Theorem repswcshw 13558

Description: A cyclically shifted "repeated symbol word". (Contributed by Alexander van der Vekens, 7-Nov-2018.)

Assertion

Ref	Expression
repswcshw	|- ( ( S e. V /\ N e. NN0 /\ I e. ZZ ) -> ( ( S repeatS N ) cyclShift I ) = ( S repeatS N ) )

Proof of Theorem repswcshw

Step	Hyp	Ref	Expression
1		0csh0 13539	. . . . 5 |- ( (/) cyclShift I ) = (/)
2		repsw0 13524	. . . . . 6 |- ( S e. V -> ( S repeatS 0 ) = (/) )
3	2	oveq1d 6665	. . . . 5 |- ( S e. V -> ( ( S repeatS 0 ) cyclShift I ) = ( (/) cyclShift I ) )
4	1, 3, 2	3eqtr4a 2682	. . . 4 |- ( S e. V -> ( ( S repeatS 0 ) cyclShift I ) = ( S repeatS 0 ) )
5	4	3ad2ant1 1082	. . 3 |- ( ( S e. V /\ N e. NN0 /\ I e. ZZ ) -> ( ( S repeatS 0 ) cyclShift I ) = ( S repeatS 0 ) )
6		oveq2 6658	. . . . 5 |- ( N = 0 -> ( S repeatS N ) = ( S repeatS 0 ) )
7	6	oveq1d 6665	. . . 4 |- ( N = 0 -> ( ( S repeatS N ) cyclShift I ) = ( ( S repeatS 0 ) cyclShift I ) )
8	7, 6	eqeq12d 2637	. . 3 |- ( N = 0 -> ( ( ( S repeatS N ) cyclShift I ) = ( S repeatS N ) <-> ( ( S repeatS 0 ) cyclShift I ) = ( S repeatS 0 ) ) )
9	5, 8	syl5ibr 236	. 2 |- ( N = 0 -> ( ( S e. V /\ N e. NN0 /\ I e. ZZ ) -> ( ( S repeatS N ) cyclShift I ) = ( S repeatS N ) ) )
10		idd 24	. . . 4 |- ( -. N = 0 -> ( S e. V -> S e. V ) )
11		df-ne 2795	. . . . 5 |- ( N =/= 0 <-> -. N = 0 )
12		elnnne0 11306	. . . . . 6 |- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) )
13	12	simplbi2com 657	. . . . 5 |- ( N =/= 0 -> ( N e. NN0 -> N e. NN ) )
14	11, 13	sylbir 225	. . . 4 |- ( -. N = 0 -> ( N e. NN0 -> N e. NN ) )
15		idd 24	. . . 4 |- ( -. N = 0 -> ( I e. ZZ -> I e. ZZ ) )
16	10, 14, 15	3anim123d 1406	. . 3 |- ( -. N = 0 -> ( ( S e. V /\ N e. NN0 /\ I e. ZZ ) -> ( S e. V /\ N e. NN /\ I e. ZZ ) ) )
17		nnnn0 11299	. . . . . . 7 |- ( N e. NN -> N e. NN0 )
18	17	anim2i 593	. . . . . 6 |- ( ( S e. V /\ N e. NN ) -> ( S e. V /\ N e. NN0 ) )
19		repsw 13522	. . . . . 6 |- ( ( S e. V /\ N e. NN0 ) -> ( S repeatS N ) e. Word V )
20	18, 19	syl 17	. . . . 5 |- ( ( S e. V /\ N e. NN ) -> ( S repeatS N ) e. Word V )
21		cshword 13537	. . . . 5 |- ( ( ( S repeatS N ) e. Word V /\ I e. ZZ ) -> ( ( S repeatS N ) cyclShift I ) = ( ( ( S repeatS N ) substr <. ( I mod ( # ` ( S repeatS N ) ) ) , ( # ` ( S repeatS N ) ) >. ) ++ ( ( S repeatS N ) substr <. 0 , ( I mod ( # ` ( S repeatS N ) ) ) >. ) ) )
22	20, 21	stoic3 1701	. . . 4 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( ( S repeatS N ) cyclShift I ) = ( ( ( S repeatS N ) substr <. ( I mod ( # ` ( S repeatS N ) ) ) , ( # ` ( S repeatS N ) ) >. ) ++ ( ( S repeatS N ) substr <. 0 , ( I mod ( # ` ( S repeatS N ) ) ) >. ) ) )
23		repswlen 13523	. . . . . . . . . 10 |- ( ( S e. V /\ N e. NN0 ) -> ( # ` ( S repeatS N ) ) = N )
24	18, 23	syl 17	. . . . . . . . 9 |- ( ( S e. V /\ N e. NN ) -> ( # ` ( S repeatS N ) ) = N )
25	24	oveq2d 6666	. . . . . . . 8 |- ( ( S e. V /\ N e. NN ) -> ( I mod ( # ` ( S repeatS N ) ) ) = ( I mod N ) )
