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Theorem Lemma2 30493
Description: .... (Contributed by Alexander van der Vekens, 15-Jun-2018.)
Assertion
Ref Expression
Lemma2  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  { y  e.  ( ( V ClWWalksN  E ) `  N
)  |  E. n  e.  ( 0 ... N
) y  =  ( W cyclShift  n ) }  =  { y  e. Word  V  |  E. n  e.  ( 0 ... N ) y  =  ( W cyclShift  n ) } )
Distinct variable groups:    n, E, y    n, N, y    n, V, y    n, W, y

Proof of Theorem Lemma2
Dummy variables  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2449 . . . 4  |-  ( y  =  x  ->  (
y  =  ( W cyclShift  n )  <->  x  =  ( W cyclShift  n ) ) )
21rexbidv 2736 . . 3  |-  ( y  =  x  ->  ( E. n  e.  (
0 ... N ) y  =  ( W cyclShift  n )  <->  E. n  e.  (
0 ... N ) x  =  ( W cyclShift  n ) ) )
32cbvrabv 2971 . 2  |-  { y  e.  ( ( V ClWWalksN  E ) `  N
)  |  E. n  e.  ( 0 ... N
) y  =  ( W cyclShift  n ) }  =  { x  e.  (
( V ClWWalksN  E ) `  N )  |  E. n  e.  ( 0 ... N ) x  =  ( W cyclShift  n ) }
4 clwwlknprop 30435 . . . . . . . 8  |-  ( w  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )
54simp2d 1001 . . . . . . 7  |-  ( w  e.  ( ( V ClWWalksN  E ) `  N
)  ->  w  e. Word  V )
65ad2antrl 727 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N
) )  /\  (
w  e.  ( ( V ClWWalksN  E ) `  N
)  /\  E. n  e.  ( 0 ... N
) w  =  ( W cyclShift  n ) ) )  ->  w  e. Word  V
)
7 simprr 756 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N
) )  /\  (
w  e.  ( ( V ClWWalksN  E ) `  N
)  /\  E. n  e.  ( 0 ... N
) w  =  ( W cyclShift  n ) ) )  ->  E. n  e.  ( 0 ... N ) w  =  ( W cyclShift  n ) )
86, 7jca 532 . . . . 5  |-  ( ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N
) )  /\  (
w  e.  ( ( V ClWWalksN  E ) `  N
)  /\  E. n  e.  ( 0 ... N
) w  =  ( W cyclShift  n ) ) )  ->  ( w  e. Word  V  /\  E. n  e.  ( 0 ... N
) w  =  ( W cyclShift  n ) ) )
9 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ( w  e. Word  V  /\  n  e.  ( 0 ... N ) )  /\  ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `
 N ) ) )  /\  w  =  ( W cyclShift  n )
)  ->  ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `
 N ) ) )
10 simpllr 758 . . . . . . . . . . . . 13  |-  ( ( ( ( w  e. Word  V  /\  n  e.  ( 0 ... N ) )  /\  ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `
 N ) ) )  /\  w  =  ( W cyclShift  n )
)  ->  n  e.  ( 0 ... N
) )
11 clwwnisshclwwn 30473 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  (
n  e.  ( 0 ... N )  -> 
( W cyclShift  n )  e.  ( ( V ClWWalksN  E ) `
 N ) ) )
129, 10, 11sylc 60 . . . . . . . . . . . 12  |-  ( ( ( ( w  e. Word  V  /\  n  e.  ( 0 ... N ) )  /\  ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `
 N ) ) )  /\  w  =  ( W cyclShift  n )
)  ->  ( W cyclShift  n )  e.  ( ( V ClWWalksN  E ) `  N
) )
13 eleq1 2503 . . . . . . . . . . . . 13  |-  ( w  =  ( W cyclShift  n )  ->  ( w  e.  ( ( V ClWWalksN  E ) `
 N )  <->  ( W cyclShift  n )  e.  ( ( V ClWWalksN  E ) `  N
) ) )
1413adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ( w  e. Word  V  /\  n  e.  ( 0 ... N ) )  /\  ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `
 N ) ) )  /\  w  =  ( W cyclShift  n )
)  ->  ( w  e.  ( ( V ClWWalksN  E ) `
 N )  <->  ( W cyclShift  n )  e.  ( ( V ClWWalksN  E ) `  N
) ) )
1512, 14mpbird 232 . . . . . . . . . . 11  |-  ( ( ( ( w  e. Word  V  /\  n  e.  ( 0 ... N ) )  /\  ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `
 N ) ) )  /\  w  =  ( W cyclShift  n )
)  ->  w  e.  ( ( V ClWWalksN  E ) `
 N ) )
1615exp31 604 . . . . . . . . . 10  |-  ( ( w  e. Word  V  /\  n  e.  ( 0 ... N ) )  ->  ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `
 N ) )  ->  ( w  =  ( W cyclShift  n )  ->  w  e.  ( ( V ClWWalksN  E ) `  N
) ) ) )
1716com23 78 . . . . . . . . 9  |-  ( ( w  e. Word  V  /\  n  e.  ( 0 ... N ) )  ->  ( w  =  ( W cyclShift  n )  ->  ( ( N  e. 
