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Theorem cshwlen 12733
Description: The length of a cyclically shifted word is the same as the length of the original word. (Contributed by AV, 16-May-2018.) (Revised by AV, 20-May-2018.) (Revised by AV, 27-Oct-2018.)
Assertion
Ref Expression
cshwlen  |-  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( # `  ( W cyclShift  N ) )  =  ( # `  W
) )

Proof of Theorem cshwlen
StepHypRef Expression
1 oveq1 6291 . . . . 5  |-  ( W  =  (/)  ->  ( W cyclShift  N )  =  (
(/) cyclShift  N ) )
2 0csh0 12727 . . . . . 6  |-  ( (/) cyclShift  N
)  =  (/)
32a1i 11 . . . . 5  |-  ( W  =  (/)  ->  ( (/) cyclShift  N
)  =  (/) )
4 eqcom 2476 . . . . . 6  |-  ( W  =  (/)  <->  (/)  =  W )
54biimpi 194 . . . . 5  |-  ( W  =  (/)  ->  (/)  =  W )
61, 3, 53eqtrd 2512 . . . 4  |-  ( W  =  (/)  ->  ( W cyclShift  N )  =  W )
76fveq2d 5870 . . 3  |-  ( W  =  (/)  ->  ( # `  ( W cyclShift  N )
)  =  ( # `  W ) )
87a1d 25 . 2  |-  ( W  =  (/)  ->  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( # `  ( W cyclShift  N ) )  =  ( # `  W
) ) )
9 cshword 12725 . . . . . 6  |-  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( W cyclShift  N )  =  ( ( W substr  <. ( N  mod  ( # `
 W ) ) ,  ( # `  W
) >. ) concat  ( W substr  <.
0 ,  ( N  mod  ( # `  W
) ) >. )
) )
109fveq2d 5870 . . . . 5  |-  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( # `  ( W cyclShift  N ) )  =  ( # `  (
( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >. ) ) ) )
1110adantr 465 . . . 4  |-  ( ( ( W  e. Word  V  /\  N  e.  ZZ )  /\  W  =/=  (/) )  -> 
( # `  ( W cyclShift  N ) )  =  ( # `  (
( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >. ) ) ) )
12 swrdcl 12609 . . . . . . 7  |-  ( W  e. Word  V  ->  ( W substr  <. ( N  mod  ( # `  W ) ) ,  ( # `  W ) >. )  e. Word  V )
13 swrdcl 12609 . . . . . . 7  |-  ( W  e. Word  V  ->  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >.
)  e. Word  V )
14 ccatlen 12559 . . . . . . 7  |-  ( ( ( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. )  e. Word  V  /\  ( W substr  <. 0 ,  ( N  mod  ( # `
 W ) )
>. )  e. Word  V )  ->  ( # `  (
( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >. ) ) )  =  ( ( # `  ( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. ) )  +  (
# `  ( W substr  <.
0 ,  ( N  mod  ( # `  W
) ) >. )
) ) )
1512, 13, 14syl2anc 661 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 ( ( W substr  <. ( N  mod  ( # `
 W ) ) ,  ( # `  W
) >. ) concat  ( W substr  <.
0 ,  ( N  mod  ( # `  W
) ) >. )
) )  =  ( ( # `  ( W substr  <. ( N  mod  ( # `  W ) ) ,  ( # `  W ) >. )
)  +  ( # `  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >. ) ) ) )
1615adantr 465 . . . . 5  |-  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( # `  (
( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. ) concat  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >. ) ) )  =  ( ( # `  ( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. ) )  +  (
# `  ( W substr  <.
0 ,  ( N  mod  ( # `  W
) ) >. )
) ) )
1716adantr 465 . . . 4  |-  ( ( ( W  e. Word  V  /\  N  e.  ZZ )  /\  W  =/=  (/) )  -> 
( # `  ( ( W substr  <. ( N  mod  ( # `  W ) ) ,  ( # `  W ) >. ) concat  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >.
