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

Theorem erclwwlkneqlen 30636
Description: If two classes are equivalent regarding  .~, then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
Hypotheses
Ref Expression
erclwwlkn.w  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
erclwwlkn.r  |-  .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
Assertion
Ref Expression
erclwwlkneqlen  |-  ( ( T  e.  X  /\  U  e.  Y )  ->  ( T  .~  U  ->  ( # `  T
)  =  ( # `  U ) ) )
Distinct variable groups:    t, E, u    t, N, u    n, V, t, u    t, W, u    T, n, t, u    U, n, t, u    n, W    n, X    n, Y
Allowed substitution hints:    .~ ( u, t, n)    E( n)    N( n)    X( u, t)    Y( u, t)

Proof of Theorem erclwwlkneqlen
StepHypRef Expression
1 erclwwlkn.w . . 3  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
2 erclwwlkn.r . . 3  |-  .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
31, 2erclwwlkneq 30635 . 2  |-  ( ( T  e.  X  /\  U  e.  Y )  ->  ( T  .~  U  <->  ( T  e.  W  /\  U  e.  W  /\  E. n  e.  ( 0 ... N ) T  =  ( U cyclShift  n ) ) ) )
4 fveq2 5789 . . . . . . . . 9  |-  ( T  =  ( U cyclShift  n )  ->  ( # `  T
)  =  ( # `  ( U cyclShift  n )
) )
5 clwwlknprop 30573 . . . . . . . . . . . . . 14  |-  ( U  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 U )  =  N ) ) )
65, 1eleq2s 2559 . . . . . . . . . . . . 13  |-  ( U  e.  W  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  U
)  =  N ) ) )
7 simpl2 992 . . . . . . . . . . . . . . 15  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e. 
NN0  /\  ( # `  U
)  =  N ) )  /\  n  e.  ( 0 ... N
) )  ->  U  e. Word  V )
8 oveq2 6198 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  =  ( # `  U
)  ->  ( 0 ... N )  =  ( 0 ... ( # `
 U ) ) )
98eleq2d 2521 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  ( # `  U
)  ->  ( n  e.  ( 0 ... N
)  <->  n  e.  (
0 ... ( # `  U
) ) ) )
109eqcoms 2463 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  U )  =  N  ->  ( n  e.  ( 0 ... N )  <->  n  e.  ( 0 ... ( # `
 U ) ) ) )
1110adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( # `  U )  =  N )  -> 
( n  e.  ( 0 ... N )  <-> 
n  e.  ( 0 ... ( # `  U
) ) ) )
12113ad2ant3 1011 . . . . . . . . . . . . . . . 16  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  U
)  =  N ) )  ->  ( n  e.  ( 0 ... N
)  <->  n  e.  (
0 ... ( # `  U
) ) ) )
1312biimpa 484 . . . . . . . . . . . . . . 15  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e. 
NN0  /\  ( # `  U
)  =  N ) )  /\  n  e.  ( 0 ... N
) )  ->  n  e.  ( 0 ... ( # `
 U ) ) )
147, 13jca 532 . . . . . . . . . . . . . 14  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e. 
