Theorem scshwfzeqfzo 13572

Description: For a nonempty word the sets of shifted words, expressd by a finite interval of integers or by a half-open integer range are identical. (Contributed by Alexander van der Vekens, 15-Jun-2018.)

Assertion

Ref	Expression
scshwfzeqfzo	|- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> { y e. Word V | E. n e. ( 0 ... N ) y = ( X cyclShift n ) } = { y e. Word V | E. n e. ( 0..^ N ) y = ( X cyclShift n ) } )

Distinct variable groups:   n, N, y   n, V, y   n, X, y

Proof of Theorem scshwfzeqfzo

Step	Hyp	Ref	Expression
1		lencl 13324	. . . . . . . . . . . 12 |- ( X e. Word V -> ( # ` X ) e. NN0 )
2		elnn0uz 11725	. . . . . . . . . . . 12 |- ( ( # ` X ) e. NN0 <-> ( # ` X ) e. ( ZZ>= ` 0 ) )
3	1, 2	sylib 208	. . . . . . . . . . 11 |- ( X e. Word V -> ( # ` X ) e. ( ZZ>= ` 0 ) )
4	3	adantr 481	. . . . . . . . . 10 |- ( ( X e. Word V /\ N = ( # ` X ) ) -> ( # ` X ) e. ( ZZ>= ` 0 ) )
5		eleq1 2689	. . . . . . . . . . 11 |- ( N = ( # ` X ) -> ( N e. ( ZZ>= ` 0 ) <-> ( # ` X ) e. ( ZZ>= ` 0 ) ) )
6	5	adantl 482	. . . . . . . . . 10 |- ( ( X e. Word V /\ N = ( # ` X ) ) -> ( N e. ( ZZ>= ` 0 ) <-> ( # ` X ) e. ( ZZ>= ` 0 ) ) )
7	4, 6	mpbird 247	. . . . . . . . 9 |- ( ( X e. Word V /\ N = ( # ` X ) ) -> N e. ( ZZ>= ` 0 ) )
8	7	3adant2 1080	. . . . . . . 8 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> N e. ( ZZ>= ` 0 ) )
9	8	adantr 481	. . . . . . 7 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> N e. ( ZZ>= ` 0 ) )
10		fzisfzounsn 12580	. . . . . . 7 |- ( N e. ( ZZ>= ` 0 ) -> ( 0 ... N ) = ( ( 0..^ N ) u. { N } ) )
11	9, 10	syl 17	. . . . . 6 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> ( 0 ... N ) = ( ( 0..^ N ) u. { N } ) )
12	11	rexeqdv 3145	. . . . 5 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> ( E. n e. ( 0 ... N ) y = ( X cyclShift n ) <-> E. n e. ( ( 0..^ N ) u. { N } ) y = ( X cyclShift n ) ) )
13		rexun 3793	. . . . 5 |- ( E. n e. ( ( 0..^ N ) u. { N } ) y = ( X cyclShift n ) <-> ( E. n e. ( 0..^ N ) y = ( X cyclShift n ) \/ E. n e. { N } y = ( X cyclShift n ) ) )
14	12, 13	syl6bb 276	. . . 4 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> ( E. n e. ( 0 ... N ) y = ( X cyclShift n ) <-> ( E. n e. ( 0..^ N ) y = ( X cyclShift n ) \/ E. n e. { N } y = ( X cyclShift n ) ) ) )
15		ax-1 6	. . . . . 6 |- ( E. n e. ( 0..^ N ) y = ( X cyclShift n ) -> ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> E. n e. ( 0..^ N ) y = ( X cyclShift n ) ) )
16		fvex 6201	. . . . . . . . . . . 12 |- ( # ` X ) e. _V
17		eleq1 2689	. . . . . . . . . . . 12 |- ( N = ( # ` X ) -> ( N e. _V <-> ( # ` X ) e. _V ) )
18	16, 17	mpbiri 248	. . . . . . . . . . 11 |- ( N = ( # ` X ) -> N e. _V )
19		oveq2 6658	. . . . . . . . . . . . 13 |- ( n = N -> ( X cyclShift n ) = ( X cyclShift N ) )
20	19	eqeq2d 2632	. . . . . . . . . . . 12 |- ( n = N -> ( y = ( X cyclShift n ) <-> y = ( X cyclShift N ) ) )
21	20	rexsng 4219	. . . . . . . . . . 11 |- ( N e. _V -> ( E. n e. { N } y = ( X cyclShift n ) <-> y = ( X cyclShift N ) ) )
22	18, 21	syl 17	. . . . . . . . . 10 |- ( N = ( # ` X ) -> ( E. n e. { N } y = ( X cyclShift n ) <-> y = ( X cyclShift N ) ) )
23	22	3ad2ant3 1084	. . . . . . . . 9 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> ( E. n e. { N } y = ( X cyclShift n ) <-> y = ( X cyclShift N ) ) )