NN0  /\  W  e.  ( ( V ClWWalksN  E ) `
 N ) )  ->  w  e.  ( ( V ClWWalksN  E ) `  N ) ) ) )
1817rexlimdva 2841 . . . . . . . 8  |-  ( w  e. Word  V  ->  ( E. n  e.  (
0 ... N ) w  =  ( W cyclShift  n )  ->  ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `
 N ) )  ->  w  e.  ( ( V ClWWalksN  E ) `  N ) ) ) )
1918imp 429 . . . . . . 7  |-  ( ( w  e. Word  V  /\  E. n  e.  ( 0 ... N ) w  =  ( W cyclShift  n ) )  ->  ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `
 N ) )  ->  w  e.  ( ( V ClWWalksN  E ) `  N ) ) )
2019impcom 430 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N
) )  /\  (
w  e. Word  V  /\  E. n  e.  ( 0 ... N ) w  =  ( W cyclShift  n ) ) )  ->  w  e.  ( ( V ClWWalksN  E ) `
 N ) )
21 simprr 756 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N
) )  /\  (
w  e. Word  V  /\  E. n  e.  ( 0 ... N ) w  =  ( W cyclShift  n ) ) )  ->  E. n  e.  ( 0 ... N
) w  =  ( W cyclShift  n ) )
2220, 21jca 532 . . . . 5  |-  ( ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N
) )  /\  (
w  e. Word  V  /\  E. n  e.  ( 0 ... N ) w  =  ( W cyclShift  n ) ) )  ->  (
w  e.  ( ( V ClWWalksN  E ) `  N
)  /\  E. n  e.  ( 0 ... N
) w  =  ( W cyclShift  n ) ) )
238, 22impbida 828 . . . 4  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  (
( w  e.  ( ( V ClWWalksN  E ) `  N )  /\  E. n  e.  ( 0 ... N ) w  =  ( W cyclShift  n ) )  <->  ( w  e. Word  V  /\  E. n  e.  ( 0 ... N
) w  =  ( W cyclShift  n ) ) ) )
24 eqeq1 2449 . . . . . 6  |-  ( x  =  w  ->  (
x  =  ( W cyclShift  n )  <->  w  =  ( W cyclShift  n ) ) )
2524rexbidv 2736 . . . . 5  |-  ( x  =  w  ->  ( E. n  e.  (
0 ... N ) x  =  ( W cyclShift  n )  <->  E. n  e.  (
0 ... N ) w  =  ( W cyclShift  n ) ) )
2625elrab 3117 . . . 4  |-  ( w  e.  { x  e.  ( ( V ClWWalksN  E ) `
 N )  |  E. n  e.  ( 0 ... N ) x  =  ( W cyclShift  n ) }  <->  ( w  e.  ( ( V ClWWalksN  E ) `
 N )  /\  E. n  e.  ( 0 ... N ) w  =  ( W cyclShift  n ) ) )
27 eqeq1 2449 . . . . . 6  |-  ( y  =  w  ->  (
y  =  ( W cyclShift  n )  <->  w  =  ( W cyclShift  n ) ) )
2827rexbidv 2736 . . . . 5  |-  ( y  =  w  ->  ( E. n  e.  (
0 ... N ) y  =  ( W cyclShift  n )  <->  E. n  e.  (
0 ... N ) w  =  ( W cyclShift  n ) ) )
2928elrab 3117 . . . 4  |-  ( w  e.  { y  e. Word  V  |  E. n  e.  ( 0 ... N
) y  =  ( W cyclShift  n ) }  <->  ( w  e. Word  V  /\  E. n  e.  ( 0 ... N
) w  =  ( W cyclShift  n ) ) )
3023, 26, 293bitr4g 288 . . 3  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  (
w  e.  { x  e.  ( ( V ClWWalksN  E ) `
 N )  |  E. n  e.  ( 0 ... N ) x  =  ( W cyclShift  n ) }  <->  w  e.  { y  e. Word  V  |  E. n  e.  (
0 ... N ) y  =  ( W cyclShift  n ) } ) )
3130eqrdv 2441 . 2  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  { x  e.  ( ( V ClWWalksN  E ) `
 N )  |  E. n  e.  ( 0 ... N ) x  =  ( W cyclShift  n ) }  =  { y  e. Word  V  |  E. n  e.  ( 0 ... N ) y  =  ( W cyclShift  n ) } )
323, 31syl5eq 2487 1  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  { y  e.  ( ( V ClWWalksN  E ) `  N
)  |  E. n  e.  ( 0 ... N
) y  =  ( W cyclShift  n ) }  =  { y  e. Word  V  |  E. n  e.  ( 0 ... N ) y  =  ( W cyclShift  n ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716   {crab 2719   _Vcvv 2972   ` cfv 5418  (class class class)co 6091   0cc0 9282   NN0cn0 10579   ...cfz 11437   #chash 12103  Word cword 12221   cyclShift ccsh 12425   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  ax-pre-sup 9360
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-rmo 2723  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-sup 7691  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-div 9994  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-hash 12104  df-word 12229  df-lsw 12230  df-concat 12231  df-substr 12233  df-csh 12426  df-clwwlk 30416  df-clwwlkn 30417
This theorem is referenced by:  hashecclwwlkn1  30508  usghashecclwwlk  30509
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