26	25, 24	opeq12d 4410	. . . . . . 7 |- ( ( S e. V /\ N e. NN ) -> <. ( I mod ( # ` ( S repeatS N ) ) ) , ( # ` ( S repeatS N ) ) >. = <. ( I mod N ) , N >. )
27	26	oveq2d 6666	. . . . . 6 |- ( ( S e. V /\ N e. NN ) -> ( ( S repeatS N ) substr <. ( I mod ( # ` ( S repeatS N ) ) ) , ( # ` ( S repeatS N ) ) >. ) = ( ( S repeatS N ) substr <. ( I mod N ) , N >. ) )
28	25	opeq2d 4409	. . . . . . 7 |- ( ( S e. V /\ N e. NN ) -> <. 0 , ( I mod ( # ` ( S repeatS N ) ) ) >. = <. 0 , ( I mod N ) >. )
29	28	oveq2d 6666	. . . . . 6 |- ( ( S e. V /\ N e. NN ) -> ( ( S repeatS N ) substr <. 0 , ( I mod ( # ` ( S repeatS N ) ) ) >. ) = ( ( S repeatS N ) substr <. 0 , ( I mod N ) >. ) )
30	27, 29	oveq12d 6668	. . . . 5 |- ( ( S e. V /\ N e. NN ) -> ( ( ( S repeatS N ) substr <. ( I mod ( # ` ( S repeatS N ) ) ) , ( # ` ( S repeatS N ) ) >. ) ++ ( ( S repeatS N ) substr <. 0 , ( I mod ( # ` ( S repeatS N ) ) ) >. ) ) = ( ( ( S repeatS N ) substr <. ( I mod N ) , N >. ) ++ ( ( S repeatS N ) substr <. 0 , ( I mod N ) >. ) ) )
31	30	3adant3 1081	. . . 4 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( ( ( S repeatS N ) substr <. ( I mod ( # ` ( S repeatS N ) ) ) , ( # ` ( S repeatS N ) ) >. ) ++ ( ( S repeatS N ) substr <. 0 , ( I mod ( # ` ( S repeatS N ) ) ) >. ) ) = ( ( ( S repeatS N ) substr <. ( I mod N ) , N >. ) ++ ( ( S repeatS N ) substr <. 0 , ( I mod N ) >. ) ) )
32	18	3adant3 1081	. . . . . . 7 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( S e. V /\ N e. NN0 ) )
33		zmodcl 12690	. . . . . . . . . 10 |- ( ( I e. ZZ /\ N e. NN ) -> ( I mod N ) e. NN0 )
34	33	ancoms 469	. . . . . . . . 9 |- ( ( N e. NN /\ I e. ZZ ) -> ( I mod N ) e. NN0 )
35	17	adantr 481	. . . . . . . . 9 |- ( ( N e. NN /\ I e. ZZ ) -> N e. NN0 )
36	34, 35	jca 554	. . . . . . . 8 |- ( ( N e. NN /\ I e. ZZ ) -> ( ( I mod N ) e. NN0 /\ N e. NN0 ) )
37	36	3adant1 1079	. . . . . . 7 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( ( I mod N ) e. NN0 /\ N e. NN0 ) )
38		nnre 11027	. . . . . . . . 9 |- ( N e. NN -> N e. RR )
39	38	leidd 10594	. . . . . . . 8 |- ( N e. NN -> N <_ N )
40	39	3ad2ant2 1083	. . . . . . 7 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> N <_ N )
41		repswswrd 13531	. . . . . . 7 |- ( ( ( S e. V /\ N e. NN0 ) /\ ( ( I mod N ) e. NN0 /\ N e. NN0 ) /\ N <_ N ) -> ( ( S repeatS N ) substr <. ( I mod N ) , N >. ) = ( S repeatS ( N - ( I mod N ) ) ) )
42	32, 37, 40, 41	syl3anc 1326	. . . . . 6 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( ( S repeatS N ) substr <. ( I mod N ) , N >. ) = ( S repeatS ( N - ( I mod N ) ) ) )
43		0nn0 11307	. . . . . . . . 9 |- 0 e. NN0
44	34, 43	jctil 560	. . . . . . . 8 |- ( ( N e. NN /\ I e. ZZ ) -> ( 0 e. NN0 /\ ( I mod N ) e. NN0 ) )
45	44	3adant1 1079	. . . . . . 7 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( 0 e. NN0 /\ ( I mod N ) e. NN0 ) )
