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

Theorem cshwmodn 12729
Description: Cyclically shifting a word is invariant regarding modulo the word's length. (Contributed by AV, 26-Oct-2018.)
Assertion
Ref Expression
cshwmodn  |-  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( W cyclShift  N )  =  ( W cyclShift  ( N  mod  ( # `  W
) ) ) )

Proof of Theorem cshwmodn
StepHypRef Expression
1 0csh0 12727 . . . 4  |-  ( (/) cyclShift  N
)  =  (/)
2 oveq1 6291 . . . 4  |-  ( W  =  (/)  ->  ( W cyclShift  N )  =  (
(/) cyclShift  N ) )
3 oveq1 6291 . . . . 5  |-  ( W  =  (/)  ->  ( W cyclShift  ( N  mod  ( # `  W ) ) )  =  ( (/) cyclShift  ( N  mod  ( # `  W
) ) ) )
4 0csh0 12727 . . . . 5  |-  ( (/) cyclShift  ( N  mod  ( # `  W ) ) )  =  (/)
53, 4syl6eq 2524 . . . 4  |-  ( W  =  (/)  ->  ( W cyclShift  ( N  mod  ( # `  W ) ) )  =  (/) )
61, 2, 53eqtr4a 2534 . . 3  |-  ( W  =  (/)  ->  ( W cyclShift  N )  =  ( W cyclShift  ( N  mod  ( # `
 W ) ) ) )
76a1d 25 . 2  |-  ( W  =  (/)  ->  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( W cyclShift  N )  =  ( W cyclShift  ( N  mod  ( # `  W
) ) ) ) )
8 lennncl 12529 . . . . . . 7  |-  ( ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
9 zre 10868 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  N  e.  RR )
10 nnrp 11229 . . . . . . . . . . . 12  |-  ( (
# `  W )  e.  NN  ->  ( # `  W
)  e.  RR+ )
11 modabs2 11998 . . . . . . . . . . . 12  |-  ( ( N  e.  RR  /\  ( # `  W )  e.  RR+ )  ->  (
( N  mod  ( # `
 W ) )  mod  ( # `  W
) )  =  ( N  mod  ( # `  W ) ) )
129, 10, 11syl2anr 478 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )  ->  ( ( N  mod  ( # `  W ) )  mod  ( # `  W ) )  =  ( N  mod  ( # `
 W ) ) )
1312opeq1d 4219 . . . . . . . . . 10  |-  ( ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )  -> 
<. ( ( N  mod  ( # `  W ) )  mod  ( # `  W ) ) ,  ( # `  W
) >.  =  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. )
1413oveq2d 6300 . . . . . . . . 9  |-  ( ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )  ->  ( W substr  <. (
( N  mod  ( # `
 W ) )  mod  ( # `  W
) ) ,  (
# `  W ) >. )  =  ( W substr  <. ( N  mod  ( # `
 W ) ) ,  ( # `  W
) >. ) )
1512opeq2d 4220 . . . . . . . . . 10  |-  ( ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )  -> 
<. 0 ,  ( ( N  mod  ( # `
 W ) )  mod  ( # `  W
) ) >.  =  <. 0 ,  ( N  mod  ( # `  W
) ) >. )
1615oveq2d 6300 . . . . . . . . 9  |-  ( ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )  ->  ( W substr  <. 0 ,  ( ( N  mod  ( # `  W
) )  mod  ( # `
 W ) )
>. )  =  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >.
) )
1714, 16oveq12d 6302 . . . . . . . 8  |-  ( ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )  ->  ( ( W substr  <. (
( N  mod  ( # `
 W ) )  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( ( N  mod  ( # `  W
) )  mod  ( # `
 W ) )
>. ) )  =  ( ( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >. ) ) )
1817ex 434 . . . . . . 7  |-  ( (
# `  W )  e.  NN  ->  ( N  e.  ZZ  ->  ( ( W substr  <. ( ( N  mod  ( # `  W
) )  mod  ( # `
 W ) ) ,  ( # `  W
) >. ) concat  ( W substr  <.
0 ,  ( ( N  mod  ( # `  W ) )  mod  ( # `  W
) ) >. )
)  =  ( ( W substr  <. ( N  mod  ( # `  W ) ) ,  ( # `  W ) >. ) concat  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >.
