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Theorem cshwidxm 12728
Description: The symbol at index (n-N) of a word of length n (not 0) cyclically shifted by N positions (not 0) is the symbol at index 0 of the original word. (Contributed by AV, 18-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.)
Assertion
Ref Expression
cshwidxm  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( W cyclShift  N ) `
 ( ( # `  W )  -  N
) )  =  ( W `  0 ) )

Proof of Theorem cshwidxm
StepHypRef Expression
1 simpl 457 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  ->  W  e. Word  V )
2 elfzelz 11677 . . . 4  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  N  e.  ZZ )
32adantl 466 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  ->  N  e.  ZZ )
4 ubmelfzo 11838 . . . 4  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( # `  W )  -  N )  e.  ( 0..^ ( # `  W ) ) )
54adantl 466 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( # `  W
)  -  N )  e.  ( 0..^ (
# `  W )
) )
6 cshwidxmod 12724 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ZZ  /\  (
( # `  W )  -  N )  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( W cyclShift  N ) `  (
( # `  W )  -  N ) )  =  ( W `  ( ( ( (
# `  W )  -  N )  +  N
)  mod  ( # `  W
) ) ) )
71, 3, 5, 6syl3anc 1223 . 2  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( W cyclShift  N ) `
 ( ( # `  W )  -  N
) )  =  ( W `  ( ( ( ( # `  W
)  -  N )  +  N )  mod  ( # `  W
) ) ) )
8 elfz1b 11737 . . . . . . . 8  |-  ( N  e.  ( 1 ... ( # `  W
) )  <->  ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) ) )
9 nncn 10533 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  CC )
10 nncn 10533 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN  ->  ( # `  W
)  e.  CC )
119, 10anim12ci 567 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN )  -> 
( ( # `  W
)  e.  CC  /\  N  e.  CC )
)
12113adant3 1011 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) )  ->  (
( # `  W )  e.  CC  /\  N  e.  CC ) )
138, 12sylbi 195 . . . . . . 7  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( # `  W )  e.  CC  /\  N  e.  CC ) )
14 npcan 9818 . . . . . . 7  |-  ( ( ( # `  W
)  e.  CC  /\  N  e.  CC )  ->  ( ( ( # `  W )  -  N
)  +  N )  =  ( # `  W
) )
1513, 14syl 16 . . . . . 6  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( ( # `  W
)  -  N )  +  N )  =  ( # `  W
) )
1615oveq1d 6290 . . . . 5  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( ( ( # `  W )  -  N
)  +  N )  mod  ( # `  W
) )  =  ( ( # `  W
)  mod  ( # `  W
) ) )
1716adantl 466 . . . 4  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( ( (
# `  W )  -  N )  +  N
)  mod  ( # `  W
) )  =  ( ( # `  W
)  mod  ( # `  W
) ) )
18 nnrp 11218 . . . . . . . 8  |-  ( (
# `  W )  e.  NN  ->  ( # `  W
)  e.  RR+ )
19 modid0 11977 . . . . . . . 8  |-  ( (
# `  W )  e.  RR+  ->  ( ( # `
 W )  mod  ( # `  W
) )  =  0 )
2018, 19syl 16 . . . . . . 7  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  mod  ( # `  W
) )  =  0 )
21203ad2ant2 1013 . . . . . 6  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) )  ->  (
( # `  W )  mod  ( # `  W
) )  =  0 )
228, 21sylbi 195 . . . . 5  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( # `  W )  mod  ( # `  W
) )  =  0 )
2322adantl 466 . . . 4  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( # `  W
)  mod  ( # `  W
) )  =  0 )
2417, 23eqtrd 2501 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( ( (
# `  W )  -  N )  +  N
)  mod  ( # `  W
) )  =  0 )
2524fveq2d 5861 . 2  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( W `  (
( ( ( # `  W )  -  N
)  +  N )  mod  ( # `  W
) ) )  =  ( W `  0
) )
267, 25eqtrd 2501 1  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( W cyclShift  N ) `
 ( ( # `  W )  -  N
) )  =  ( W `  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    <_ cle 9618    - cmin 9794   NNcn 10525   ZZcz 10853   RR+crp 11209   ...cfz 11661  ..^cfzo 11781    mod cmo 11952   #chash 12360  Word cword 12487   cyclShift ccsh 12709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-hash 12361  df-word 12495  df-concat 12497  df-substr 12499  df-csh 12710
This theorem is referenced by: (None)
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