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Theorem erclwwlkeqlen 24644
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 24643 . 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 5872 . . . . . . . . 9  |-  ( U  =  ( W cyclShift  n )  ->  ( # `  U
)  =  ( # `  ( W cyclShift  n )
) )
4 clwwlkprop 24602 . . . . . . . . . . . . 13  |-  ( W  e.  ( V ClWWalks  E )  ->  ( V  e. 
_V  /\  E  e.  _V  /\  W  e. Word  V
) )
5 pm3.2 447 . . . . . . . . . . . . . 14  |-  ( W  e. Word  V  ->  (
n  e.  ( 0 ... ( # `  W
) )  ->  ( W  e. Word  V  /\  n  e.  ( 0 ... ( # `
 W ) ) ) ) )
653ad2ant3 1019 . . . . . . . . . . . . 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 726 . . . . . . . . . . 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 11700 . . . . . . . . . . 11  |-  ( n  e.  ( 0 ... ( # `  W
) )  ->  n  e.  ZZ )
11 cshwlen 12749 . . . . . . . . . . 11  |-  ( ( W  e. Word  V  /\  n  e.  ZZ )  ->  ( # `  ( W cyclShift  n ) )  =  ( # `  W
) )
1210, 11sylan2 474 . . . . . . . . . 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 2530 . . . . . . . 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 2959 . . . . . 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 1193 . . 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 973    = wceq 1379    e. wcel 1767   E.wrex 2818   _Vcvv 3118   class class class wbr 4453   {copab 4510   ` cfv 5594  (class class class)co 6295   0cc0 9504   ZZcz 10876   ...cfz 11684   #chash 12385  Word cword 12514   cyclShift ccsh 12738   ClWWalks cclwwlk 24580
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-hash 12386  df-word 12522  df-concat 12524  df-substr 12526  df-csh 12739  df-clwwlk 24583
This theorem is referenced by:  erclwwlksym  24646  erclwwlktr  24647
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