NN0  /\  ( # `  U
)  =  N ) )  /\  n  e.  ( 0 ... N
) )  ->  ( U  e. Word  V  /\  n  e.  ( 0 ... ( # `
 U ) ) ) )
1514ex 434 . . . . . . . . . . . . 13  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  U
)  =  N ) )  ->  ( n  e.  ( 0 ... N
)  ->  ( U  e. Word  V  /\  n  e.  ( 0 ... ( # `
 U ) ) ) ) )
166, 15syl 16 . . . . . . . . . . . 12  |-  ( U  e.  W  ->  (
n  e.  ( 0 ... N )  -> 
( U  e. Word  V  /\  n  e.  (
0 ... ( # `  U
) ) ) ) )
1716ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( T  e.  W  /\  U  e.  W
)  /\  ( T  e.  X  /\  U  e.  Y ) )  -> 
( n  e.  ( 0 ... N )  ->  ( U  e. Word  V  /\  n  e.  ( 0 ... ( # `  U ) ) ) ) )
1817imp 429 . . . . . . . . . 10  |-  ( ( ( ( T  e.  W  /\  U  e.  W )  /\  ( T  e.  X  /\  U  e.  Y )
)  /\  n  e.  ( 0 ... N
) )  ->  ( U  e. Word  V  /\  n  e.  ( 0 ... ( # `
 U ) ) ) )
19 elfzelz 11554 . . . . . . . . . . 11  |-  ( n  e.  ( 0 ... ( # `  U
) )  ->  n  e.  ZZ )
20 cshwlen 12538 . . . . . . . . . . 11  |-  ( ( U  e. Word  V  /\  n  e.  ZZ )  ->  ( # `  ( U cyclShift  n ) )  =  ( # `  U
) )
2119, 20sylan2 474 . . . . . . . . . 10  |-  ( ( U  e. Word  V  /\  n  e.  ( 0 ... ( # `  U
) ) )  -> 
( # `  ( U cyclShift  n ) )  =  ( # `  U
) )
2218, 21syl 16 . . . . . . . . 9  |-  ( ( ( ( T  e.  W  /\  U  e.  W )  /\  ( T  e.  X  /\  U  e.  Y )
)  /\  n  e.  ( 0 ... N
) )  ->  ( # `
 ( U cyclShift  n ) )  =  ( # `  U ) )
234, 22sylan9eqr 2514 . . . . . . . 8  |-  ( ( ( ( ( T  e.  W  /\  U  e.  W )  /\  ( T  e.  X  /\  U  e.  Y )
)  /\  n  e.  ( 0 ... N
) )  /\  T  =  ( U cyclShift  n ) )  ->  ( # `  T
)  =  ( # `  U ) )
2423ex 434 . . . . . . 7  |-  ( ( ( ( T  e.  W  /\  U  e.  W )  /\  ( T  e.  X  /\  U  e.  Y )
)  /\  n  e.  ( 0 ... N
) )  ->  ( T  =  ( U cyclShift  n )  ->  ( # `  T
)  =  ( # `  U ) ) )
2524rexlimdva 2937 . . . . . 6  |-  ( ( ( T  e.  W  /\  U  e.  W
)  /\  ( T  e.  X  /\  U  e.  Y ) )  -> 
( E. n  e.  ( 0 ... N
) T  =  ( U cyclShift  n )  ->  ( # `
 T )  =  ( # `  U
) ) )
2625ex 434 . . . . 5  |-  ( ( T  e.  W  /\  U  e.  W )  ->  ( ( T  e.  X  /\  U  e.  Y )  ->  ( E. n  e.  (
0 ... N ) T  =  ( U cyclShift  n )  ->  ( # `  T
)  =  ( # `  U ) ) ) )
2726com23 78 . . . 4  |-  ( ( T  e.  W  /\  U  e.  W )  ->  ( E. n  e.  ( 0 ... N
) T  =  ( U cyclShift  n )  ->  (
( T  e.  X  /\  U  e.  Y
)  ->  ( # `  T
)  =  ( # `  U ) ) ) )
28273impia 1185 . . 3  |-  ( ( T  e.  W  /\  U  e.  W  /\  E. n  e.  ( 0 ... N ) T  =  ( U cyclShift  n ) )  ->  ( ( T  e.  X  /\  U  e.  Y )  ->  ( # `  T
)  =  ( # `  U ) ) )
2928com12 31 . 2  |-  ( ( T  e.  X  /\  U  e.  Y )  ->  ( ( T  e.  W  /\  U  e.  W  /\  E. n  e.  ( 0 ... N
) T  =  ( U cyclShift  n ) )  -> 
( # `  T )  =  ( # `  U
) ) )
303, 29sylbid 215 1  |-  ( ( T  e.  X  /\  U  e.  Y )  ->  ( T  .~  U  ->  ( # `  T
)  =  ( # `  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2796   _Vcvv 3068   class class class wbr 4390   {copab 4447   ` cfv 5516  (class class class)co 6190   0cc0 9383   NN0cn0 10680   ZZcz 10747   ...cfz 11538   #chash 12204  Word cword 12323   cyclShift ccsh 12527   ClWWalksN cclwwlkn 30552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-fz 11539  df-fzo 11650  df-fl 11743  df-mod 11810  df-hash 12205  df-word 12331  df-concat 12333  df-substr 12335  df-csh 12528  df-clwwlk 30554  df-clwwlkn 30555
This theorem is referenced by:  erclwwlknsym  30638  erclwwlkntr  30639
  Copyright terms: Public domain W3C validator