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Theorem cshwn 12545
Description: A word cyclically shifted by its length is the word itself. (Contributed by AV, 16-May-2018.) (Revised by AV, 20-May-2018.) (Revised by AV, 26-Oct-2018.)
Assertion
Ref Expression
cshwn  |-  ( W  e. Word  V  ->  ( W cyclShift  ( # `  W
) )  =  W )

Proof of Theorem cshwn
StepHypRef Expression
1 0csh0 12541 . . . 4  |-  ( (/) cyclShift  (
# `  W )
)  =  (/)
2 oveq1 6200 . . . 4  |-  ( (/)  =  W  ->  ( (/) cyclShift  (
# `  W )
)  =  ( W cyclShift  ( # `  W ) ) )
3 id 22 . . . 4  |-  ( (/)  =  W  ->  (/)  =  W )
41, 2, 33eqtr3a 2516 . . 3  |-  ( (/)  =  W  ->  ( W cyclShift  ( # `  W ) )  =  W )
54a1d 25 . 2  |-  ( (/)  =  W  ->  ( W  e. Word  V  ->  ( W cyclShift  ( # `  W
) )  =  W ) )
6 lencl 12360 . . . . . . 7  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
76nn0zd 10849 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  ZZ )
8 cshwmodn 12543 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( # `  W )  e.  ZZ )  -> 
( W cyclShift  ( # `  W
) )  =  ( W cyclShift  ( ( # `  W
)  mod  ( # `  W
) ) ) )
97, 8mpdan 668 . . . . 5  |-  ( W  e. Word  V  ->  ( W cyclShift  ( # `  W
) )  =  ( W cyclShift  ( ( # `  W
)  mod  ( # `  W
) ) ) )
109adantl 466 . . . 4  |-  ( (
(/)  =/=  W  /\  W  e. Word  V )  ->  ( W cyclShift  ( # `  W
) )  =  ( W cyclShift  ( ( # `  W
)  mod  ( # `  W
) ) ) )
11 necom 2717 . . . . . . . . 9  |-  ( (/)  =/=  W  <->  W  =/=  (/) )
12 lennncl 12361 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
1311, 12sylan2b 475 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  (/) 
=/=  W )  -> 
( # `  W )  e.  NN )
1413nnrpd 11130 . . . . . . 7  |-  ( ( W  e. Word  V  /\  (/) 
=/=  W )  -> 
( # `  W )  e.  RR+ )
1514ancoms 453 . . . . . 6  |-  ( (
(/)  =/=  W  /\  W  e. Word  V )  ->  ( # `  W
)  e.  RR+ )
16 modid0 11843 . . . . . 6  |-  ( (
# `  W )  e.  RR+  ->  ( ( # `
 W )  mod  ( # `  W
) )  =  0 )
1715, 16syl 16 . . . . 5  |-  ( (
(/)  =/=  W  /\  W  e. Word  V )  ->  ( ( # `  W
)  mod  ( # `  W
) )  =  0 )
1817oveq2d 6209 . . . 4  |-  ( (
(/)  =/=  W  /\  W  e. Word  V )  ->  ( W cyclShift  ( ( # `
 W )  mod  ( # `  W
) ) )  =  ( W cyclShift  0 ) )
19 cshw0 12542 . . . . 5  |-  ( W  e. Word  V  ->  ( W cyclShift  0 )  =  W )
2019adantl 466 . . . 4  |-  ( (
(/)  =/=  W  /\  W  e. Word  V )  ->  ( W cyclShift  0 )  =  W )
2110, 18, 203eqtrd 2496 . . 3  |-  ( (
(/)  =/=  W  /\  W  e. Word  V )  ->  ( W cyclShift  ( # `  W
) )  =  W )
2221ex 434 . 2  |-  ( (/)  =/=  W  ->  ( W  e. Word  V  ->  ( W cyclShift  (
# `  W )
)  =  W ) )
235, 22pm2.61ine 2761 1  |-  ( W  e. Word  V  ->  ( W cyclShift  ( # `  W
) )  =  W )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   (/)c0 3738   ` cfv 5519  (class class class)co 6193   0cc0 9386   NNcn 10426   ZZcz 10750   RR+crp 11095    mod cmo 11818   #chash 12213  Word cword 12332   cyclShift ccsh 12536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-fz 11548  df-fzo 11659  df-fl 11752  df-mod 11819  df-hash 12214  df-word 12340  df-concat 12342  df-substr 12344  df-csh 12537
This theorem is referenced by:  2cshwid  12559  cshweqdif2  12564  clwwisshclwwn  30613  erclwwlksym0  30619  scshwfzeqfzo  30633
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