46		zre 11381	. . . . . . . . . 10 |- ( I e. ZZ -> I e. RR )
47		nnrp 11842	. . . . . . . . . 10 |- ( N e. NN -> N e. RR+ )
48		modcl 12672	. . . . . . . . . 10 |- ( ( I e. RR /\ N e. RR+ ) -> ( I mod N ) e. RR )
49	46, 47, 48	syl2anr 495	. . . . . . . . 9 |- ( ( N e. NN /\ I e. ZZ ) -> ( I mod N ) e. RR )
50	38	adantr 481	. . . . . . . . 9 |- ( ( N e. NN /\ I e. ZZ ) -> N e. RR )
51		modlt 12679	. . . . . . . . . 10 |- ( ( I e. RR /\ N e. RR+ ) -> ( I mod N ) < N )
52	46, 47, 51	syl2anr 495	. . . . . . . . 9 |- ( ( N e. NN /\ I e. ZZ ) -> ( I mod N ) < N )
53	49, 50, 52	ltled 10185	. . . . . . . 8 |- ( ( N e. NN /\ I e. ZZ ) -> ( I mod N ) <_ N )
54	53	3adant1 1079	. . . . . . 7 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( I mod N ) <_ N )
55		repswswrd 13531	. . . . . . 7 |- ( ( ( S e. V /\ N e. NN0 ) /\ ( 0 e. NN0 /\ ( I mod N ) e. NN0 ) /\ ( I mod N ) <_ N ) -> ( ( S repeatS N ) substr <. 0 , ( I mod N ) >. ) = ( S repeatS ( ( I mod N ) - 0 ) ) )
56	32, 45, 54, 55	syl3anc 1326	. . . . . 6 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( ( S repeatS N ) substr <. 0 , ( I mod N ) >. ) = ( S repeatS ( ( I mod N ) - 0 ) ) )
57	42, 56	oveq12d 6668	. . . . 5 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( ( ( S repeatS N ) substr <. ( I mod N ) , N >. ) ++ ( ( S repeatS N ) substr <. 0 , ( I mod N ) >. ) ) = ( ( S repeatS ( N - ( I mod N ) ) ) ++ ( S repeatS ( ( I mod N ) - 0 ) ) ) )
58		simp1 1061	. . . . . 6 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> S e. V )
59	33	nn0red 11352	. . . . . . . . . 10 |- ( ( I e. ZZ /\ N e. NN ) -> ( I mod N ) e. RR )
60	59	ancoms 469	. . . . . . . . 9 |- ( ( N e. NN /\ I e. ZZ ) -> ( I mod N ) e. RR )
61	60, 50, 52	ltled 10185	. . . . . . . 8 |- ( ( N e. NN /\ I e. ZZ ) -> ( I mod N ) <_ N )
62	61	3adant1 1079	. . . . . . 7 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( I mod N ) <_ N )
63	34	3adant1 1079	. . . . . . . 8 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( I mod N ) e. NN0 )
64	17	3ad2ant2 1083	. . . . . . . 8 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> N e. NN0 )
65		nn0sub 11343	. . . . . . . 8 |- ( ( ( I mod N ) e. NN0 /\ N e. NN0 ) -> ( ( I mod N ) <_ N <-> ( N - ( I mod N ) ) e. NN0 ) )
66	63, 64, 65	syl2anc 693	. . . . . . 7 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( ( I mod N ) <_ N <-> ( N - ( I mod N ) ) e. NN0 ) )
67	62, 66	mpbid 222	. . . . . 6 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( N - ( I mod N ) ) e. NN0 )
68	33	nn0ge0d 11354	. . . . . . . . 9 |- ( ( I e. ZZ /\ N e. NN ) -> 0 <_ ( I mod N ) )
69	68	ancoms 469	. . . . . . . 8 |- ( ( N e. NN /\ I e. ZZ ) -> 0 <_ ( I mod N ) )
70	69	3adant1 1079	. . . . . . 7 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> 0 <_ ( I mod N ) )
71	63, 43	jctil 560	. . . . . . . 8 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( 0 e. NN0 /\ ( I mod N ) e. NN0 ) )
