Theorem cshco 13582

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.

Step	Hyp	Ref	Expression
1		ffn 6045	. . . 4 |- ( F : A --> B -> F Fn A )
2	1	3ad2ant3 1084	. . 3 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> F Fn A )
3		cshwfn 13547	. . . 4 |- ( ( W e. Word A /\ N e. ZZ ) -> ( W cyclShift N ) Fn ( 0..^ ( # ` W ) ) )
4	3	3adant3 1081	. . 3 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> ( W cyclShift N ) Fn ( 0..^ ( # ` W ) ) )
5		cshwrn 13548	. . . 4 |- ( ( W e. Word A /\ N e. ZZ ) -> ran ( W cyclShift N ) C_ A )
6	5	3adant3 1081	. . 3 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> ran ( W cyclShift N ) C_ A )
7		fnco 5999	. . 3 |- ( ( F Fn A /\ ( W cyclShift N ) Fn ( 0..^ ( # ` W ) ) /\ ran ( W cyclShift N ) C_ A ) -> ( F o. ( W cyclShift N ) ) Fn ( 0..^ ( # ` W ) ) )
8	2, 4, 6, 7	syl3anc 1326	. 2 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> ( F o. ( W cyclShift N ) ) Fn ( 0..^ ( # ` W ) ) )
9		wrdco 13577	. . . . 5 |- ( ( W e. Word A /\ F : A --> B ) -> ( F o. W ) e. Word B )
10	9	3adant2 1080	. . . 4 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> ( F o. W ) e. Word B )
11		simp2 1062	. . . 4 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> N e. ZZ )
12		cshwfn 13547	. . . 4 |- ( ( ( F o. W ) e. Word B /\ N e. ZZ ) -> ( ( F o. W ) cyclShift N ) Fn ( 0..^ ( # ` ( F o. W ) ) ) )
13	10, 11, 12	syl2anc 693	. . 3 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> ( ( F o. W ) cyclShift N ) Fn ( 0..^ ( # ` ( F o. W ) ) ) )
14		lenco 13578	. . . . . 6 |- ( ( W e. Word A /\ F : A --> B ) -> ( # ` ( F o. W ) ) = ( # ` W ) )
15	14	3adant2 1080	. . . . 5 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> ( # ` ( F o. W ) ) = ( # ` W ) )
16	15	oveq2d 6666	. . . 4 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> ( 0..^ ( # ` ( F o. W ) ) ) = ( 0..^ ( # ` W ) ) )
17	16	fneq2d 5982	. . 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 ) ) ) )
18	13, 17	mpbid 222	. 2 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> ( ( F o. W ) cyclShift N ) Fn ( 0..^ ( # ` W ) ) )
19	15	adantr 481	. . . . . . 7 |- ( ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) /\ i e. ( 0..^ ( # ` W ) ) ) -> ( # ` ( F o. W ) ) = ( # ` W ) )
20	19	oveq2d 6666	. . . . . 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 ) ) )
21	20	fveq2d 6195	. . . . 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 ) ) ) )
22	21	fveq2d 6195	. . . 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 13319	. . . . . . 7 |- ( W e. Word A -> W Fn ( 0..^ ( # ` W ) ) )
24	23	3ad2ant1 1082	. . . . . 6 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> W Fn ( 0..^ ( # ` W ) ) )
25	24	adantr 481	. . . . 5 |- ( ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) /\ i e. ( 0..^ ( # ` W ) ) ) -> W Fn ( 0..^ ( # ` W ) ) )
26		elfzoelz 12470	. . . . . . . 8 |- ( i e. ( 0..^ ( # ` W ) ) -> i e. ZZ )
27		zaddcl 11417	. . . . . . . 8 |- ( ( i e. ZZ /\ N e. ZZ ) -> ( i + N ) e. ZZ )
28	26, 11, 27	syl2anr 495	. . . . . . 7 |- ( ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) /\ i e. ( 0..^ ( # ` W ) ) ) -> ( i + N ) e. ZZ )
29		elfzo0 12508	. . . . . . . . 9 |- ( i e. ( 0..^ ( # ` W ) ) <-> ( i e. NN0 /\ ( # ` W ) e. NN /\ i < ( # ` W ) ) )
30	29	simp2bi 1077	. . . . . . . 8 |- ( i e. ( 0..^ ( # ` W ) ) -> ( # ` W ) e. NN )
31	30	adantl 482	. . . . . . 7 |- ( ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) /\ i e. ( 0..^ ( # ` W ) ) ) -> ( # ` W ) e. NN )
