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Theorem eleclclwwlknlem2 24632
Description: Lemma 2 for eleclclwwlkn 24647. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
Hypothesis
Ref Expression
erclwwlkn1.w  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
Assertion
Ref Expression
eleclclwwlknlem2  |-  ( ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W
) )  ->  ( E. m  e.  (
0 ... N ) Y  =  ( x cyclShift  m
)  <->  E. n  e.  ( 0 ... N ) Y  =  ( X cyclShift  n ) ) )
Distinct variable groups:    m, n, N    m, V, n    m, X, n    m, Y, n   
k, m, n    x, m, n
Allowed substitution hints:    E( x, k, m, n)    N( x, k)    V( x, k)    W( x, k, m, n)    X( x, k)    Y( x, k)

Proof of Theorem eleclclwwlknlem2
StepHypRef Expression
1 simpl 457 . . . . 5  |-  ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  ->  k  e.  ( 0 ... N
) )
21anim1i 568 . . . 4  |-  ( ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W
) )  ->  (
k  e.  ( 0 ... N )  /\  ( X  e.  W  /\  x  e.  W
) ) )
32adantr 465 . . 3  |-  ( ( ( ( k  e.  ( 0 ... N
)  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W ) )  /\  E. m  e.  ( 0 ... N ) Y  =  ( x cyclShift  m
) )  ->  (
k  e.  ( 0 ... N )  /\  ( X  e.  W  /\  x  e.  W
) ) )
4 simpr 461 . . . . 5  |-  ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  ->  X  =  ( x cyclShift  k
) )
54adantr 465 . . . 4  |-  ( ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W
) )  ->  X  =  ( x cyclShift  k
) )
65anim1i 568 . . 3  |-  ( ( ( ( k  e.  ( 0 ... N
)  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W ) )  /\  E. m  e.  ( 0 ... N ) Y  =  ( x cyclShift  m
) )  ->  ( X  =  ( x cyclShift  k )  /\  E. m  e.  ( 0 ... N
) Y  =  ( x cyclShift  m ) ) )
7 erclwwlkn1.w . . . 4  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
87eleclclwwlknlem1 24631 . . 3  |-  ( ( k  e.  ( 0 ... N )  /\  ( X  e.  W  /\  x  e.  W
) )  ->  (
( X  =  ( x cyclShift  k )  /\  E. m  e.  ( 0 ... N ) Y  =  ( x cyclShift  m
) )  ->  E. n  e.  ( 0 ... N
) Y  =  ( X cyclShift  n ) ) )
93, 6, 8sylc 60 . 2  |-  ( ( ( ( k  e.  ( 0 ... N
)  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W ) )  /\  E. m  e.  ( 0 ... N ) Y  =  ( x cyclShift  m
) )  ->  E. n  e.  ( 0 ... N
) Y  =  ( X cyclShift  n ) )
10 clwwlknprop 24586 . . . . . . . . . . 11  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 x )  =  N ) ) )
1110, 7eleq2s 2575 . . . . . . . . . 10  |-  ( x  e.  W  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) ) )
12 fznn0sub2 11791 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... N )  ->  ( N  -  k )  e.  ( 0 ... N
) )
13 oveq1 6302 . . . . . . . . . . . . . 14  |-  ( (
# `  x )  =  N  ->  ( (
# `  x )  -  k )  =  ( N  -  k
) )
1413eleq1d 2536 . . . . . . . . . . . . 13  |-  ( (
# `  x )  =  N  ->  ( ( ( # `  x
)  -  k )  e.  ( 0 ... N )  <->  ( N  -  k )  e.  ( 0 ... N
) ) )
1512, 14syl5ibr 221 . . . . . . . . . . . 12  |-  ( (
# `  x )  =  N  ->  ( k  e.  ( 0 ... N )  ->  (
( # `  x )  -  k )  e.  ( 0 ... N
) ) )
1615adantl 466 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  -> 
( k  e.  ( 0 ... N )  ->  ( ( # `  x )  -  k
)  e.  ( 0 ... N ) ) )
17163ad2ant3 1019 . . . . . . . . . 10  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( k  e.  ( 0 ... N
)  ->  ( ( # `
 x )  -  k )  e.  ( 0 ... N ) ) )
1811, 17syl 16 . . . . . . . . 9  |-  ( x  e.  W  ->  (
