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Theorem eclclwwlkn1 30511
Description: An equivalence class according to  .~. (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by Alexander van der Vekens, 15-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
eclclwwlkn1  |-  ( B  e.  X  ->  ( B  e.  ( W /.  .~  )  <->  E. x  e.  W  B  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) } ) )
Distinct variable groups:    t, E, u    t, N, u    n, V, t, u    t, W, u    x, n, t, u, N    y, n, t, u, x, W    x,  .~ , y    x, W    x, E    x, N    x, V    x, X    x, B, y   
y, W    y, X
Allowed substitution hints:    B( u, t, n)    .~ ( u, t, n)    E( y, n)    N( y)    V( y)    X( u, t, n)

Proof of Theorem eclclwwlkn1
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, 2eclclwwlkn0 30510 . 2  |-  ( B  e.  X  ->  ( B  e.  ( W /.  .~  )  <->  E. x  e.  W  B  =  { y  |  x  .~  y } ) )
41, 2erclwwlknsym 30505 . . . . . . . 8  |-  ( x  .~  y  ->  y  .~  x )
51, 2erclwwlknsym 30505 . . . . . . . 8  |-  ( y  .~  x  ->  x  .~  y )
64, 5impbii 188 . . . . . . 7  |-  ( x  .~  y  <->  y  .~  x )
76a1i 11 . . . . . 6  |-  ( ( B  e.  X  /\  x  e.  W )  ->  ( x  .~  y  <->  y  .~  x ) )
87abbidv 2562 . . . . 5  |-  ( ( B  e.  X  /\  x  e.  W )  ->  { y  |  x  .~  y }  =  { y  |  y  .~  x } )
9 vex 2980 . . . . . . . 8  |-  y  e. 
_V
10 vex 2980 . . . . . . . 8  |-  x  e. 
_V
111, 2erclwwlkneq 30502 . . . . . . . 8  |-  ( ( y  e.  _V  /\  x  e.  _V )  ->  ( y  .~  x  <->  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) ) )
129, 10, 11mp2an 672 . . . . . . 7  |-  ( y  .~  x  <->  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N
) y  =  ( x cyclShift  n ) ) )
1312a1i 11 . . . . . 6  |-  ( ( B  e.  X  /\  x  e.  W )  ->  ( y  .~  x  <->  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) ) )
1413abbidv 2562 . . . . 5  |-  ( ( B  e.  X  /\  x  e.  W )  ->  { y  |  y  .~  x }  =  { y  |  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) } )
15 3anan12 978 . . . . . . . 8  |-  ( ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) )  <->  ( x  e.  W  /\  (
y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) ) )
16 ibar 504 . . . . . . . . . 10  |-  ( x  e.  W  ->  (
( y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) )  <->  ( x  e.  W  /\  (
y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) ) ) )
1716bicomd 201 . . . . . . . . 9  |-  ( x  e.  W  ->  (
( x  e.  W  /\  ( y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) ) )  <-> 
( y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) ) ) )
1817adantl 466 . . . . . . . 8  |-  ( ( B  e.  X  /\  x  e.  W )  ->  ( ( x  e.  W  /\  ( y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) )  <->  ( y  e.  W  /\  E. n  e.  ( 0 ... N
) y  =  ( x cyclShift  n ) ) ) )
1915, 18syl5bb 257 . . . . . . 7  |-  ( ( B  e.  X  /\  x  e.  W )  ->  ( ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N
) y  =  ( x cyclShift  n ) )  <->  ( y  e.  W  /\  E. n  e.  ( 0 ... N
) y  =  ( x cyclShift  n ) ) ) )
2019abbidv 2562 . . . . . 6  |-  ( ( B  e.  X  /\  x  e.  W )  ->  { y  |  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) }  =  { y  |  ( y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) } )
21 df-rab 2729 . . . . . 6  |-  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) }  =  {
y  |  ( y  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) }
2220, 21syl6eqr 2493 . . . . 5  |-  ( ( B  e.  X  /\  x  e.  W )  ->  { y  |  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) }  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) } )
238, 14, 223eqtrd 2479 . . . 4  |-  ( ( B  e.  X  /\  x  e.  W )  ->  { y  |  x  .~  y }  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) } )
2423eqeq2d 2454 . . 3  |-  ( ( B  e.  X  /\  x  e.  W )  ->  ( B  =  {
y  |  x  .~  y }  <->  B  =  {
y  e.  W  |  E. n  e.  (
0 ... N ) y  =  ( x cyclShift  n
) } ) )
2524rexbidva 2737 . 2  |-  ( B  e.  X  ->  ( E. x  e.  W  B  =  { y  |  x  .~  y } 
<->  E. x  e.  W  B  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) } ) )
263, 25bitrd 253 1  |-  ( B  e.  X  ->  ( B  e.  ( W /.  .~  )  <->  E. x  e.  W  B  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2721   {crab 2724   _Vcvv 2977   class class class wbr 4297   {copab 4354   ` cfv 5423  (class class class)co 6096   /.cqs 7105   0cc0 9287   ...cfz 11442   cyclShift ccsh 12430   ClWWalksN cclwwlkn 30419
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-ec 7108  df-qs 7112  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-hash 12109  df-word 12234  df-concat 12236  df-substr 12238  df-csh 12431  df-clwwlk 30421  df-clwwlkn 30422
This theorem is referenced by:  eleclclwwlkn  30512  hashecclwwlkn1  30513  usghashecclwwlk  30514
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