Theorem cshimadifsn 13575

Description: The image of a cyclically shifted word under its domain without its left bound is the image of a cyclically shifted word under its domain without the number of shifted symbols. (Contributed by AV, 19-Mar-2021.)

Assertion

Ref	Expression
cshimadifsn	|- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( F " ( ( 0..^ N ) \ { J } ) ) = ( ( F cyclShift J ) " ( 1..^ N ) ) )

Proof of Theorem cshimadifsn

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wrdfn 13319	. . . . . 6 |- ( F e. Word S -> F Fn ( 0..^ ( # ` F ) ) )
2		fnfun 5988	. . . . . 6 |- ( F Fn ( 0..^ ( # ` F ) ) -> Fun F )
3	1, 2	syl 17	. . . . 5 |- ( F e. Word S -> Fun F )
4	3	3ad2ant1 1082	. . . 4 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> Fun F )
5		wrddm 13312	. . . . . 6 |- ( F e. Word S -> dom F = ( 0..^ ( # ` F ) ) )
6		difssd 3738	. . . . . . . . 9 |- ( ( dom F = ( 0..^ ( # ` F ) ) /\ N = ( # ` F ) ) -> ( ( 0..^ ( # ` F ) ) \ { J } ) C_ ( 0..^ ( # ` F ) ) )
7		oveq2 6658	. . . . . . . . . . 11 |- ( N = ( # ` F ) -> ( 0..^ N ) = ( 0..^ ( # ` F ) ) )
8	7	difeq1d 3727	. . . . . . . . . 10 |- ( N = ( # ` F ) -> ( ( 0..^ N ) \ { J } ) = ( ( 0..^ ( # ` F ) ) \ { J } ) )
9	8	adantl 482	. . . . . . . . 9 |- ( ( dom F = ( 0..^ ( # ` F ) ) /\ N = ( # ` F ) ) -> ( ( 0..^ N ) \ { J } ) = ( ( 0..^ ( # ` F ) ) \ { J } ) )
10		simpl 473	. . . . . . . . 9 |- ( ( dom F = ( 0..^ ( # ` F ) ) /\ N = ( # ` F ) ) -> dom F = ( 0..^ ( # ` F ) ) )
11	6, 9, 10	3sstr4d 3648	. . . . . . . 8 |- ( ( dom F = ( 0..^ ( # ` F ) ) /\ N = ( # ` F ) ) -> ( ( 0..^ N ) \ { J } ) C_ dom F )
12	11	a1d 25	. . . . . . 7 |- ( ( dom F = ( 0..^ ( # ` F ) ) /\ N = ( # ` F ) ) -> ( J e. ( 0..^ N ) -> ( ( 0..^ N ) \ { J } ) C_ dom F ) )
13	12	ex 450	. . . . . 6 |- ( dom F = ( 0..^ ( # ` F ) ) -> ( N = ( # ` F ) -> ( J e. ( 0..^ N ) -> ( ( 0..^ N ) \ { J } ) C_ dom F ) ) )
14	5, 13	syl 17	. . . . 5 |- ( F e. Word S -> ( N = ( # ` F ) -> ( J e. ( 0..^ N ) -> ( ( 0..^ N ) \ { J } ) C_ dom F ) ) )
15	14	3imp 1256	. . . 4 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( ( 0..^ N ) \ { J } ) C_ dom F )
16	4, 15	jca 554	. . 3 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( Fun F /\ ( ( 0..^ N ) \ { J } ) C_ dom F ) )
17		dfimafn 6245	. . 3 |- ( ( Fun F /\ ( ( 0..^ N ) \ { J } ) C_ dom F ) -> ( F " ( ( 0..^ N ) \ { J } ) ) = { z | E. x e. ( ( 0..^ N ) \ { J } ) ( F ` x ) = z } )
18	16, 17	syl 17	. 2 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( F " ( ( 0..^ N ) \ { J } ) ) = { z | E. x e. ( ( 0..^ N ) \ { J } ) ( F ` x ) = z } )
19		modsumfzodifsn 12743	. . . . . . 7 |- ( ( J e. ( 0..^ N ) /\ y e. ( 1..^ N ) ) -> ( ( y + J ) mod N ) e. ( ( 0..^ N ) \ { J } ) )
20	19	3ad2antl3 1225	. . . . . 6 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ y e. ( 1..^ N ) ) -> ( ( y + J ) mod N ) e. ( ( 0..^ N ) \ { J } ) )
