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Theorem cshwidxm 12909
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 459 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  ->  W  e. Word  V )
2 elfzelz 11800 . . . 4  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  N  e.  ZZ )
32adantl 468 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  ->  N  e.  ZZ )
4 ubmelfzo 11979 . . . 4  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( # `  W )  -  N )  e.  ( 0..^ ( # `  W ) ) )
54adantl 468 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( # `  W
)  -  N )  e.  ( 0..^ (
# `  W )
) )
6 cshwidxmod 12905 . . 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 1268 . 2  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( W cyclShift  N ) `
 ( ( # `  W )  -  N
) )  =  ( W `  ( ( ( ( # `  W
)  -  N )  +  N )  mod  ( # `  W
) ) ) )
8 elfz1b 11864 . . . . . . . 8  |-  ( N  e.  ( 1 ... ( # `  W
) )  <->  ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) ) )
9 nncn 10617 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  CC )
10 nncn 10617 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN  ->  ( # `  W
)  e.  CC )
119, 10anim12ci 571 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN )  -> 
( ( # `  W
)  e.  CC  /\  N  e.  CC )
)
12113adant3 1028 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) )  ->  (
( # `  W )  e.  CC  /\  N  e.  CC ) )
138, 12sylbi 199 . . . . . . 7  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( # `  W )  e.  CC  /\  N  e.  CC ) )
14 npcan 9884 . . . . . . 7  |-  ( ( ( # `  W
)  e.  CC  /\  N  e.  CC )  ->  ( ( ( # `  W )  -  N
)  +  N )  =  ( # `  W
) )
1513, 14syl 17 . . . . . 6  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( ( # `  W
)  -  N )  +  N )  =  ( # `  W
) )
1615oveq1d 6305 . . . . 5  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( ( ( # `  W )  -  N
)  +  N )  mod  ( # `  W
) )  =  ( ( # `  W
)  mod  ( # `  W
) ) )
1716adantl 468 . . . 4  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( ( (
# `  W )  -  N )  +  N
)  mod  ( # `  W
) )  =  ( ( # `  W
)  mod  ( # `  W
) ) )
18 nnrp 11311 . . . . . . . 8  |-  ( (
# `  W )  e.  NN  ->  ( # `  W
)  e.  RR+ )
19 modid0 12122 . . . . . . . 8  |-  ( (
# `  W )  e.  RR+  ->  ( ( # `
 W )  mod  ( # `  W
) )  =  0 )
2018, 19syl 17 . . . . . . 7  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  mod  ( # `  W
) )  =  0 )
21203ad2ant2 1030 . . . . . 6  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) )  ->  (
( # `  W )  mod  ( # `  W
) )  =  0 )
228, 21sylbi 199 . . . . 5  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( # `  W )  mod  ( # `  W
) )  =  0 )
2322adantl 468 . . . 4  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( # `  W
)  mod  ( # `  W
) )  =  0 )
2417, 23eqtrd 2485 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( ( (
# `  W )  -  N )  +  N
)  mod  ( # `  W
) )  =  0 )
2524fveq2d 5869 . 2  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( W `  (
( ( ( # `  W )  -  N
)  +  N )  mod  ( # `  W
) ) )  =  ( W `  0
) )
267, 25eqtrd 2485 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 371    /\ w3a 985    = wceq 1444    e. wcel 1887   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   CCcc 9537   0cc0 9539   1c1 9540    + caddc 9542    <_ cle 9676    - cmin 9860   NNcn 10609   ZZcz 10937   RR+crp 11302   ...cfz 11784  ..^cfzo 11915    mod cmo 12096   #chash 12515  Word cword 12656   cyclShift ccsh 12890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-hash 12516  df-word 12664  df-concat 12666  df-substr 12668  df-csh 12891
This theorem is referenced by: (None)
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