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Theorem cshwidxn 13555
Description: The symbol at index (n-1) of a word of length n (not 0) cyclically shifted by N positions (not 0) is the symbol at index (N-1) 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
cshwidxn  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( W cyclShift  N ) `
 ( ( # `  W )  -  1 ) )  =  ( W `  ( N  -  1 ) ) )

Proof of Theorem cshwidxn
StepHypRef Expression
1 simpl 473 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  ->  W  e. Word  V )
2 elfzelz 12342 . . . 4  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  N  e.  ZZ )
32adantl 482 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  ->  N  e.  ZZ )
4 elfz1b 12409 . . . . . 6  |-  ( N  e.  ( 1 ... ( # `  W
) )  <->  ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) ) )
5 simp2 1062 . . . . . 6  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) )  ->  ( # `
 W )  e.  NN )
64, 5sylbi 207 . . . . 5  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  ( # `
 W )  e.  NN )
76adantl 482 . . . 4  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( # `  W )  e.  NN )
8 fzo0end 12560 . . . 4  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  - 
1 )  e.  ( 0..^ ( # `  W
) ) )
97, 8syl 17 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( # `  W
)  -  1 )  e.  ( 0..^ (
# `  W )
) )
10 cshwidxmod 13549 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ZZ  /\  (
( # `  W )  -  1 )  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( W cyclShift  N ) `  (
( # `  W )  -  1 ) )  =  ( W `  ( ( ( (
# `  W )  -  1 )  +  N )  mod  ( # `
 W ) ) ) )
111, 3, 9, 10syl3anc 1326 . 2  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( W cyclShift  N ) `
 ( ( # `  W )  -  1 ) )  =  ( W `  ( ( ( ( # `  W
)  -  1 )  +  N )  mod  ( # `  W
) ) ) )
12 nncn 11028 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN  ->  ( # `  W
)  e.  CC )
1312adantl 482 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN )  -> 
( # `  W )  e.  CC )
14 1cnd 10056 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN )  -> 
1  e.  CC )
15 nncn 11028 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  CC )
1615adantr 481 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN )  ->  N  e.  CC )
1713, 14, 163jca 1242 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN )  -> 
( ( # `  W
)  e.  CC  /\  1  e.  CC  /\  N  e.  CC ) )
18173adant3 1081 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) )  ->  (
( # `  W )  e.  CC  /\  1  e.  CC  /\  N  e.  CC ) )
194, 18sylbi 207 . . . . . . 7  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( # `  W )  e.  CC  /\  1  e.  CC  /\  N  e.  CC ) )
20 subadd23 10293 . . . . . . 7  |-  ( ( ( # `  W
)  e.  CC  /\  1  e.  CC  /\  N  e.  CC )  ->  (
( ( # `  W
)  -  1 )  +  N )  =  ( ( # `  W
)  +  ( N  -  1 ) ) )
2119, 20syl 17 . . . . . 6  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( ( # `  W
)  -  1 )  +  N )  =  ( ( # `  W
)  +  ( N  -  1 ) ) )
2221oveq1d 6665 . . . . 5  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( ( ( # `  W )  -  1 )  +  N )  mod  ( # `  W
) )  =  ( ( ( # `  W
)  +  ( N  -  1 ) )  mod  ( # `  W
) ) )
23 nnm1nn0 11334 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
24233ad2ant1 1082 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) )  ->  ( N  -  1 )  e.  NN0 )
25 simp3 1063 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) )  ->  N  <_  ( # `  W
) )
26 nnz 11399 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  ZZ )
27 nnz 11399 . . . . . . . . . . . 12  |-  ( (
# `  W )  e.  NN  ->  ( # `  W
)  e.  ZZ )
2826, 27anim12i 590 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN )  -> 
( N  e.  ZZ  /\  ( # `  W
)  e.  ZZ ) )
29283adant3 1081 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) )  ->  ( N  e.  ZZ  /\  ( # `
 W )  e.  ZZ ) )
30 zlem1lt 11429 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( # `  W )  e.  ZZ )  -> 
( N  <_  ( # `
 W )  <->  ( N  -  1 )  < 
( # `  W ) ) )
3129, 30syl 17 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) )  ->  ( N  <_  ( # `  W
)  <->  ( N  - 
1 )  <  ( # `
 W ) ) )
3225, 31mpbid 222 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) )  ->  ( N  -  1 )  <  ( # `  W
) )
3324, 5, 323jca 1242 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( # `  W )  e.  NN  /\  N  <_  ( # `  W
) )  ->  (
( N  -  1 )  e.  NN0  /\  ( # `  W )  e.  NN  /\  ( N  -  1 )  <  ( # `  W
) ) )
344, 33sylbi 207 . . . . . 6  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( N  -  1 )  e.  NN0  /\  ( # `  W )  e.  NN  /\  ( N  -  1 )  <  ( # `  W
) ) )
35 addmodid 12718 . . . . . 6  |-  ( ( ( N  -  1 )  e.  NN0  /\  ( # `  W )  e.  NN  /\  ( N  -  1 )  <  ( # `  W
) )  ->  (
( ( # `  W
)  +  ( N  -  1 ) )  mod  ( # `  W
) )  =  ( N  -  1 ) )
3634, 35syl 17 . . . . 5  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( ( # `  W
)  +  ( N  -  1 ) )  mod  ( # `  W
) )  =  ( N  -  1 ) )
3722, 36eqtrd 2656 . . . 4  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  (
( ( ( # `  W )  -  1 )  +  N )  mod  ( # `  W
) )  =  ( N  -  1 ) )
3837fveq2d 6195 . . 3  |-  ( N  e.  ( 1 ... ( # `  W
) )  ->  ( W `  ( (
( ( # `  W
)  -  1 )  +  N )  mod  ( # `  W
) ) )  =  ( W `  ( N  -  1 ) ) )
3938adantl 482 . 2  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( W `  (
( ( ( # `  W )  -  1 )  +  N )  mod  ( # `  W
) ) )  =  ( W `  ( N  -  1 ) ) )
4011, 39eqtrd 2656 1  |-  ( ( W  e. Word  V  /\  N  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( W cyclShift  N ) `
 ( ( # `  W )  -  1 ) )  =  ( W `  ( N  -  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 196    /\ wa 384    /\ w3a 1037    = wceq 1483    e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937    + caddc 9939    < clt 10074    <_ cle 10075    - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326  ..^cfzo 12465    mod cmo 12668   #chash 13117  Word cword 13291   cyclShift ccsh 13534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-csh 13535
This theorem is referenced by:  lswcshw  13561
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