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Theorem cshwidxm 12442
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 11451 . . . 4  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  N  e.  ZZ )
32adantl 466 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  ->  N  e.  ZZ )
4 ubmelfzo 11601 . . . 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 12438 . . 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 1218 . 2  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( W cyclShift  N ) `
 ( ( # `  W )  -  N
) )  =  ( W `  ( ( ( ( # `  W
)  -  N )  +  N )  mod  ( # `  W
) ) ) )
8 elfz1b 11525 . . . . . . . 8  |-  ( N  e.  ( 1 ... ( # `  W
) )  <->  ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) ) )
9 nncn 10328 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  CC )
10 nncn 10328 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN  ->  ( # `  W
)  e.  CC )
119, 10anim12ci 567 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN )  -> 
( ( # `  W
)  e.  CC  /\  N  e.  CC )
)
12113adant3 1008 . . . . . . . 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 9617 . . . . . . 7  |-  ( ( ( # `  W
)  e.  CC  /\  N  e.  CC )  ->  ( ( ( # `  W )  -  N
)  +  N )  =  ( # `  W
) )
1513, 14syl 16 . . . . . 6  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( ( # `  W
)  -  N )  +  N )  =  ( # `  W
) )
1615oveq1d 6104 . . . . 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 10998 . . . . . . . 8  |-  ( (
# `  W )  e.  NN  ->  ( # `  W
)  e.  RR+ )
19 modid0 11731 . . . . . . . 8  |-  ( (
# `  W )  e.  RR+  ->  ( ( # `
 W )  mod  ( # `  W
) )  =  0 )
2018, 19syl 16 . . . . . . 7  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  mod  ( # `  W
) )  =  0 )
21203ad2ant2 1010 . . . . . 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 2473 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( ( (
# `  W )  -  N )  +  N
)  mod  ( # `  W
) )  =  0 )
2524fveq2d 5693 . 2  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( W `  (
( ( ( # `  W )  -  N
)  +  N )  mod  ( # `  W
) ) )  =  ( W `  0
) )
267, 25eqtrd 2473 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 965    = wceq 1369    e. wcel 1756   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   CCcc 9278   0cc0 9280   1c1 9281    + caddc 9283    <_ cle 9417    - cmin 9593   NNcn 10320   ZZcz 10644   RR+crp 10989   ...cfz 11435  ..^cfzo 11546    mod cmo 11706   #chash 12101  Word cword 12219   cyclShift ccsh 12423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-fz 11436  df-fzo 11547  df-fl 11640  df-mod 11707  df-hash 12102  df-word 12227  df-concat 12229  df-substr 12231  df-csh 12424
This theorem is referenced by: (None)
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