) >. )
1312oveq2d 6107 . . . . . . . . . . . . 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 1009 . . . . . . . . . . 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 12264 . . . . . . . . . . . . . . . . . . 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 12294 . . . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . 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 1011 . . . . . . . . . . . 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 12351 . . . . . . . . . . 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 2476 . . . . . . . . 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 6098 . . . . . . 7  |-  ( x  =  ( p concat  <" (
p `  0 ) "> )  ->  (
x substr  <. 0 ,  N >. )  =  ( ( p concat  <" ( p `
 0 ) "> ) substr  <. 0 ,  N >. ) )
3534eqeq2d 2454 . . . . . 6  |-  ( x  =  ( p concat  <" (
p `  0 ) "> )  ->  (
p  =  ( x substr  <. 0 ,  N >. )  <-> 
p  =  ( ( p concat  <" ( p `
 0 ) "> ) substr  <. 0 ,  N >. ) ) )
3635rspcev 3073 . . . . 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 30457 . . . . . . 7  |-  ( x  e.  D  ->  ( F `  x )  =  ( x substr  <. 0 ,  N >. ) )
3938eqeq2d 2454 . . . . . 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 2732 . . . 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 2799 . 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 5858 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   {crab 2719   {cpr 3879   <.cop 3883    e. cmpt 4350   ran crn 4841   -->wf 5414   -onto->wfo 5416   ` cfv 5418  (class class class)co 6091   0cc0 9282   1c1 9283    + caddc 9285    - cmin 9595   NNcn 10322  ..^cfzo 11548   #chash 12103  Word cword 12221   lastS clsw 12222   concat cconcat 12223   <"cs1 12224   substr csubstr 12225   WWalksN cwwlkn 30312   ClWWalksN cclwwlkn 30414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-lsw 12230  df-concat 12231  df-s1 12232  df-substr 12233  df-wwlk 30313  df-wwlkn 30314  df-clwwlk 30416  df-clwwlkn 30417
This theorem is referenced by:  clwwlkf1o  30460
  Copyright terms: Public domain W3C validator