24	23	adantr 481	. . . . . . . 8 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> ( E. n e. { N } y = ( X cyclShift n ) <-> y = ( X cyclShift N ) ) )
25		oveq2 6658	. . . . . . . . . . . . 13 |- ( N = ( # ` X ) -> ( X cyclShift N ) = ( X cyclShift ( # ` X ) ) )
26	25	3ad2ant3 1084	. . . . . . . . . . . 12 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> ( X cyclShift N ) = ( X cyclShift ( # ` X ) ) )
27		cshwn 13543	. . . . . . . . . . . . 13 |- ( X e. Word V -> ( X cyclShift ( # ` X ) ) = X )
28	27	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> ( X cyclShift ( # ` X ) ) = X )
29	26, 28	eqtrd 2656	. . . . . . . . . . 11 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> ( X cyclShift N ) = X )
30	29	eqeq2d 2632	. . . . . . . . . 10 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> ( y = ( X cyclShift N ) <-> y = X ) )
31	30	adantr 481	. . . . . . . . 9 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> ( y = ( X cyclShift N ) <-> y = X ) )
32		cshw0 13540	. . . . . . . . . . . . . . 15 |- ( X e. Word V -> ( X cyclShift 0 ) = X )
33	32	3ad2ant1 1082	. . . . . . . . . . . . . 14 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> ( X cyclShift 0 ) = X )
34		lennncl 13325	. . . . . . . . . . . . . . . . . 18 |- ( ( X e. Word V /\ X =/= (/) ) -> ( # ` X ) e. NN )
35	34	3adant3 1081	. . . . . . . . . . . . . . . . 17 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> ( # ` X ) e. NN )
36		eleq1 2689	. . . . . . . . . . . . . . . . . 18 |- ( N = ( # ` X ) -> ( N e. NN <-> ( # ` X ) e. NN ) )
37	36	3ad2ant3 1084	. . . . . . . . . . . . . . . . 17 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> ( N e. NN <-> ( # ` X ) e. NN ) )
38	35, 37	mpbird 247	. . . . . . . . . . . . . . . 16 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> N e. NN )
39		lbfzo0 12507	. . . . . . . . . . . . . . . 16 |- ( 0 e. ( 0..^ N ) <-> N e. NN )
40	38, 39	sylibr 224	. . . . . . . . . . . . . . 15 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> 0 e. ( 0..^ N ) )
41		oveq2 6658	. . . . . . . . . . . . . . . . . . . 20 |- ( 0 = n -> ( X cyclShift 0 ) = ( X cyclShift n ) )
42	41	eqeq1d 2624	. . . . . . . . . . . . . . . . . . 19 |- ( 0 = n -> ( ( X cyclShift 0 ) = X <-> ( X cyclShift n ) = X ) )
43	42	eqcoms 2630	. . . . . . . . . . . . . . . . . 18 |- ( n = 0 -> ( ( X cyclShift 0 ) = X <-> ( X cyclShift n ) = X ) )
44		eqcom 2629	. . . . . . . . . . . . . . . . . 18 |- ( ( X cyclShift n ) = X <-> X = ( X cyclShift n ) )
45	43, 44	syl6bb 276	. . . . . . . . . . . . . . . . 17 |- ( n = 0 -> ( ( X cyclShift 0 ) = X <-> X = ( X cyclShift n ) ) )
46	45	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ n = 0 ) -> ( ( X cyclShift 0 ) = X <-> X = ( X cyclShift n ) ) )
47	46	biimpd 219	. . . . . . . . . . . . . . 15 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ n = 0 ) -> ( ( X cyclShift 0 ) = X -> X = ( X cyclShift n ) ) )
48	40, 47	rspcimedv 3311	. . . . . . . . . . . . . 14 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> ( ( X cyclShift 0 ) = X -> E. n e. ( 0..^ N ) X = ( X cyclShift n ) ) )
49	33, 48	mpd 15	. . . . . . . . . . . . 13 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> E. n e. ( 0..^ N ) X = ( X cyclShift n ) )
