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Theorem cshco 12796
Description: Mapping of words commutes with the "cyclical shift" operation. (Contributed by AV, 12-Nov-2018.)
Assertion
Ref Expression
cshco  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( F  o.  ( W cyclShift  N ) )  =  ( ( F  o.  W ) cyclShift  N ) )

Proof of Theorem cshco
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 ffn 5713 . . . 4  |-  ( F : A --> B  ->  F  Fn  A )
213ad2ant3 1017 . . 3  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  ->  F  Fn  A )
3 cshwfn 12766 . . . 4  |-  ( ( W  e. Word  A  /\  N  e.  ZZ )  ->  ( W cyclShift  N )  Fn  ( 0..^ ( # `  W ) ) )
433adant3 1014 . . 3  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( W cyclShift  N )  Fn  ( 0..^ ( # `  W ) ) )
5 cshwrn 12767 . . . 4  |-  ( ( W  e. Word  A  /\  N  e.  ZZ )  ->  ran  ( W cyclShift  N ) 
C_  A )
653adant3 1014 . . 3  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  ->  ran  ( W cyclShift  N )  C_  A )
7 fnco 5671 . . 3  |-  ( ( F  Fn  A  /\  ( W cyclShift  N )  Fn  ( 0..^ ( # `  W ) )  /\  ran  ( W cyclShift  N )  C_  A )  ->  ( F  o.  ( W cyclShift  N ) )  Fn  (
0..^ ( # `  W
) ) )
82, 4, 6, 7syl3anc 1226 . 2  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( F  o.  ( W cyclShift  N ) )  Fn  ( 0..^ ( # `  W ) ) )
9 wrdco 12791 . . . . 5  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( F  o.  W )  e. Word  B
)
1093adant2 1013 . . . 4  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( F  o.  W
)  e. Word  B )
11 simp2 995 . . . 4  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  ->  N  e.  ZZ )
12 cshwfn 12766 . . . 4  |-  ( ( ( F  o.  W
)  e. Word  B  /\  N  e.  ZZ )  ->  ( ( F  o.  W ) cyclShift  N )  Fn  ( 0..^ ( # `  ( F  o.  W
) ) ) )
1310, 11, 12syl2anc 659 . . 3  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( ( F  o.  W ) cyclShift  N )  Fn  ( 0..^ ( # `  ( F  o.  W
) ) ) )
14 lenco 12792 . . . . . 6  |-  ( ( W  e. Word  A  /\  F : A --> B )  ->  ( # `  ( F  o.  W )
)  =  ( # `  W ) )
15143adant2 1013 . . . . 5  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( # `  ( F  o.  W ) )  =  ( # `  W
) )
1615oveq2d 6286 . . . 4  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( 0..^ ( # `  ( F  o.  W
) ) )  =  ( 0..^ ( # `  W ) ) )
1716fneq2d 5654 . . 3  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( ( ( F  o.  W ) cyclShift  N
)  Fn  ( 0..^ ( # `  ( F  o.  W )
) )  <->  ( ( F  o.  W ) cyclShift  N )  Fn  ( 0..^ ( # `  W
) ) ) )
1813, 17mpbid 210 . 2  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( ( F  o.  W ) cyclShift  N )  Fn  ( 0..^ ( # `  W ) ) )
1915adantr 463 . . . . . . 7  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( # `  ( F  o.  W ) )  =  ( # `  W
) )
2019oveq2d 6286 . . . . . 6  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( i  +  N )  mod  ( # `
 ( F  o.  W ) ) )  =  ( ( i  +  N )  mod  ( # `  W
) ) )
2120fveq2d 5852 . . . . 5  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  (
( i  +  N
)  mod  ( # `  ( F  o.  W )
) ) )  =  ( W `  (
( i  +  N
)  mod  ( # `  W
) ) ) )
2221fveq2d 5852 . . . 4  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( F `  ( W `  ( (
i  +  N )  mod  ( # `  ( F  o.  W )
) ) ) )  =  ( F `  ( W `  ( ( i  +  N )  mod  ( # `  W
) ) ) ) )
23 wrdfn 12550 . . . . . . 7  |-  ( W  e. Word  A  ->  W  Fn  ( 0..^ ( # `  W ) ) )
24233ad2ant1 1015 . . . . . 6  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  ->  W  Fn  ( 0..^ ( # `  W
) ) )
2524adantr 463 . . . . 5  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  ->  W  Fn  ( 0..^ ( # `  W
) ) )
26 elfzoelz 11804 . . . . . . . 8  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  i  e.  ZZ )
27 zaddcl 10900 . . . . . . . 8  |-  ( ( i  e.  ZZ  /\  N  e.  ZZ )  ->  ( i  +  N
)  e.  ZZ )
2826, 11, 27syl2anr 476 . . . . . . 7  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( i  +  N
)  e.  ZZ )
29 elfzo0 11840 . . . . . . . . 9  |-  ( i  e.  ( 0..^ (
# `  W )
)  <->  ( i  e. 