k  e.  ( 0 ... N )  -> 
( ( # `  x
)  -  k )  e.  ( 0 ... N ) ) )
1918adantl 466 . . . . . . . 8  |-  ( ( X  e.  W  /\  x  e.  W )  ->  ( k  e.  ( 0 ... N )  ->  ( ( # `  x )  -  k
)  e.  ( 0 ... N ) ) )
2019com12 31 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  (
( X  e.  W  /\  x  e.  W
)  ->  ( ( # `
 x )  -  k )  e.  ( 0 ... N ) ) )
2120adantr 465 . . . . . 6  |-  ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  ->  (
( X  e.  W  /\  x  e.  W
)  ->  ( ( # `
 x )  -  k )  e.  ( 0 ... N ) ) )
2221imp 429 . . . . 5  |-  ( ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W
) )  ->  (
( # `  x )  -  k )  e.  ( 0 ... N
) )
2322adantr 465 . . . 4  |-  ( ( ( ( k  e.  ( 0 ... N
)  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W ) )  /\  E. n  e.  ( 0 ... N ) Y  =  ( X cyclShift  n ) )  ->  ( ( # `
 x )  -  k )  e.  ( 0 ... N ) )
24 simpr 461 . . . . . 6  |-  ( ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W
) )  ->  ( X  e.  W  /\  x  e.  W )
)
2524ancomd 451 . . . . 5  |-  ( ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W
) )  ->  (
x  e.  W  /\  X  e.  W )
)
2625adantr 465 . . . 4  |-  ( ( ( ( k  e.  ( 0 ... N
)  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W ) )  /\  E. n  e.  ( 0 ... N ) Y  =  ( X cyclShift  n ) )  ->  ( x  e.  W  /\  X  e.  W ) )
2723, 26jca 532 . . 3  |-  ( ( ( ( k  e.  ( 0 ... N
)  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W ) )  /\  E. n  e.  ( 0 ... N ) Y  =  ( X cyclShift  n ) )  ->  ( (
( # `  x )  -  k )  e.  ( 0 ... N
)  /\  ( x  e.  W  /\  X  e.  W ) ) )
28 simpl2 1000 . . . . . . . . . . . . 13  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 x )  =  N ) )  /\  k  e.  ( 0 ... N ) )  ->  x  e. Word  V
)
29 oveq2 6303 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  x
)  ->  ( 0 ... N )  =  ( 0 ... ( # `
 x ) ) )
3029eleq2d 2537 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  x
)  ->  ( k  e.  ( 0 ... N
)  <->  k  e.  ( 0 ... ( # `  x ) ) ) )
3130eqcoms 2479 . . . . . . . . . . . . . . . 16  |-  ( (
# `  x )  =  N  ->  ( k  e.  ( 0 ... N )  <->  k  e.  ( 0 ... ( # `
 x ) ) ) )
3231adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  -> 
( k  e.  ( 0 ... N )  <-> 
k  e.  ( 0 ... ( # `  x
) ) ) )
33323ad2ant3 1019 . . . . . . . . . . . . . 14  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( k  e.  ( 0 ... N
)  <->  k  e.  ( 0 ... ( # `  x ) ) ) )
3433biimpa 484 . . . . . . . . . . . . 13  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 x )  =  N ) )  /\  k  e.  ( 0 ... N ) )  ->  k  e.  ( 0 ... ( # `  x ) ) )
3528, 34jca 532 . . . . . . . . . . . 12  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 x )  =  N ) )  /\  k  e.  ( 0 ... N ) )  ->  ( x  e. Word  V  /\  k  e.  ( 0 ... ( # `  x ) ) ) )
3635ex 434 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( k  e.  ( 0 ... N
)  ->  ( x  e. Word  V  /\  k  e.  ( 0 ... ( # `
 x ) ) ) ) )
3711, 36syl 16 . . . . . . . . . 10  |-  ( x  e.  W  ->  (
k  e.  ( 0 ... N )  -> 
( x  e. Word  V  /\  k  e.  (
0 ... ( # `  x
) ) ) ) )
3837adantl 466 . . . . . . . . 9  |-  ( ( X  e.  W  /\  x  e.  W )  ->  ( k  e.  ( 0 ... N )  ->  ( x  e. Word  V  /\  k  e.  ( 0 ... ( # `  x ) ) ) ) )
3938com12 31 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  (
( X  e.  W  /\  x  e.  W
)  ->  ( x  e. Word  V  /\  k  e.  ( 0 ... ( # `