50	49	adantr 481	. . . . . . . . . . . 12 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> E. n e. ( 0..^ N ) X = ( X cyclShift n ) )
51	50	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) /\ y = X ) -> E. n e. ( 0..^ N ) X = ( X cyclShift n ) )
52		eqeq1 2626	. . . . . . . . . . . . 13 |- ( y = X -> ( y = ( X cyclShift n ) <-> X = ( X cyclShift n ) ) )
53	52	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) /\ y = X ) -> ( y = ( X cyclShift n ) <-> X = ( X cyclShift n ) ) )
54	53	rexbidv 3052	. . . . . . . . . . 11 |- ( ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) /\ y = X ) -> ( E. n e. ( 0..^ N ) y = ( X cyclShift n ) <-> E. n e. ( 0..^ N ) X = ( X cyclShift n ) ) )
55	51, 54	mpbird 247	. . . . . . . . . 10 |- ( ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) /\ y = X ) -> E. n e. ( 0..^ N ) y = ( X cyclShift n ) )
56	55	ex 450	. . . . . . . . 9 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> ( y = X -> E. n e. ( 0..^ N ) y = ( X cyclShift n ) ) )
57	31, 56	sylbid 230	. . . . . . . 8 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> ( y = ( X cyclShift N ) -> E. n e. ( 0..^ N ) y = ( X cyclShift n ) ) )
58	24, 57	sylbid 230	. . . . . . 7 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> ( E. n e. { N } y = ( X cyclShift n ) -> E. n e. ( 0..^ N ) y = ( X cyclShift n ) ) )
59	58	com12 32	. . . . . 6 |- ( E. n e. { N } y = ( X cyclShift n ) -> ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> E. n e. ( 0..^ N ) y = ( X cyclShift n ) ) )
60	15, 59	jaoi 394	. . . . 5 |- ( ( E. n e. ( 0..^ N ) y = ( X cyclShift n ) \/ E. n e. { N } y = ( X cyclShift n ) ) -> ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> E. n e. ( 0..^ N ) y = ( X cyclShift n ) ) )
61	60	com12 32	. . . 4 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> ( ( E. n e. ( 0..^ N ) y = ( X cyclShift n ) \/ E. n e. { N } y = ( X cyclShift n ) ) -> E. n e. ( 0..^ N ) y = ( X cyclShift n ) ) )
62	14, 61	sylbid 230	. . 3 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> ( E. n e. ( 0 ... N ) y = ( X cyclShift n ) -> E. n e. ( 0..^ N ) y = ( X cyclShift n ) ) )
63		fzossfz 12488	. . . 4 |- ( 0..^ N ) C_ ( 0 ... N )
64		ssrexv 3667	. . . 4 |- ( ( 0..^ N ) C_ ( 0 ... N ) -> ( E. n e. ( 0..^ N ) y = ( X cyclShift n ) -> E. n e. ( 0 ... N ) y = ( X cyclShift n ) ) )
65	63, 64	mp1i 13	. . 3 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> ( E. n e. ( 0..^ N ) y = ( X cyclShift n ) -> E. n e. ( 0 ... N ) y = ( X cyclShift n ) ) )
66	62, 65	impbid 202	. 2 |- ( ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) /\ y e. Word V ) -> ( E. n e. ( 0 ... N ) y = ( X cyclShift n ) <-> E. n e. ( 0..^ N ) y = ( X cyclShift n ) ) )
67	66	rabbidva 3188	1 |- ( ( X e. Word V /\ X =/= (/) /\ N = ( # ` X ) ) -> { y e. Word V | E. n e. ( 0 ... N ) y = ( X cyclShift n ) } = { y e. Word V | E. n e. ( 0..^ N ) y = ( X cyclShift n ) } )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   ` cfv 5888  (class class class)co 6650   0cc0 9936   NNcn 11020   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #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-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:  hashecclwwlksn1  26954  umgrhashecclwwlk  26955