MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cshwsidrepsw Structured version   Unicode version

Theorem cshwsidrepsw 14432
Description: If cyclically shifting a word of length being a prime number by a number of positions which is not divisible by the prime number results in the word itself, the word is a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)
Assertion
Ref Expression
cshwsidrepsw  |-  ( ( W  e. Word  V  /\  ( # `  W )  e.  Prime )  ->  (
( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W )  ->  W  =  ( ( W ` 
0 ) repeatS  ( # `  W
) ) ) )

Proof of Theorem cshwsidrepsw
Dummy variables  i 
j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  ( # `  W )  e.  Prime )  ->  ( # `
 W )  e. 
Prime )
21adantr 465 . . . . . . . 8  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  ->  ( # `
 W )  e. 
Prime )
3 simp1 996 . . . . . . . . 9  |-  ( ( L  e.  ZZ  /\  ( L  mod  ( # `  W ) )  =/=  0  /\  ( W cyclShift  L )  =  W )  ->  L  e.  ZZ )
43adantl 466 . . . . . . . 8  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  ->  L  e.  ZZ )
5 simpr2 1003 . . . . . . . 8  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  ->  ( L  mod  ( # `  W
) )  =/=  0
)
62, 4, 53jca 1176 . . . . . . 7  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  ->  (
( # `  W )  e.  Prime  /\  L  e.  ZZ  /\  ( L  mod  ( # `  W
) )  =/=  0
) )
76adantr 465 . . . . . 6  |-  ( ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( # `  W )  e.  Prime  /\  L  e.  ZZ  /\  ( L  mod  ( # `  W ) )  =/=  0 ) )
8 simpr 461 . . . . . 6  |-  ( ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) )  ->  i  e.  ( 0..^ ( # `  W
) ) )
9 modprmn0modprm0 14187 . . . . . 6  |-  ( ( ( # `  W
)  e.  Prime  /\  L  e.  ZZ  /\  ( L  mod  ( # `  W
) )  =/=  0
)  ->  ( i  e.  ( 0..^ ( # `  W ) )  ->  E. j  e.  (
0..^ ( # `  W
) ) ( ( i  +  ( j  x.  L ) )  mod  ( # `  W
) )  =  0 ) )
107, 8, 9sylc 60 . . . . 5  |-  ( ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) )  ->  E. j  e.  ( 0..^ ( # `  W
) ) ( ( i  +  ( j  x.  L ) )  mod  ( # `  W
) )  =  0 )
11 elfzonn0 11831 . . . . . . . . . 10  |-  ( j  e.  ( 0..^ (
# `  W )
)  ->  j  e.  NN0 )
1211ad2antrr 725 . . . . . . . . 9  |-  ( ( ( j  e.  ( 0..^ ( # `  W
) )  /\  (
( i  +  ( j  x.  L ) )  mod  ( # `  W ) )  =  0 )  /\  (
( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) ) )  ->  j  e.  NN0 )
13 simpl 457 . . . . . . . . . . . . 13  |-  ( ( W  e. Word  V  /\  ( # `  W )  e.  Prime )  ->  W  e. Word  V )
1413, 3anim12i 566 . . . . . . . . . . . 12  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  ->  ( W  e. Word  V  /\  L  e.  ZZ ) )
1514adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) )  ->  ( W  e. Word  V  /\  L  e.  ZZ ) )
1615adantl 466 . . . . . . . . . 10  |-  ( ( ( j  e.  ( 0..^ ( # `  W
) )  /\  (
( i  +  ( j  x.  L ) )  mod  ( # `  W ) )  =  0 )  /\  (
( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) ) )  ->  ( W  e. Word  V  /\  L  e.  ZZ ) )
17 simpr3 1004 . . . . . . . . . . . 12  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  ->  ( W cyclShift  L )  =  W )
1817anim1i 568 . . . . . . . . . . 11  |-  ( ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( W cyclShift  L )  =  W  /\  i  e.  ( 0..^ ( # `  W
) ) ) )
1918adantl 466 . . . . . . . . . 10  |-  ( ( ( j  e.  ( 0..^ ( # `  W
) )  /\  (
( i  +  ( j  x.  L ) )  mod  ( # `  W ) )  =  0 )  /\  (
( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) ) )  ->  ( ( W cyclShift  L )  =  W  /\  i  e.  ( 0..^ ( # `  W
) ) ) )
20 cshweqrep 12748 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  L  e.  ZZ )  ->  ( ( ( W cyclShift  L )  =  W  /\  i  e.  ( 0..^ ( # `  W
) ) )  ->  A. k  e.  NN0  ( W `  i )  =  ( W `  ( ( i  +  ( k  x.  L
) )  mod  ( # `
 W ) ) ) ) )
2116, 19, 20sylc 60 . . . . . . . . 9  |-  ( ( ( j  e.  ( 0..^ ( # `  W
) )  /\  (
( i  +  ( j  x.  L ) )  mod  ( # `  W ) )  =  0 )  /\  (
( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) ) )  ->  A. k  e.  NN0  ( W `  i )  =  ( W `  ( ( i  +  ( k  x.  L ) )  mod  ( # `  W
) ) ) )
22 oveq1 6289 . . . . . . . . . . . . . 14  |-  ( k  =  j  ->  (
