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Theorem eleclclwwlknlem2 30489
Description: Lemma 2 for eleclclwwlkn 30505. (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 30488 . . 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 30433 . . . . . . . . . . 11  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 x )  =  N ) ) )
1110, 7eleq2s 2534 . . . . . . . . . 10  |-  ( x  e.  W  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) ) )
12 fznn0sub2 11487 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... N )  ->  ( N  -  k )  e.  ( 0 ... N
) )
13 oveq1 6097 . . . . . . . . . . . . . 14  |-  ( (
# `  x )  =  N  ->  ( (
# `  x )  -  k )  =  ( N  -  k
) )
1413eleq1d 2508 . . . . . . . . . . . . 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 1011 . . . . . . . . . 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 992 . . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  x
)  ->  ( 0 ... N )  =  ( 0 ... ( # `
 x ) ) )
3029eleq2d 2509 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  x
)  ->  ( k  e.  ( 0 ... N
)  <->  k  e.  ( 0 ... ( # `  x ) ) ) )
3130eqcoms 2445 . . . . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . . 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 2447 . . . . . . 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 6097 . . . . . . . 8  |-  ( X  =  ( x cyclShift  k
)  ->  ( X cyclShift  ( ( # `  x
)  -  k ) )  =  ( ( x cyclShift  k ) cyclShift  ( ( # `
 x )  -  k ) ) )
4544eqcoms 2445 . . . . . . 7  |-  ( ( x cyclShift  k )  =  X  ->  ( X cyclShift  ( (
# `  x )  -  k ) )  =  ( ( x cyclShift  k ) cyclShift  ( ( # `  x )  -  k
) ) )
46 elfzelz 11452 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( # `  x
) )  ->  k  e.  ZZ )
47 2cshwid 12447 . . . . . . . 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 2496 . . . . . 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 2447 . . . 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 30488 . . 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 828 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 965    = wceq 1369    e. wcel 1756   E.wrex 2715   _Vcvv 2971   ` cfv 5417  (class class class)co 6090   0cc0 9281    - cmin 9594   NN0cn0 10578   ZZcz 10645   ...cfz 11436   #chash 12102  Word cword 12220   cyclShift ccsh 12424   ClWWalksN cclwwlkn 30412
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-hash 12103  df-word 12228  df-concat 12230  df-substr 12232  df-csh 12425  df-clwwlk 30414  df-clwwlkn 30415
This theorem is referenced by:  eleclclwwlkn  30505
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