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Theorem erclwwlkeqlen 30407
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, 11-Jun-2018.)
Hypothesis
Ref Expression
erclwwlk.r  |-  .~  =  { <. u ,  w >.  |  ( u  e.  ( V ClWWalks  E )  /\  w  e.  ( V ClWWalks  E )  /\  E. n  e.  ( 0 ... ( # `  w
) ) u  =  ( w cyclShift  n )
) }
Assertion
Ref Expression
erclwwlkeqlen  |-  ( ( U  e.  X  /\  W  e.  Y )  ->  ( U  .~  W  ->  ( # `  U
)  =  ( # `  W ) ) )
Distinct variable groups:    n, E, u, w    n, V, u, w    U, n, u, w   
n, W, u, w   
n, X    n, Y
Allowed substitution hints:    .~ ( w, u, n)    X( w, u)    Y( w, u)

Proof of Theorem erclwwlkeqlen
StepHypRef Expression
1 erclwwlk.r . . 3  |-  .~  =  { <. u ,  w >.  |  ( u  e.  ( V ClWWalks  E )  /\  w  e.  ( V ClWWalks  E )  /\  E. n  e.  ( 0 ... ( # `  w
) ) u  =  ( w cyclShift  n )
) }
21erclwwlkeq 30406 . 2  |-  ( ( U  e.  X  /\  W  e.  Y )  ->  ( U  .~  W  <->  ( U  e.  ( V ClWWalks  E )  /\  W  e.  ( V ClWWalks  E )  /\  E. n  e.  ( 0 ... ( # `  W ) ) U  =  ( W cyclShift  n ) ) ) )
3 fveq2 5688 . . . . . . . . 9  |-  ( U  =  ( W cyclShift  n )  ->  ( # `  U
)  =  ( # `  ( W cyclShift  n )
) )
4 clwwlkprop 30358 . . . . . . . . . . . . 13  |-  ( W  e.  ( V ClWWalks  E )  ->  ( V  e. 
_V  /\  E  e.  _V  /\  W  e. Word  V
) )
5 pm3.2 445 . . . . . . . . . . . . . 14  |-  ( W  e. Word  V  ->  (
n  e.  ( 0 ... ( # `  W
) )  ->  ( W  e. Word  V  /\  n  e.  ( 0 ... ( # `
 W ) ) ) ) )
653ad2ant3 1006 . . . . . . . . . . . . 13  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  W  e. Word  V )  ->  (
n  e.  ( 0 ... ( # `  W
) )  ->  ( W  e. Word  V  /\  n  e.  ( 0 ... ( # `
 W ) ) ) ) )
74, 6syl 16 . . . . . . . . . . . 12  |-  ( W  e.  ( V ClWWalks  E )  ->  ( n  e.  ( 0 ... ( # `
 W ) )  ->  ( W  e. Word  V  /\  n  e.  ( 0 ... ( # `  W ) ) ) ) )
87ad2antlr 721 . . . . . . . . . . 11  |-  ( ( ( U  e.  ( V ClWWalks  E )  /\  W  e.  ( V ClWWalks  E )
)  /\  ( U  e.  X  /\  W  e.  Y ) )  -> 
( n  e.  ( 0 ... ( # `  W ) )  -> 
( W  e. Word  V  /\  n  e.  (
0 ... ( # `  W
) ) ) ) )
98imp 429 . . . . . . . . . 10  |-  ( ( ( ( U  e.  ( V ClWWalks  E )  /\  W  e.  ( V ClWWalks  E ) )  /\  ( U  e.  X  /\  W  e.  Y
) )  /\  n  e.  ( 0 ... ( # `
 W ) ) )  ->  ( W  e. Word  V  /\  n  e.  ( 0 ... ( # `
 W ) ) ) )
10 elfzelz 11449 . . . . . . . . . . 11  |-  ( n  e.  ( 0 ... ( # `  W
) )  ->  n  e.  ZZ )
11 cshwlen 12432 . . . . . . . . . . 11  |-  ( ( W  e. Word  V  /\  n  e.  ZZ )  ->  ( # `  ( W cyclShift  n ) )  =  ( # `  W
) )