 x ) ) ) ) )
4039adantr 465 . . . . . . 7  |-  ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  ->  (
( X  e.  W  /\  x  e.  W
)  ->  ( x  e. Word  V  /\  k  e.  ( 0 ... ( # `
 x ) ) ) ) )
4140imp 429 . . . . . 6  |-  ( ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W
) )  ->  (
x  e. Word  V  /\  k  e.  ( 0 ... ( # `  x
) ) ) )
424eqcomd 2475 . . . . . . 7  |-  ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  ->  (
x cyclShift  k )  =  X )
4342adantr 465 . . . . . 6  |-  ( ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W
) )  ->  (
x cyclShift  k )  =  X )
44 oveq1 6302 . . . . . . . 8  |-  ( X  =  ( x cyclShift  k
)  ->  ( X cyclShift  ( ( # `  x
)  -  k ) )  =  ( ( x cyclShift  k ) cyclShift  ( ( # `
 x )  -  k ) ) )
4544eqcoms 2479 . . . . . . 7  |-  ( ( x cyclShift  k )  =  X  ->  ( X cyclShift  ( (
# `  x )  -  k ) )  =  ( ( x cyclShift  k ) cyclShift  ( ( # `  x )  -  k
) ) )
46 elfzelz 11700 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( # `  x
) )  ->  k  e.  ZZ )
47 2cshwid 12762 . . . . . . . 8  |-  ( ( x  e. Word  V  /\  k  e.  ZZ )  ->  ( ( x cyclShift  k
) cyclShift  ( ( # `  x
)  -  k ) )  =  x )
4846, 47sylan2 474 . . . . . . 7  |-  ( ( x  e. Word  V  /\  k  e.  ( 0 ... ( # `  x
) ) )  -> 
( ( x cyclShift  k
) cyclShift  ( ( # `  x
)  -  k ) )  =  x )
4945, 48sylan9eqr 2530 . . . . . 6  |-  ( ( ( x  e. Word  V  /\  k  e.  (
0 ... ( # `  x
) ) )  /\  ( x cyclShift  k )  =  X )  ->  ( X cyclShift  ( ( # `  x
)  -  k ) )  =  x )
5041, 43, 49syl2anc 661 . . . . 5  |-  ( ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W
) )  ->  ( X cyclShift  ( ( # `  x
)  -  k ) )  =  x )
5150eqcomd 2475 . . . 4  |-  ( ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W
) )  ->  x  =  ( X cyclShift  ( (
# `  x )  -  k ) ) )
5251anim1i 568 . . 3  |-  ( ( ( ( k  e.  ( 0 ... N
)  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W ) )  /\  E. n  e.  ( 0 ... N ) Y  =  ( X cyclShift  n ) )  ->  ( x  =  ( X cyclShift  ( (
# `  x )  -  k ) )  /\  E. n  e.  ( 0 ... N
) Y  =  ( X cyclShift  n ) ) )
537eleclclwwlknlem1 24631 . . 3  |-  ( ( ( ( # `  x
)  -  k )  e.  ( 0 ... N )  /\  (
x  e.  W  /\  X  e.  W )
)  ->  ( (
x  =  ( X cyclShift  ( ( # `  x
)  -  k ) )  /\  E. n  e.  ( 0 ... N
) Y  =  ( X cyclShift  n ) )  ->  E. m  e.  (
0 ... N ) Y  =  ( x cyclShift  m
) ) )
5427, 52, 53sylc 60 . 2  |-  ( ( ( ( k  e.  ( 0 ... N
)  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W ) )  /\  E. n  e.  ( 0 ... N ) Y  =  ( X cyclShift  n ) )  ->  E. m  e.  ( 0 ... N
) Y  =  ( x cyclShift  m ) )
559, 54impbida 830 1  |-  ( ( ( k  e.  ( 0 ... N )  /\  X  =  ( x cyclShift  k ) )  /\  ( X  e.  W  /\  x  e.  W
) )  ->  ( E. m  e.  (
0 ... N ) Y  =  ( x cyclShift  m
)  <->  E. n  e.  ( 0 ... N ) Y  =  ( X cyclShift  n ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818   _Vcvv 3118   ` cfv 5594  (class class class)co 6295   0cc0 9504    - cmin 9817   NN0cn0 10807   ZZcz 10876   ...cfz 11684   #chash 12385  Word cword 12515   cyclShift ccsh 12739   ClWWalksN cclwwlkn 24563
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-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 12523  df-concat 12525  df-substr 12527  df-csh 12740  df-clwwlk 24565  df-clwwlkn 24566
This theorem is referenced by:  eleclclwwlkn  24647
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