21		oveq2 6658	. . . . . . . . . 10 |- ( ( # ` F ) = N -> ( ( y + J ) mod ( # ` F ) ) = ( ( y + J ) mod N ) )
22	21	eqcoms 2630	. . . . . . . . 9 |- ( N = ( # ` F ) -> ( ( y + J ) mod ( # ` F ) ) = ( ( y + J ) mod N ) )
23	22	eleq1d 2686	. . . . . . . 8 |- ( N = ( # ` F ) -> ( ( ( y + J ) mod ( # ` F ) ) e. ( ( 0..^ N ) \ { J } ) <-> ( ( y + J ) mod N ) e. ( ( 0..^ N ) \ { J } ) ) )
24	23	3ad2ant2 1083	. . . . . . 7 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( ( ( y + J ) mod ( # ` F ) ) e. ( ( 0..^ N ) \ { J } ) <-> ( ( y + J ) mod N ) e. ( ( 0..^ N ) \ { J } ) ) )
25	24	adantr 481	. . . . . 6 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ y e. ( 1..^ N ) ) -> ( ( ( y + J ) mod ( # ` F ) ) e. ( ( 0..^ N ) \ { J } ) <-> ( ( y + J ) mod N ) e. ( ( 0..^ N ) \ { J } ) ) )
26	20, 25	mpbird 247	. . . . 5 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ y e. ( 1..^ N ) ) -> ( ( y + J ) mod ( # ` F ) ) e. ( ( 0..^ N ) \ { J } ) )
27		modfzo0difsn 12742	. . . . . . 7 |- ( ( J e. ( 0..^ N ) /\ x e. ( ( 0..^ N ) \ { J } ) ) -> E. y e. ( 1..^ N ) x = ( ( y + J ) mod N ) )
28	27	3ad2antl3 1225	. . . . . 6 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ x e. ( ( 0..^ N ) \ { J } ) ) -> E. y e. ( 1..^ N ) x = ( ( y + J ) mod N ) )
29		oveq2 6658	. . . . . . . . . . 11 |- ( N = ( # ` F ) -> ( ( y + J ) mod N ) = ( ( y + J ) mod ( # ` F ) ) )
30	29	eqcomd 2628	. . . . . . . . . 10 |- ( N = ( # ` F ) -> ( ( y + J ) mod ( # ` F ) ) = ( ( y + J ) mod N ) )
31	30	eqeq2d 2632	. . . . . . . . 9 |- ( N = ( # ` F ) -> ( x = ( ( y + J ) mod ( # ` F ) ) <-> x = ( ( y + J ) mod N ) ) )
32	31	rexbidv 3052	. . . . . . . 8 |- ( N = ( # ` F ) -> ( E. y e. ( 1..^ N ) x = ( ( y + J ) mod ( # ` F ) ) <-> E. y e. ( 1..^ N ) x = ( ( y + J ) mod N ) ) )
33	32	3ad2ant2 1083	. . . . . . 7 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( E. y e. ( 1..^ N ) x = ( ( y + J ) mod ( # ` F ) ) <-> E. y e. ( 1..^ N ) x = ( ( y + J ) mod N ) ) )
34	33	adantr 481	. . . . . 6 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ x e. ( ( 0..^ N ) \ { J } ) ) -> ( E. y e. ( 1..^ N ) x = ( ( y + J ) mod ( # ` F ) ) <-> E. y e. ( 1..^ N ) x = ( ( y + J ) mod N ) ) )
35	28, 34	mpbird 247	. . . . 5 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ x e. ( ( 0..^ N ) \ { J } ) ) -> E. y e. ( 1..^ N ) x = ( ( y + J ) mod ( # ` F ) ) )
36		fveq2 6191	. . . . . . . 8 |- ( x = ( ( y + J ) mod ( # ` F ) ) -> ( F ` x ) = ( F ` ( ( y + J ) mod ( # ` F ) ) ) )
37	36	3ad2ant3 1084	. . . . . . 7 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ y e. ( 1..^ N ) /\ x = ( ( y + J ) mod ( # ` F ) ) ) -> ( F ` x ) = ( F ` ( ( y + J ) mod ( # ` F ) ) ) )
38		simpl1 1064	. . . . . . . . 9 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ y e. ( 1..^ N ) ) -> F e. Word S )