NN0  /\  ( # `  W
)  e.  NN  /\  i  <  ( # `  W
) ) )
3029simp2bi 1010 . . . . . . . 8  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( # `  W
)  e.  NN )
3130adantl 464 . . . . . . 7  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( # `  W )  e.  NN )
32 zmodfzo 12001 . . . . . . 7  |-  ( ( ( i  +  N
)  e.  ZZ  /\  ( # `  W )  e.  NN )  -> 
( ( i  +  N )  mod  ( # `
 W ) )  e.  ( 0..^ (
# `  W )
) )
3328, 31, 32syl2anc 659 . . . . . 6  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( i  +  N )  mod  ( # `
 W ) )  e.  ( 0..^ (
# `  W )
) )
3415oveq2d 6286 . . . . . . . 8  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( ( i  +  N )  mod  ( # `
 ( F  o.  W ) ) )  =  ( ( i  +  N )  mod  ( # `  W
) ) )
3534eleq1d 2523 . . . . . . 7  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( ( ( i  +  N )  mod  ( # `  ( F  o.  W )
) )  e.  ( 0..^ ( # `  W
) )  <->  ( (
i  +  N )  mod  ( # `  W
) )  e.  ( 0..^ ( # `  W
) ) ) )
3635adantr 463 . . . . . 6  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( i  +  N )  mod  ( # `  ( F  o.  W )
) )  e.  ( 0..^ ( # `  W
) )  <->  ( (
i  +  N )  mod  ( # `  W
) )  e.  ( 0..^ ( # `  W
) ) ) )
3733, 36mpbird 232 . . . . 5  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( i  +  N )  mod  ( # `
 ( F  o.  W ) ) )  e.  ( 0..^ (
# `  W )
) )
38 fvco2 5923 . . . . 5  |-  ( ( W  Fn  ( 0..^ ( # `  W
) )  /\  (
( i  +  N
)  mod  ( # `  ( F  o.  W )
) )  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( F  o.  W ) `  (
( i  +  N
)  mod  ( # `  ( F  o.  W )
) ) )  =  ( F `  ( W `  ( (
i  +  N )  mod  ( # `  ( F  o.  W )
) ) ) ) )
3925, 37, 38syl2anc 659 . . . 4  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( F  o.  W ) `  (
( i  +  N
)  mod  ( # `  ( F  o.  W )
) ) )  =  ( F `  ( W `  ( (
i  +  N )  mod  ( # `  ( F  o.  W )
) ) ) ) )
40 simpl1 997 . . . . 5  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  ->  W  e. Word  A )
4111adantr 463 . . . . 5  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  ->  N  e.  ZZ )
42 simpr 459 . . . . 5  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
i  e.  ( 0..^ ( # `  W
) ) )
43 cshwidxmod 12768 . . . . . 6  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  i  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( W cyclShift  N ) `  i
)  =  ( W `
 ( ( i  +  N )  mod  ( # `  W
) ) ) )
4443fveq2d 5852 . . . . 5  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  i  e.  ( 0..^ ( # `  W ) ) )  ->  ( F `  ( ( W cyclShift  N ) `
 i ) )  =  ( F `  ( W `  ( ( i  +  N )  mod  ( # `  W
) ) ) ) )
4540, 41, 42, 44syl3anc 1226 . . . 4  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( F `  (
( W cyclShift  N ) `  i ) )  =  ( F `  ( W `  ( (
i  +  N )  mod  ( # `  W
) ) ) ) )
4622, 39, 453eqtr4rd 2506 . . 3  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( F `  (
( W cyclShift  N ) `  i ) )  =  ( ( F  o.  W ) `  (
( i  +  N
)  mod  ( # `  ( F  o.  W )
) ) ) )
47 fvco2 5923 . . . 4  |-  ( ( ( W cyclShift  N )  Fn  ( 0..^ ( # `  W ) )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( F  o.  ( W cyclShift  N ) ) `
 i )  =  ( F `  (
( W cyclShift  N ) `  i ) ) )
484, 47sylan 469 . . 3  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( F  o.  ( W cyclShift  N ) ) `
 i )  =  ( F `  (
( W cyclShift  N ) `  i ) ) )
4910adantr 463 . . . 4  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( F  o.  W
)  e. Word  B )
5015eqcomd 2462 . . . . . . 7  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( # `  W )  =  ( # `  ( F  o.  W )
) )
5150oveq2d 6286 . . . . . 6  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( 0..^ ( # `  W ) )  =  ( 0..^ ( # `  ( F  o.  W
) ) ) )
5251eleq2d 2524 . . . . 5  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( i  e.  ( 0..^ ( # `  W
) )  <->  i  e.  ( 0..^ ( # `  ( F  o.  W )
) ) ) )
5352biimpa 482 . . . 4  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
i  e.  ( 0..^ ( # `  ( F  o.  W )
) ) )
54 cshwidxmod 12768 . . . 4  |-  ( ( ( F  o.  W
)  e. Word  B  /\  N  e.  ZZ  /\  i  e.  ( 0..^ ( # `  ( F  o.  W
) ) ) )  ->  ( ( ( F  o.  W ) cyclShift  N ) `  i
)  =  ( ( F  o.  W ) `
 ( ( i  +  N )  mod  ( # `  ( F  o.  W )
) ) ) )
5549, 41, 53, 54syl3anc 1226 . . 3  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( F  o.  W ) cyclShift  N
) `  i )  =  ( ( F  o.  W ) `  ( ( i  +  N )  mod  ( # `
 ( F  o.  W ) ) ) ) )
5646, 48, 553eqtr4d 2505 . 2  |-  ( ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  /\  i  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( F  o.  ( W cyclShift  N ) ) `
 i )  =  ( ( ( F  o.  W ) cyclShift  N
) `  i )
)
578, 18, 56eqfnfvd 5960 1  |-  ( ( W  e. Word  A  /\  N  e.  ZZ  /\  F : A --> B )  -> 
( F  o.  ( W cyclShift  N ) )  =  ( ( F  o.  W ) cyclShift  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    C_ wss 3461   class class class wbr 4439   ran crn 4989    o. ccom 4992    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   0cc0 9481    + caddc 9484    < clt 9617   NNcn 10531   NN0cn0 10791   ZZcz 10860  ..^cfzo 11799    mod cmo 11978   #chash 12390  Word cword 12521   cyclShift ccsh 12753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-hash 12391  df-word 12529  df-concat 12531  df-substr 12533  df-csh 12754
This theorem is referenced by: (None)
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