k  x.  L )  =  ( j  x.  L ) )
2322oveq2d 6298 . . . . . . . . . . . . 13  |-  ( k  =  j  ->  (
i  +  ( k  x.  L ) )  =  ( i  +  ( j  x.  L
) ) )
2423oveq1d 6297 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
( i  +  ( k  x.  L ) )  mod  ( # `  W ) )  =  ( ( i  +  ( j  x.  L
) )  mod  ( # `
 W ) ) )
2524fveq2d 5868 . . . . . . . . . . 11  |-  ( k  =  j  ->  ( W `  ( (
i  +  ( k  x.  L ) )  mod  ( # `  W
) ) )  =  ( W `  (
( i  +  ( j  x.  L ) )  mod  ( # `  W ) ) ) )
2625eqeq2d 2481 . . . . . . . . . 10  |-  ( k  =  j  ->  (
( W `  i
)  =  ( W `
 ( ( i  +  ( k  x.  L ) )  mod  ( # `  W
) ) )  <->  ( W `  i )  =  ( W `  ( ( i  +  ( j  x.  L ) )  mod  ( # `  W
) ) ) ) )
2726rspcva 3212 . . . . . . . . 9  |-  ( ( j  e.  NN0  /\  A. k  e.  NN0  ( W `  i )  =  ( W `  ( ( i  +  ( k  x.  L
) )  mod  ( # `
 W ) ) ) )  ->  ( W `  i )  =  ( W `  ( ( i  +  ( j  x.  L
) )  mod  ( # `
 W ) ) ) )
2812, 21, 27syl2anc 661 . . . . . . . 8  |-  ( ( ( j  e.  ( 0..^ ( # `  W
) )  /\  (
( i  +  ( j  x.  L ) )  mod  ( # `  W ) )  =  0 )  /\  (
( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) ) )  ->  ( W `  i )  =  ( W `  ( ( i  +  ( j  x.  L ) )  mod  ( # `  W
) ) ) )
29 fveq2 5864 . . . . . . . . . 10  |-  ( ( ( i  +  ( j  x.  L ) )  mod  ( # `  W ) )  =  0  ->  ( W `  ( ( i  +  ( j  x.  L
) )  mod  ( # `
 W ) ) )  =  ( W `
 0 ) )
3029adantl 466 . . . . . . . . 9  |-  ( ( j  e.  ( 0..^ ( # `  W
) )  /\  (
( i  +  ( j  x.  L ) )  mod  ( # `  W ) )  =  0 )  ->  ( W `  ( (
i  +  ( j  x.  L ) )  mod  ( # `  W
) ) )  =  ( W `  0
) )
3130adantr 465 . . . . . . . 8  |-  ( ( ( j  e.  ( 0..^ ( # `  W
) )  /\  (
( i  +  ( j  x.  L ) )  mod  ( # `  W ) )  =  0 )  /\  (
( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) ) )  ->  ( W `  ( ( i  +  ( j  x.  L
) )  mod  ( # `
 W ) ) )  =  ( W `
 0 ) )
3228, 31eqtrd 2508 . . . . . . 7  |-  ( ( ( j  e.  ( 0..^ ( # `  W
) )  /\  (
( i  +  ( j  x.  L ) )  mod  ( # `  W ) )  =  0 )  /\  (
( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) ) )  ->  ( W `  i )  =  ( W `  0 ) )
3332ex 434 . . . . . 6  |-  ( ( j  e.  ( 0..^ ( # `  W
) )  /\  (
( i  +  ( j  x.  L ) )  mod  ( # `  W ) )  =  0 )  ->  (
( ( ( W  e. Word  V  /\  ( # `
 W )  e. 
Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `  W
) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  i
)  =  ( W `
 0 ) ) )
3433rexlimiva 2951 . . . . 5  |-  ( E. j  e.  ( 0..^ ( # `  W
) ) ( ( i  +  ( j  x.  L ) )  mod  ( # `  W
) )  =  0  ->  ( ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) )  ->  ( W `  i )  =  ( W `  0 ) ) )
3510, 34mpcom 36 . . . 4  |-  ( ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  /\  i  e.  ( 0..^ ( # `  W ) ) )  ->  ( W `  i )  =  ( W `  0 ) )
3635ralrimiva 2878 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )
37 repswsymballbi 12711 . . . 4  |-  ( W  e. Word  V  ->  ( W  =  ( ( W `  0 ) repeatS  (
# `  W )
)  <->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
3837ad2antrr 725 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  ->  ( W  =  ( ( W `  0 ) repeatS  (
# `  W )
)  <->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
3936, 38mpbird 232 . 2  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  ->  W  =  ( ( W `
 0 ) repeatS  ( # `
 W ) ) )
4039ex 434 1  |-  ( ( W  e. Word  V  /\  ( # `  W )  e.  Prime )  ->  (
( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W )  ->  W  =  ( ( W ` 
0 ) repeatS  ( # `  W
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   ` cfv 5586  (class class class)co 6282   0cc0 9488    + caddc 9491    x. cmul 9493   NN0cn0 10791   ZZcz 10860  ..^cfzo 11788    mod cmo 11960   #chash 12369  Word cword 12496   repeatS creps 12503   cyclShift ccsh 12718   Primecprime 14072
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-hash 12370  df-word 12504  df-concat 12506  df-substr 12508  df-reps 12511  df-csh 12719  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-dvds 13844  df-gcd 14000  df-prm 14073  df-phi 14151
This theorem is referenced by:  cshwsidrepswmod0  14433
  Copyright terms: Public domain W3C validator