39		elfzoelz 12470	. . . . . . . . . . 11 |- ( J e. ( 0..^ N ) -> J e. ZZ )
40	39	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> J e. ZZ )
41	40	adantr 481	. . . . . . . . 9 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ y e. ( 1..^ N ) ) -> J e. ZZ )
42		oveq2 6658	. . . . . . . . . . . . 13 |- ( N = ( # ` F ) -> ( 1..^ N ) = ( 1..^ ( # ` F ) ) )
43	42	eleq2d 2687	. . . . . . . . . . . 12 |- ( N = ( # ` F ) -> ( y e. ( 1..^ N ) <-> y e. ( 1..^ ( # ` F ) ) ) )
44		fzo0ss1 12498	. . . . . . . . . . . . 13 |- ( 1..^ ( # ` F ) ) C_ ( 0..^ ( # ` F ) )
45	44	sseli 3599	. . . . . . . . . . . 12 |- ( y e. ( 1..^ ( # ` F ) ) -> y e. ( 0..^ ( # ` F ) ) )
46	43, 45	syl6bi 243	. . . . . . . . . . 11 |- ( N = ( # ` F ) -> ( y e. ( 1..^ N ) -> y e. ( 0..^ ( # ` F ) ) ) )
47	46	3ad2ant2 1083	. . . . . . . . . 10 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( y e. ( 1..^ N ) -> y e. ( 0..^ ( # ` F ) ) ) )
48	47	imp 445	. . . . . . . . 9 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ y e. ( 1..^ N ) ) -> y e. ( 0..^ ( # ` F ) ) )
49		cshwidxmod 13549	. . . . . . . . . 10 |- ( ( F e. Word S /\ J e. ZZ /\ y e. ( 0..^ ( # ` F ) ) ) -> ( ( F cyclShift J ) ` y ) = ( F ` ( ( y + J ) mod ( # ` F ) ) ) )
50	49	eqcomd 2628	. . . . . . . . 9 |- ( ( F e. Word S /\ J e. ZZ /\ y e. ( 0..^ ( # ` F ) ) ) -> ( F ` ( ( y + J ) mod ( # ` F ) ) ) = ( ( F cyclShift J ) ` y ) )
51	38, 41, 48, 50	syl3anc 1326	. . . . . . . 8 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ y e. ( 1..^ N ) ) -> ( F ` ( ( y + J ) mod ( # ` F ) ) ) = ( ( F cyclShift J ) ` y ) )
52	51	3adant3 1081	. . . . . . 7 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ y e. ( 1..^ N ) /\ x = ( ( y + J ) mod ( # ` F ) ) ) -> ( F ` ( ( y + J ) mod ( # ` F ) ) ) = ( ( F cyclShift J ) ` y ) )
53	37, 52	eqtrd 2656	. . . . . 6 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ y e. ( 1..^ N ) /\ x = ( ( y + J ) mod ( # ` F ) ) ) -> ( F ` x ) = ( ( F cyclShift J ) ` y ) )
54	53	eqeq1d 2624	. . . . 5 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ y e. ( 1..^ N ) /\ x = ( ( y + J ) mod ( # ` F ) ) ) -> ( ( F ` x ) = z <-> ( ( F cyclShift J ) ` y ) = z ) )
55	26, 35, 54	rexxfrd2 4885	. . . 4 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( E. x e. ( ( 0..^ N ) \ { J } ) ( F ` x ) = z <-> E. y e. ( 1..^ N ) ( ( F cyclShift J ) ` y ) = z ) )
56	55	abbidv 2741	. . 3 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> { z | E. x e. ( ( 0..^ N ) \ { J } ) ( F ` x ) = z } = { z | E. y e. ( 1..^ N ) ( ( F cyclShift J ) ` y ) = z } )
57	39	anim2i 593	. . . . . . . 8 |- ( ( F e. Word S /\ J e. ( 0..^ N ) ) -> ( F e. Word S /\ J e. ZZ ) )
58	57	3adant2 1080	. . . . . . 7 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( F e. Word S /\ J e. ZZ ) )
59		cshwfn 13547	. . . . . . 7 |- ( ( F e. Word S /\ J e. ZZ ) -> ( F cyclShift J ) Fn ( 0..^ ( # ` F ) ) )