) ) )  =  ( ( # `  ( W substr  <. ( N  mod  ( # `  W ) ) ,  ( # `  W ) >. )
)  +  ( # `  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >. ) ) ) )
18 lennncl 12529 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
19 pm3.21 448 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )  ->  ( W  e. Word  V  ->  ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
) ) )
2019ex 434 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN  ->  ( N  e.  ZZ  ->  ( W  e. Word  V  ->  ( W  e. Word  V  /\  ( (
# `  W )  e.  NN  /\  N  e.  ZZ ) ) ) ) )
2118, 20syl 16 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( N  e.  ZZ  ->  ( W  e. Word  V  -> 
( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
) ) ) )
2221ex 434 . . . . . . . 8  |-  ( W  e. Word  V  ->  ( W  =/=  (/)  ->  ( N  e.  ZZ  ->  ( W  e. Word  V  ->  ( W  e. Word  V  /\  ( (
# `  W )  e.  NN  /\  N  e.  ZZ ) ) ) ) ) )
2322com24 87 . . . . . . 7  |-  ( W  e. Word  V  ->  ( W  e. Word  V  ->  ( N  e.  ZZ  ->  ( W  =/=  (/)  ->  ( W  e. Word  V  /\  (
( # `  W )  e.  NN  /\  N  e.  ZZ ) ) ) ) ) )
2423pm2.43i 47 . . . . . 6  |-  ( W  e. Word  V  ->  ( N  e.  ZZ  ->  ( W  =/=  (/)  ->  ( W  e. Word  V  /\  (
( # `  W )  e.  NN  /\  N  e.  ZZ ) ) ) ) )
2524imp31 432 . . . . 5  |-  ( ( ( W  e. Word  V  /\  N  e.  ZZ )  /\  W  =/=  (/) )  -> 
( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
) )
26 simpl 457 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  W  e. Word  V )
27 pm3.22 449 . . . . . . . . . 10  |-  ( ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )  ->  ( N  e.  ZZ  /\  ( # `  W
)  e.  NN ) )
2827adantl 466 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  ( N  e.  ZZ  /\  ( # `  W )  e.  NN ) )
29 zmodfzp1 11987 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  ( # `  W )  e.  NN )  -> 
( N  mod  ( # `
 W ) )  e.  ( 0 ... ( # `  W
) ) )
3028, 29syl 16 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  ( N  mod  ( # `  W
) )  e.  ( 0 ... ( # `  W ) ) )
31 lencl 12528 . . . . . . . . . 10  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
32 nn0fz0 11773 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN0  <->  ( # `  W
)  e.  ( 0 ... ( # `  W
) ) )
3332biimpi 194 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  ( 0 ... ( # `  W
) ) )
3431, 33syl 16 . . . . . . . . 9  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  ( 0 ... ( # `
 W ) ) )
3534adantr 465 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  ( # `  W
)  e.  ( 0 ... ( # `  W
) ) )
36 swrdlen 12613 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  ( N  mod  ( # `  W ) )  e.  ( 0 ... ( # `
 W ) )  /\  ( # `  W
)  e.  ( 0 ... ( # `  W
) ) )  -> 
( # `  ( W substr  <. ( N  mod  ( # `
 W ) ) ,  ( # `  W
) >. ) )  =  ( ( # `  W
)  -  ( N  mod  ( # `  W
) ) ) )
3726, 30, 35, 36syl3anc 1228 . . . . . . 7  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  ( # `  ( W substr  <. ( N  mod  ( # `  W ) ) ,  ( # `  W ) >. )
)  =  ( (
# `  W )  -  ( N  mod  ( # `  W ) ) ) )
38 zmodcl 11983 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  ( # `  W )  e.  NN )  -> 
( N  mod  ( # `
 W ) )  e.  NN0 )
3938ancoms 453 . . . . . . . . . 10  |-  ( ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )  ->  ( N  mod  ( # `
 W ) )  e.  NN0 )
4039adantl 466 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  ( N  mod  ( # `  W
) )  e.  NN0 )
41 0elfz 11772 . . . . . . . . 9  |-  ( ( N  mod  ( # `  W ) )  e. 
