Theorem cshwidxmodr 13550

Description: The symbol at a given index of a cyclically shifted nonempty word is the symbol at the shifted index of the original word. (Contributed by AV, 17-Mar-2021.)

Assertion

Ref	Expression
cshwidxmodr	|- ( ( W e. Word V /\ N e. ZZ /\ I e. ( 0..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` ( ( I - N ) mod ( # ` W ) ) ) = ( W ` I ) )

Proof of Theorem cshwidxmodr

Step	Hyp	Ref	Expression
1		elfzo0 12508	. . . . . . 7 |- ( I e. ( 0..^ ( # ` W ) ) <-> ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) )
2		nn0z 11400	. . . . . . . . . . 11 |- ( I e. NN0 -> I e. ZZ )
3	2	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) -> I e. ZZ )
4		zsubcl 11419	. . . . . . . . . 10 |- ( ( I e. ZZ /\ N e. ZZ ) -> ( I - N ) e. ZZ )
5	3, 4	sylan 488	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) /\ N e. ZZ ) -> ( I - N ) e. ZZ )
6		simpl2 1065	. . . . . . . . 9 |- ( ( ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) /\ N e. ZZ ) -> ( # ` W ) e. NN )
7	5, 6	jca 554	. . . . . . . 8 |- ( ( ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) /\ N e. ZZ ) -> ( ( I - N ) e. ZZ /\ ( # ` W ) e. NN ) )
8	7	ex 450	. . . . . . 7 |- ( ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) -> ( N e. ZZ -> ( ( I - N ) e. ZZ /\ ( # ` W ) e. NN ) ) )
9	1, 8	sylbi 207	. . . . . 6 |- ( I e. ( 0..^ ( # ` W ) ) -> ( N e. ZZ -> ( ( I - N ) e. ZZ /\ ( # ` W ) e. NN ) ) )
10	9	impcom 446	. . . . 5 |- ( ( N e. ZZ /\ I e. ( 0..^ ( # ` W ) ) ) -> ( ( I - N ) e. ZZ /\ ( # ` W ) e. NN ) )
11	10	3adant1 1079	. . . 4 |- ( ( W e. Word V /\ N e. ZZ /\ I e. ( 0..^ ( # ` W ) ) ) -> ( ( I - N ) e. ZZ /\ ( # ` W ) e. NN ) )
12		zmodfzo 12693	. . . 4 |- ( ( ( I - N ) e. ZZ /\ ( # ` W ) e. NN ) -> ( ( I - N ) mod ( # ` W ) ) e. ( 0..^ ( # ` W ) ) )
13	11, 12	syl 17	. . 3 |- ( ( W e. Word V /\ N e. ZZ /\ I e. ( 0..^ ( # ` W ) ) ) -> ( ( I - N ) mod ( # ` W ) ) e. ( 0..^ ( # ` W ) ) )
14		cshwidxmod 13549	. . 3 |- ( ( W e. Word V /\ N e. ZZ /\ ( ( I - N ) mod ( # ` W ) ) e. ( 0..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` ( ( I - N ) mod ( # ` W ) ) ) = ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) )
15	13, 14	syld3an3 1371	. 2 |- ( ( W e. Word V /\ N e. ZZ /\ I e. ( 0..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` ( ( I - N ) mod ( # ` W ) ) ) = ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) )
16		elfzoelz 12470	. . . . . . . . . . . . . . 15 |- ( I e. ( 0..^ ( # ` W ) ) -> I e. ZZ )
17	16	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0..^ ( # ` W ) ) ) -> I e. ZZ )
18	17, 4	sylan 488	. . . . . . . . . . . . 13 |- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( I - N ) e. ZZ )
19	18	zred 11482	. . . . . . . . . . . 12 |- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( I - N ) e. RR )
20		zre 11381	. . . . . . . . . . . . 13 |- ( N e. ZZ -> N e. RR )
21	20	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0..^ ( # ` W ) ) ) /\ N e. ZZ ) -> N e. RR )
22		nnrp 11842	. . . . . . . . . . . . 13 |- ( ( # ` W ) e. NN -> ( # ` W ) e. RR+ )
23	22	ad3antlr 767	. . . . . . . . . . . 12 |- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( # ` W ) e. RR+ )
24		modaddmod 12709	. . . . . . . . . . . 12 |- ( ( ( I - N ) e. RR /\ N e. RR /\ ( # ` W ) e. RR+ ) -> ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) = ( ( ( I - N ) + N ) mod ( # ` W ) ) )
25	19, 21, 23, 24	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) = ( ( ( I - N ) + N ) mod ( # ` W ) ) )
26		nn0cn 11302	. . . . . . . . . . . . . 14 |- ( I e. NN0 -> I e. CC )
27	26	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0..^ ( # ` W ) ) ) -> I e. CC )
28		zcn 11382	. . . . . . . . . . . . 13 |- ( N e. ZZ -> N e. CC )
29		npcan 10290	. . . . . . . . . . . . 13 |- ( ( I e. CC /\ N e. CC ) -> ( ( I - N ) + N ) = I )
30	27, 28, 29	syl2an 494	. . . . . . . . . . . 12 |- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( ( I - N ) + N ) = I )
31	30	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( ( ( I - N ) + N ) mod ( # ` W ) ) = ( I mod ( # ` W ) ) )
32		zmodidfzoimp 12700	. . . . . . . . . . . 12 |- ( I e. ( 0..^ ( # ` W ) ) -> ( I mod ( # ` W ) ) = I )
33	32	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( I mod ( # ` W ) ) = I )
34	25, 31, 33	3eqtrd 2660	. . . . . . . . . 10 |- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) = I )
35	34	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) )
36	35	ex 450	. . . . . . . 8 |- ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0..^ ( # ` W ) ) ) -> ( N e. ZZ -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) ) )
37	36	ex 450	. . . . . . 7 |- ( ( I e. NN0 /\ ( # ` W ) e. NN ) -> ( I e. ( 0..^ ( # ` W ) ) -> ( N e. ZZ -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) ) ) )
38	37	3adant3 1081	. . . . . 6 |- ( ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) -> ( I e. ( 0..^ ( # ` W ) ) -> ( N e. ZZ -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) ) ) )
39	1, 38	sylbi 207	. . . . 5 |- ( I e. ( 0..^ ( # ` W ) ) -> ( I e. ( 0..^ ( # ` W ) ) -> ( N e. ZZ -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) ) ) )
40	39	pm2.43i 52	. . . 4 |- ( I e. ( 0..^ ( # ` W ) ) -> ( N e. ZZ -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) ) )
41	40	impcom 446	. . 3 |- ( ( N e. ZZ /\ I e. ( 0..^ ( # ` W ) ) ) -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) )
42	41	3adant1 1079	. 2 |- ( ( W e. Word V /\ N e. ZZ /\ I e. ( 0..^ ( # ` W ) ) ) -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) )
43	15, 42	eqtrd 2656	1 |- ( ( W e. Word V /\ N e. ZZ /\ I e. ( 0..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` ( ( I - N ) mod ( # ` W ) ) ) = ( W ` I ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   < clt 10074   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   RR+crp 11832  ..^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)