60	58, 59	syl 17	. . . . . 6 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( F cyclShift J ) Fn ( 0..^ ( # ` F ) ) )
61		fnfun 5988	. . . . . . . 8 |- ( ( F cyclShift J ) Fn ( 0..^ ( # ` F ) ) -> Fun ( F cyclShift J ) )
62	61	adantl 482	. . . . . . 7 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ ( F cyclShift J ) Fn ( 0..^ ( # ` F ) ) ) -> Fun ( F cyclShift J ) )
63	42, 44	syl6eqss 3655	. . . . . . . . . 10 |- ( N = ( # ` F ) -> ( 1..^ N ) C_ ( 0..^ ( # ` F ) ) )
64	63	3ad2ant2 1083	. . . . . . . . 9 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( 1..^ N ) C_ ( 0..^ ( # ` F ) ) )
65	64	adantr 481	. . . . . . . 8 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ ( F cyclShift J ) Fn ( 0..^ ( # ` F ) ) ) -> ( 1..^ N ) C_ ( 0..^ ( # ` F ) ) )
66		fndm 5990	. . . . . . . . 9 |- ( ( F cyclShift J ) Fn ( 0..^ ( # ` F ) ) -> dom ( F cyclShift J ) = ( 0..^ ( # ` F ) ) )
67	66	adantl 482	. . . . . . . 8 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ ( F cyclShift J ) Fn ( 0..^ ( # ` F ) ) ) -> dom ( F cyclShift J ) = ( 0..^ ( # ` F ) ) )
68	65, 67	sseqtr4d 3642	. . . . . . 7 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ ( F cyclShift J ) Fn ( 0..^ ( # ` F ) ) ) -> ( 1..^ N ) C_ dom ( F cyclShift J ) )
69	62, 68	jca 554	. . . . . 6 |- ( ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) /\ ( F cyclShift J ) Fn ( 0..^ ( # ` F ) ) ) -> ( Fun ( F cyclShift J ) /\ ( 1..^ N ) C_ dom ( F cyclShift J ) ) )
70	60, 69	mpdan 702	. . . . 5 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( Fun ( F cyclShift J ) /\ ( 1..^ N ) C_ dom ( F cyclShift J ) ) )
71		dfimafn 6245	. . . . 5 |- ( ( Fun ( F cyclShift J ) /\ ( 1..^ N ) C_ dom ( F cyclShift J ) ) -> ( ( F cyclShift J ) " ( 1..^ N ) ) = { z | E. y e. ( 1..^ N ) ( ( F cyclShift J ) ` y ) = z } )
72	70, 71	syl 17	. . . 4 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( ( F cyclShift J ) " ( 1..^ N ) ) = { z | E. y e. ( 1..^ N ) ( ( F cyclShift J ) ` y ) = z } )
73	72	eqcomd 2628	. . 3 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> { z | E. y e. ( 1..^ N ) ( ( F cyclShift J ) ` y ) = z } = ( ( F cyclShift J ) " ( 1..^ N ) ) )
74	56, 73	eqtrd 2656	. 2 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> { z | E. x e. ( ( 0..^ N ) \ { J } ) ( F ` x ) = z } = ( ( F cyclShift J ) " ( 1..^ N ) ) )
75	18, 74	eqtrd 2656	1 |- ( ( F e. Word S /\ N = ( # ` F ) /\ J e. ( 0..^ N ) ) -> ( F " ( ( 0..^ N ) \ { J } ) ) = ( ( F cyclShift J ) " ( 1..^ N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   \ cdif 3571   C_ wss 3574   {csn 4177   dom cdm 5114   "cima 5117   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   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-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:  cshimadifsn0  13576