32		zmodfzo 12693	. . . . . . 7 |- ( ( ( i + N ) e. ZZ /\ ( # ` W ) e. NN ) -> ( ( i + N ) mod ( # ` W ) ) e. ( 0..^ ( # ` W ) ) )
33	28, 31, 32	syl2anc 693	. . . . . 6 |- ( ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) /\ i e. ( 0..^ ( # ` W ) ) ) -> ( ( i + N ) mod ( # ` W ) ) e. ( 0..^ ( # ` W ) ) )
34	15	oveq2d 6666	. . . . . . . 8 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> ( ( i + N ) mod ( # ` ( F o. W ) ) ) = ( ( i + N ) mod ( # ` W ) ) )
35	34	eleq1d 2686	. . . . . . 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 ) ) ) )
36	35	adantr 481	. . . . . 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 ) ) ) )
37	33, 36	mpbird 247	. . . . 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 6273	. . . . 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 ) ) ) ) ) )
39	25, 37, 38	syl2anc 693	. . . 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 1064	. . . . 5 |- ( ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) /\ i e. ( 0..^ ( # ` W ) ) ) -> W e. Word A )
41	11	adantr 481	. . . . 5 |- ( ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) /\ i e. ( 0..^ ( # ` W ) ) ) -> N e. ZZ )
42		simpr 477	. . . . 5 |- ( ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) /\ i e. ( 0..^ ( # ` W ) ) ) -> i e. ( 0..^ ( # ` W ) ) )
43		cshwidxmod 13549	. . . . . 6 |- ( ( W e. Word A /\ N e. ZZ /\ i e. ( 0..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` i ) = ( W ` ( ( i + N ) mod ( # ` W ) ) ) )
44	43	fveq2d 6195	. . . . 5 |- ( ( W e. Word A /\ N e. ZZ /\ i e. ( 0..^ ( # ` W ) ) ) -> ( F ` ( ( W cyclShift N ) ` i ) ) = ( F ` ( W ` ( ( i + N ) mod ( # ` W ) ) ) ) )
45	40, 41, 42, 44	syl3anc 1326	. . . 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 ) ) ) ) )
46	22, 39, 45	3eqtr4rd 2667	. . 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 6273	. . . 4 |- ( ( ( W cyclShift N ) Fn ( 0..^ ( # ` W ) ) /\ i e. ( 0..^ ( # ` W ) ) ) -> ( ( F o. ( W cyclShift N ) ) ` i ) = ( F ` ( ( W cyclShift N ) ` i ) ) )
48	4, 47	sylan 488	. . 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 ) ) )
49	10	adantr 481	. . . 4 |- ( ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) /\ i e. ( 0..^ ( # ` W ) ) ) -> ( F o. W ) e. Word B )
50	15	eqcomd 2628	. . . . . . 7 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> ( # ` W ) = ( # ` ( F o. W ) ) )
51	50	oveq2d 6666	. . . . . 6 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> ( 0..^ ( # ` W ) ) = ( 0..^ ( # ` ( F o. W ) ) ) )
52	51	eleq2d 2687	. . . . 5 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> ( i e. ( 0..^ ( # ` W ) ) <-> i e. ( 0..^ ( # ` ( F o. W ) ) ) ) )
53	52	biimpa 501	. . . 4 |- ( ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) /\ i e. ( 0..^ ( # ` W ) ) ) -> i e. ( 0..^ ( # ` ( F o. W ) ) ) )
54		cshwidxmod 13549	. . . 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 ) ) ) ) )
55	49, 41, 53, 54	syl3anc 1326	. . 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 ) ) ) ) )
56	46, 48, 55	3eqtr4d 2666	. 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 ) )
57	8, 18, 56	eqfnfvd 6314	1 |- ( ( W e. Word A /\ N e. ZZ /\ F : A --> B ) -> ( F o. ( W cyclShift N ) ) = ( ( F o. W ) cyclShift N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   ran crn 5115   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   + caddc 9939   < clt 10074   NNcn 11020   NN0cn0 11292   ZZcz 11377  ..^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-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: (None)