) ) ) )
198, 18syl 16 . . . . . 6  |-  ( ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( N  e.  ZZ  ->  ( ( W substr  <. (
( N  mod  ( # `
 W ) )  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( ( N  mod  ( # `  W
) )  mod  ( # `
 W ) )
>. ) )  =  ( ( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >. ) ) ) )
2019impancom 440 . . . . 5  |-  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( W  =/=  (/)  ->  (
( W substr  <. ( ( N  mod  ( # `  W ) )  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( ( N  mod  ( # `  W
) )  mod  ( # `
 W ) )
>. ) )  =  ( ( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >. ) ) ) )
2120impcom 430 . . . 4  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ZZ ) )  -> 
( ( W substr  <. (
( N  mod  ( # `
 W ) )  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( ( N  mod  ( # `  W
) )  mod  ( # `
 W ) )
>. ) )  =  ( ( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >. ) ) )
22 simprl 755 . . . . 5  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ZZ ) )  ->  W  e. Word  V )
23 simprr 756 . . . . . . 7  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ZZ ) )  ->  N  e.  ZZ )
248ex 434 . . . . . . . . 9  |-  ( W  e. Word  V  ->  ( W  =/=  (/)  ->  ( # `  W
)  e.  NN ) )
2524adantr 465 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( W  =/=  (/)  ->  ( # `
 W )  e.  NN ) )
2625impcom 430 . . . . . . 7  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ZZ ) )  -> 
( # `  W )  e.  NN )
2723, 26zmodcld 11984 . . . . . 6  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ZZ ) )  -> 
( N  mod  ( # `
 W ) )  e.  NN0 )
2827nn0zd 10964 . . . . 5  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ZZ ) )  -> 
( N  mod  ( # `
 W ) )  e.  ZZ )
29 cshword 12725 . . . . 5  |-  ( ( W  e. Word  V  /\  ( N  mod  ( # `  W ) )  e.  ZZ )  ->  ( W cyclShift  ( N  mod  ( # `
 W ) ) )  =  ( ( W substr  <. ( ( N  mod  ( # `  W
) )  mod  ( # `
 W ) ) ,  ( # `  W
) >. ) concat  ( W substr  <.
0 ,  ( ( N  mod  ( # `  W ) )  mod  ( # `  W
) ) >. )
) )
3022, 28, 29syl2anc 661 . . . 4  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ZZ ) )  -> 
( W cyclShift  ( N  mod  ( # `  W ) ) )  =  ( ( W substr  <. (
( N  mod  ( # `
 W ) )  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( ( N  mod  ( # `  W
) )  mod  ( # `
 W ) )
>. ) ) )
31 cshword 12725 . . . . 5  |-  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( W cyclShift  N )  =  ( ( W substr  <. ( N  mod  ( # `
 W ) ) ,  ( # `  W
) >. ) concat  ( W substr  <.
0 ,  ( N  mod  ( # `  W
) ) >. )
) )
3231adantl 466 . . . 4  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ZZ ) )  -> 
( W cyclShift  N )  =  ( ( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >. ) ) )
3321, 30, 323eqtr4rd 2519 . . 3  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ZZ ) )  -> 
( W cyclShift  N )  =  ( W cyclShift  ( N  mod  ( # `  W
) ) ) )
3433ex 434 . 2  |-  ( W  =/=  (/)  ->  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( W cyclShift  N )  =  ( W cyclShift  ( N  mod  ( # `
 W ) ) ) ) )
357, 34pm2.61ine 2780 1  |-  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( W cyclShift  N )  =  ( W cyclShift  ( N  mod  ( # `  W
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   (/)c0 3785   <.cop 4033   ` cfv 5588  (class class class)co 6284   RRcr 9491   0cc0 9492   NNcn 10536   ZZcz 10864   RR+crp 11220    mod cmo 11964   #chash 12373  Word cword 12500   concat cconcat 12502   substr csubstr 12504   cyclShift ccsh 12722
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 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-hash 12374  df-word 12508  df-concat 12510  df-substr 12512  df-csh 12723
This theorem is referenced by:  cshwsublen  12730  cshwn  12731
  Copyright terms: Public domain W3C validator