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Theorem cshwsidrepswmod0 14788
Description: If cyclically shifting a word of length being a prime number results in the word itself, the shift must be either by 0 (modulo the length of the word) or the word must be a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)
Assertion
Ref Expression
cshwsidrepswmod0  |-  ( ( W  e. Word  V  /\  ( # `  W )  e.  Prime  /\  L  e.  ZZ )  ->  (
( W cyclShift  L )  =  W  ->  ( ( L  mod  ( # `  W
) )  =  0  \/  W  =  ( ( W `  0
) repeatS  ( # `  W
) ) ) ) )

Proof of Theorem cshwsidrepswmod0
StepHypRef Expression
1 orc 383 . . . 4  |-  ( ( L  mod  ( # `  W ) )  =  0  ->  ( ( L  mod  ( # `  W
) )  =  0  \/  W  =  ( ( W `  0
) repeatS  ( # `  W
) ) ) )
21a1d 25 . . 3  |-  ( ( L  mod  ( # `  W ) )  =  0  ->  ( ( W cyclShift  L )  =  W  ->  ( ( L  mod  ( # `  W
) )  =  0  \/  W  =  ( ( W `  0
) repeatS  ( # `  W
) ) ) ) )
32a1d 25 . 2  |-  ( ( L  mod  ( # `  W ) )  =  0  ->  ( ( W  e. Word  V  /\  ( # `
 W )  e. 
Prime  /\  L  e.  ZZ )  ->  ( ( W cyclShift  L )  =  W  ->  ( ( L  mod  ( # `  W
) )  =  0  \/  W  =  ( ( W `  0
) repeatS  ( # `  W
) ) ) ) ) )
4 3simpa 994 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( # `  W )  e.  Prime  /\  L  e.  ZZ )  ->  ( W  e. Word  V  /\  ( # `
 W )  e. 
Prime ) )
54ad2antlr 725 . . . . 5  |-  ( ( ( ( L  mod  ( # `  W ) )  =/=  0  /\  ( W  e. Word  V  /\  ( # `  W
)  e.  Prime  /\  L  e.  ZZ ) )  /\  ( W cyclShift  L )  =  W )  ->  ( W  e. Word  V  /\  ( # `
 W )  e. 
Prime ) )
6 simplr3 1041 . . . . 5  |-  ( ( ( ( L  mod  ( # `  W ) )  =/=  0  /\  ( W  e. Word  V  /\  ( # `  W
)  e.  Prime  /\  L  e.  ZZ ) )  /\  ( W cyclShift  L )  =  W )  ->  L  e.  ZZ )
7 simpll 752 . . . . 5  |-  ( ( ( ( L  mod  ( # `  W ) )  =/=  0  /\  ( W  e. Word  V  /\  ( # `  W
)  e.  Prime  /\  L  e.  ZZ ) )  /\  ( W cyclShift  L )  =  W )  ->  ( L  mod  ( # `  W
) )  =/=  0
)
8 simpr 459 . . . . 5  |-  ( ( ( ( L  mod  ( # `  W ) )  =/=  0  /\  ( W  e. Word  V  /\  ( # `  W
)  e.  Prime  /\  L  e.  ZZ ) )  /\  ( W cyclShift  L )  =  W )  ->  ( W cyclShift  L )  =  W )
9 cshwsidrepsw 14787 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( # `  W )  e.  Prime )  ->  (
( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W )  ->  W  =  ( ( W ` 
0 ) repeatS  ( # `  W
) ) ) )
109imp 427 . . . . 5  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )  /\  ( L  e.  ZZ  /\  ( L  mod  ( # `
 W ) )  =/=  0  /\  ( W cyclShift  L )  =  W ) )  ->  W  =  ( ( W `
 0 ) repeatS  ( # `
 W ) ) )
115, 6, 7, 8, 10syl13anc 1232 . . . 4  |-  ( ( ( ( L  mod  ( # `  W ) )  =/=  0  /\  ( W  e. Word  V  /\  ( # `  W
)  e.  Prime  /\  L  e.  ZZ ) )  /\  ( W cyclShift  L )  =  W )  ->  W  =  ( ( W `
 0 ) repeatS  ( # `
 W ) ) )
1211olcd 391 . . 3  |-  ( ( ( ( L  mod  ( # `  W ) )  =/=  0  /\  ( W  e. Word  V  /\  ( # `  W
)  e.  Prime  /\  L  e.  ZZ ) )  /\  ( W cyclShift  L )  =  W )  ->  (
( L  mod  ( # `
 W ) )  =  0  \/  W  =  ( ( W `
 0 ) repeatS  ( # `
 W ) ) ) )
1312exp31 602 . 2  |-  ( ( L  mod  ( # `  W ) )  =/=  0  ->  ( ( W  e. Word  V  /\  ( # `
 W )  e. 
Prime  /\  L  e.  ZZ )  ->  ( ( W cyclShift  L )  =  W  ->  ( ( L  mod  ( # `  W
) )  =  0  \/  W  =  ( ( W `  0
) repeatS  ( # `  W
) ) ) ) ) )
143, 13pm2.61ine 2716 1  |-  ( ( W  e. Word  V  /\  ( # `  W )  e.  Prime  /\  L  e.  ZZ )  ->  (
( W cyclShift  L )  =  W  ->  ( ( L  mod  ( # `  W
) )  =  0  \/  W  =  ( ( W `  0
) repeatS  ( # `  W
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   ` cfv 5569  (class class class)co 6278   0cc0 9522   ZZcz 10905    mod cmo 12034   #chash 12452  Word cword 12583   repeatS creps 12590   cyclShift ccsh 12815   Primecprime 14426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-hash 12453  df-word 12591  df-concat 12593  df-substr 12595  df-reps 12598  df-csh 12816  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-dvds 14196  df-gcd 14354  df-prm 14427  df-phi 14505
This theorem is referenced by:  cshwshashlem1  14789
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