1210, 11sylan2 471 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  n  e.  ( 0 ... ( # `  W
) ) )  -> 
( # `  ( W cyclShift  n ) )  =  ( # `  W
) )
139, 12syl 16 . . . . . . . . 9  |-  ( ( ( ( U  e.  ( V ClWWalks  E )  /\  W  e.  ( V ClWWalks  E ) )  /\  ( U  e.  X  /\  W  e.  Y
) )  /\  n  e.  ( 0 ... ( # `
 W ) ) )  ->  ( # `  ( W cyclShift  n ) )  =  ( # `  W
) )
143, 13sylan9eqr 2495 . . . . . . . 8  |-  ( ( ( ( ( U  e.  ( V ClWWalks  E )  /\  W  e.  ( V ClWWalks  E ) )  /\  ( U  e.  X  /\  W  e.  Y
) )  /\  n  e.  ( 0 ... ( # `
 W ) ) )  /\  U  =  ( W cyclShift  n )
)  ->  ( # `  U
)  =  ( # `  W ) )
1514ex 434 . . . . . . 7  |-  ( ( ( ( U  e.  ( V ClWWalks  E )  /\  W  e.  ( V ClWWalks  E ) )  /\  ( U  e.  X  /\  W  e.  Y
) )  /\  n  e.  ( 0 ... ( # `
 W ) ) )  ->  ( U  =  ( W cyclShift  n )  ->  ( # `  U
)  =  ( # `  W ) ) )
1615rexlimdva 2839 . . . . . 6  |-  ( ( ( U  e.  ( V ClWWalks  E )  /\  W  e.  ( V ClWWalks  E )
)  /\  ( U  e.  X  /\  W  e.  Y ) )  -> 
( E. n  e.  ( 0 ... ( # `
 W ) ) U  =  ( W cyclShift  n )  ->  ( # `
 U )  =  ( # `  W
) ) )
1716ex 434 . . . . 5  |-  ( ( U  e.  ( V ClWWalks  E )  /\  W  e.  ( V ClWWalks  E )
)  ->  ( ( U  e.  X  /\  W  e.  Y )  ->  ( E. n  e.  ( 0 ... ( # `
 W ) ) U  =  ( W cyclShift  n )  ->  ( # `
 U )  =  ( # `  W
) ) ) )
1817com23 78 . . . 4  |-  ( ( U  e.  ( V ClWWalks  E )  /\  W  e.  ( V ClWWalks  E )
)  ->  ( E. n  e.  ( 0 ... ( # `  W
) ) U  =  ( W cyclShift  n )  ->  ( ( U  e.  X  /\  W  e.  Y )  ->  ( # `
 U )  =  ( # `  W
) ) ) )
19183impia 1179 . . 3  |-  ( ( U  e.  ( V ClWWalks  E )  /\  W  e.  ( V ClWWalks  E )  /\  E. n  e.  ( 0 ... ( # `  W ) ) U  =  ( W cyclShift  n ) )  ->  ( ( U  e.  X  /\  W  e.  Y )  ->  ( # `  U
)  =  ( # `  W ) ) )
2019com12 31 . 2  |-  ( ( U  e.  X  /\  W  e.  Y )  ->  ( ( U  e.  ( V ClWWalks  E )  /\  W  e.  ( V ClWWalks  E )  /\  E. n  e.  ( 0 ... ( # `  W
) ) U  =  ( W cyclShift  n )
)  ->  ( # `  U
)  =  ( # `  W ) ) )
212, 20sylbid 215 1  |-  ( ( U  e.  X  /\  W  e.  Y )  ->  ( U  .~  W  ->  ( # `  U
)  =  ( # `  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   E.wrex 2714   _Vcvv 2970   class class class wbr 4289   {copab 4346   ` cfv 5415  (class class class)co 6090   0cc0 9278   ZZcz 10642   ...cfz 11433   #chash 12099  Word cword 12217   cyclShift ccsh 12421   ClWWalks cclwwlk 30338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-hash 12100  df-word 12225  df-concat 12227  df-substr 12229  df-csh 12422  df-clwwlk 30341
This theorem is referenced by:  erclwwlksym  30409  erclwwlktr  30410
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