72		nn0sub 11343	. . . . . . . 8 |- ( ( 0 e. NN0 /\ ( I mod N ) e. NN0 ) -> ( 0 <_ ( I mod N ) <-> ( ( I mod N ) - 0 ) e. NN0 ) )
73	71, 72	syl 17	. . . . . . 7 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( 0 <_ ( I mod N ) <-> ( ( I mod N ) - 0 ) e. NN0 ) )
74	70, 73	mpbid 222	. . . . . 6 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( ( I mod N ) - 0 ) e. NN0 )
75		repswccat 13532	. . . . . 6 |- ( ( S e. V /\ ( N - ( I mod N ) ) e. NN0 /\ ( ( I mod N ) - 0 ) e. NN0 ) -> ( ( S repeatS ( N - ( I mod N ) ) ) ++ ( S repeatS ( ( I mod N ) - 0 ) ) ) = ( S repeatS ( ( N - ( I mod N ) ) + ( ( I mod N ) - 0 ) ) ) )
76	58, 67, 74, 75	syl3anc 1326	. . . . 5 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( ( S repeatS ( N - ( I mod N ) ) ) ++ ( S repeatS ( ( I mod N ) - 0 ) ) ) = ( S repeatS ( ( N - ( I mod N ) ) + ( ( I mod N ) - 0 ) ) ) )
77		nncn 11028	. . . . . . . . . . 11 |- ( N e. NN -> N e. CC )
78	77	adantl 482	. . . . . . . . . 10 |- ( ( I e. ZZ /\ N e. NN ) -> N e. CC )
79	33	nn0cnd 11353	. . . . . . . . . 10 |- ( ( I e. ZZ /\ N e. NN ) -> ( I mod N ) e. CC )
80		0cnd 10033	. . . . . . . . . 10 |- ( ( I e. ZZ /\ N e. NN ) -> 0 e. CC )
81	78, 79, 80	npncand 10416	. . . . . . . . 9 |- ( ( I e. ZZ /\ N e. NN ) -> ( ( N - ( I mod N ) ) + ( ( I mod N ) - 0 ) ) = ( N - 0 ) )
82	77	subid1d 10381	. . . . . . . . . 10 |- ( N e. NN -> ( N - 0 ) = N )
83	82	adantl 482	. . . . . . . . 9 |- ( ( I e. ZZ /\ N e. NN ) -> ( N - 0 ) = N )
84	81, 83	eqtrd 2656	. . . . . . . 8 |- ( ( I e. ZZ /\ N e. NN ) -> ( ( N - ( I mod N ) ) + ( ( I mod N ) - 0 ) ) = N )
85	84	ancoms 469	. . . . . . 7 |- ( ( N e. NN /\ I e. ZZ ) -> ( ( N - ( I mod N ) ) + ( ( I mod N ) - 0 ) ) = N )
86	85	3adant1 1079	. . . . . 6 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( ( N - ( I mod N ) ) + ( ( I mod N ) - 0 ) ) = N )
87	86	oveq2d 6666	. . . . 5 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( S repeatS ( ( N - ( I mod N ) ) + ( ( I mod N ) - 0 ) ) ) = ( S repeatS N ) )
88	57, 76, 87	3eqtrd 2660	. . . 4 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( ( ( S repeatS N ) substr <. ( I mod N ) , N >. ) ++ ( ( S repeatS N ) substr <. 0 , ( I mod N ) >. ) ) = ( S repeatS N ) )
89	22, 31, 88	3eqtrd 2660	. . 3 |- ( ( S e. V /\ N e. NN /\ I e. ZZ ) -> ( ( S repeatS N ) cyclShift I ) = ( S repeatS N ) )
90	16, 89	syl6 35	. 2 |- ( -. N = 0 -> ( ( S e. V /\ N e. NN0 /\ I e. ZZ ) -> ( ( S repeatS N ) cyclShift I ) = ( S repeatS N ) ) )
91	9, 90	pm2.61i 176	1 |- ( ( S e. V /\ N e. NN0 /\ I e. ZZ ) -> ( ( S repeatS N ) cyclShift I ) = ( S repeatS N ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   RR+crp 11832   mod cmo 12668   #chash 13117  Word cword 13291   ++ cconcat 13293   substr csubstr 13295   repeatS creps 13298   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-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-reps 13306  df-csh 13535

This theorem is referenced by:  cshwrepswhash1  15809