NN0  ->  0  e.  ( 0 ... ( N  mod  ( # `  W
) ) ) )
4240, 41syl 16 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  0  e.  ( 0 ... ( N  mod  ( # `  W
) ) ) )
43 swrdlen 12613 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  0  e.  ( 0 ... ( N  mod  ( # `  W ) ) )  /\  ( N  mod  ( # `  W
) )  e.  ( 0 ... ( # `  W ) ) )  ->  ( # `  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >.
) )  =  ( ( N  mod  ( # `
 W ) )  -  0 ) )
4426, 42, 30, 43syl3anc 1228 . . . . . . 7  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  ( # `  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >.
) )  =  ( ( N  mod  ( # `
 W ) )  -  0 ) )
4537, 44oveq12d 6302 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  ( ( # `
 ( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. ) )  +  (
# `  ( W substr  <.
0 ,  ( N  mod  ( # `  W
) ) >. )
) )  =  ( ( ( # `  W
)  -  ( N  mod  ( # `  W
) ) )  +  ( ( N  mod  ( # `  W ) )  -  0 ) ) )
4638nn0cnd 10854 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( # `  W )  e.  NN )  -> 
( N  mod  ( # `
 W ) )  e.  CC )
4746ancoms 453 . . . . . . . . 9  |-  ( ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )  ->  ( N  mod  ( # `
 W ) )  e.  CC )
4847adantl 466 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  ( N  mod  ( # `  W
) )  e.  CC )
4948subid1d 9919 . . . . . . 7  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  ( ( N  mod  ( # `  W
) )  -  0 )  =  ( N  mod  ( # `  W
) ) )
5049oveq2d 6300 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  ( (
( # `  W )  -  ( N  mod  ( # `  W ) ) )  +  ( ( N  mod  ( # `
 W ) )  -  0 ) )  =  ( ( (
# `  W )  -  ( N  mod  ( # `  W ) ) )  +  ( N  mod  ( # `  W ) ) ) )
5131nn0cnd 10854 . . . . . . 7  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  CC )
52 npcan 9829 . . . . . . 7  |-  ( ( ( # `  W
)  e.  CC  /\  ( N  mod  ( # `  W ) )  e.  CC )  ->  (
( ( # `  W
)  -  ( N  mod  ( # `  W
) ) )  +  ( N  mod  ( # `
 W ) ) )  =  ( # `  W ) )
5351, 47, 52syl2an 477 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  ( (
( # `  W )  -  ( N  mod  ( # `  W ) ) )  +  ( N  mod  ( # `  W ) ) )  =  ( # `  W
) )
5445, 50, 533eqtrd 2512 . . . . 5  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  e.  NN  /\  N  e.  ZZ )
)  ->  ( ( # `
 ( W substr  <. ( N  mod  ( # `  W
) ) ,  (
# `  W ) >. ) )  +  (
# `  ( W substr  <.
0 ,  ( N  mod  ( # `  W
) ) >. )
) )  =  (
# `  W )
)
5525, 54syl 16 . . . 4  |-  ( ( ( W  e. Word  V  /\  N  e.  ZZ )  /\  W  =/=  (/) )  -> 
( ( # `  ( W substr  <. ( N  mod  ( # `  W ) ) ,  ( # `  W ) >. )
)  +  ( # `  ( W substr  <. 0 ,  ( N  mod  ( # `  W ) ) >. ) ) )  =  ( # `  W
) )
5611, 17, 553eqtrd 2512 . . 3  |-  ( ( ( W  e. Word  V  /\  N  e.  ZZ )  /\  W  =/=  (/) )  -> 
( # `  ( W cyclShift  N ) )  =  ( # `  W
) )
5756expcom 435 . 2  |-  ( W  =/=  (/)  ->  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( # `
 ( W cyclShift  N ) )  =  ( # `  W ) ) )
588, 57pm2.61ine 2780 1  |-  ( ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( # `  ( W cyclShift  N ) )  =  ( # `  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   CCcc 9490   0cc0 9492    + caddc 9495    - cmin 9805   NNcn 10536   NN0cn0 10795   ZZcz 10864   ...cfz 11672    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:  cshwf  12734  2cshw  12744  lswcshw  12746  cshwleneq  12748  clwwisshclwwlem  24510  clwwnisshclwwn  24513  erclwwlkeqlen  24516  erclwwlkneqlen  24528
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