Theorem cshwrepswhash1 15809

Description: The size of the set of (different!) words resulting by cyclically shifting a nonempty "repeated symbol word" is 1. (Contributed by AV, 18-May-2018.) (Revised by AV, 8-Nov-2018.)

Hypothesis

Ref	Expression
cshwrepswhash1.m	|- M = { w e. Word V | E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w }

Assertion

Ref	Expression
cshwrepswhash1	|- ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) -> ( # ` M ) = 1 )

Distinct variable groups:   n, V, w   n, W, w   A, n, w   n, N, w

Allowed substitution hints:   M( w, n)

Proof of Theorem cshwrepswhash1

Dummy variables i u r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnnn0 11299	. . . . . . . 8 |- ( N e. NN -> N e. NN0 )
2		repsdf2 13525	. . . . . . . 8 |- ( ( A e. V /\ N e. NN0 ) -> ( W = ( A repeatS N ) <-> ( W e. Word V /\ ( # ` W ) = N /\ A. i e. ( 0..^ N ) ( W ` i ) = A ) ) )
3	1, 2	sylan2 491	. . . . . . 7 |- ( ( A e. V /\ N e. NN ) -> ( W = ( A repeatS N ) <-> ( W e. Word V /\ ( # ` W ) = N /\ A. i e. ( 0..^ N ) ( W ` i ) = A ) ) )
4		simp1 1061	. . . . . . . . . 10 |- ( ( W e. Word V /\ ( # ` W ) = N /\ A. i e. ( 0..^ N ) ( W ` i ) = A ) -> W e. Word V )
5	4	adantl 482	. . . . . . . . 9 |- ( ( ( A e. V /\ N e. NN ) /\ ( W e. Word V /\ ( # ` W ) = N /\ A. i e. ( 0..^ N ) ( W ` i ) = A ) ) -> W e. Word V )
6		eleq1 2689	. . . . . . . . . . . . . . . 16 |- ( N = ( # ` W ) -> ( N e. NN <-> ( # ` W ) e. NN ) )
7	6	eqcoms 2630	. . . . . . . . . . . . . . 15 |- ( ( # ` W ) = N -> ( N e. NN <-> ( # ` W ) e. NN ) )
8		lbfzo0 12507	. . . . . . . . . . . . . . . 16 |- ( 0 e. ( 0..^ ( # ` W ) ) <-> ( # ` W ) e. NN )
9	8	biimpri 218	. . . . . . . . . . . . . . 15 |- ( ( # ` W ) e. NN -> 0 e. ( 0..^ ( # ` W ) ) )
10	7, 9	syl6bi 243	. . . . . . . . . . . . . 14 |- ( ( # ` W ) = N -> ( N e. NN -> 0 e. ( 0..^ ( # ` W ) ) ) )
11	10	3ad2ant2 1083	. . . . . . . . . . . . 13 |- ( ( W e. Word V /\ ( # ` W ) = N /\ A. i e. ( 0..^ N ) ( W ` i ) = A ) -> ( N e. NN -> 0 e. ( 0..^ ( # ` W ) ) ) )
12	11	com12 32	. . . . . . . . . . . 12 |- ( N e. NN -> ( ( W e. Word V /\ ( # ` W ) = N /\ A. i e. ( 0..^ N ) ( W ` i ) = A ) -> 0 e. ( 0..^ ( # ` W ) ) ) )
13	12	adantl 482	. . . . . . . . . . 11 |- ( ( A e. V /\ N e. NN ) -> ( ( W e. Word V /\ ( # ` W ) = N /\ A. i e. ( 0..^ N ) ( W ` i ) = A ) -> 0 e. ( 0..^ ( # ` W ) ) ) )
14	13	imp 445	. . . . . . . . . 10 |- ( ( ( A e. V /\ N e. NN ) /\ ( W e. Word V /\ ( # ` W ) = N /\ A. i e. ( 0..^ N ) ( W ` i ) = A ) ) -> 0 e. ( 0..^ ( # ` W ) ) )
15		cshw0 13540	. . . . . . . . . . 11 |- ( W e. Word V -> ( W cyclShift 0 ) = W )
16	5, 15	syl 17	. . . . . . . . . 10 |- ( ( ( A e. V /\ N e. NN ) /\ ( W e. Word V /\ ( # ` W ) = N /\ A. i e. ( 0..^ N ) ( W ` i ) = A ) ) -> ( W cyclShift 0 ) = W )
17		oveq2 6658	. . . . . . . . . . . 12 |- ( n = 0 -> ( W cyclShift n ) = ( W cyclShift 0 ) )
18	17	eqeq1d 2624	. . . . . . . . . . 11 |- ( n = 0 -> ( ( W cyclShift n ) = W <-> ( W cyclShift 0 ) = W ) )
19	18	rspcev 3309	. . . . . . . . . 10 |- ( ( 0 e. ( 0..^ ( # ` W ) ) /\ ( W cyclShift 0 ) = W ) -> E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = W )
20	14, 16, 19	syl2anc 693	. . . . . . . . 9 |- ( ( ( A e. V /\ N e. NN ) /\ ( W e. Word V /\ ( # ` W ) = N /\ A. i e. ( 0..^ N ) ( W ` i ) = A ) ) -> E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = W )
21		eqeq2 2633	. . . . . . . . . . 11 |- ( w = W -> ( ( W cyclShift n ) = w <-> ( W cyclShift n ) = W ) )
22	21	rexbidv 3052	. . . . . . . . . 10 |- ( w = W -> ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w <-> E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = W ) )
23	22	rspcev 3309	. . . . . . . . 9 |- ( ( W e. Word V /\ E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = W ) -> E. w e. Word V E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w )
24	5, 20, 23	syl2anc 693	. . . . . . . 8 |- ( ( ( A e. V /\ N e. NN ) /\ ( W e. Word V /\ ( # ` W ) = N /\ A. i e. ( 0..^ N ) ( W ` i ) = A ) ) -> E. w e. Word V E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w )
25	24	ex 450	. . . . . . 7 |- ( ( A e. V /\ N e. NN ) -> ( ( W e. Word V /\ ( # ` W ) = N /\ A. i e. ( 0..^ N ) ( W ` i ) = A ) -> E. w e. Word V E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w ) )
26	3, 25	sylbid 230	. . . . . 6 |- ( ( A e. V /\ N e. NN ) -> ( W = ( A repeatS N ) -> E. w e. Word V E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w ) )
27	26	3impia 1261	. . . . 5 |- ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) -> E. w e. Word V E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w )
28		repsw 13522	. . . . . . . 8 |- ( ( A e. V /\ N e. NN0 ) -> ( A repeatS N ) e. Word V )
29	1, 28	sylan2 491	. . . . . . 7 |- ( ( A e. V /\ N e. NN ) -> ( A repeatS N ) e. Word V )
30	29	3adant3 1081	. . . . . 6 |- ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) -> ( A repeatS N ) e. Word V )
31		simpll3 1102	. . . . . . . . . . . 12 |- ( ( ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) /\ u e. Word V ) /\ n e. ( 0..^ ( # ` W ) ) ) -> W = ( A repeatS N ) )
32	31	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) /\ u e. Word V ) /\ n e. ( 0..^ ( # ` W ) ) ) -> ( W cyclShift n ) = ( ( A repeatS N ) cyclShift n ) )
33		simp1 1061	. . . . . . . . . . . . 13 |- ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) -> A e. V )
34	33	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) /\ u e. Word V ) /\ n e. ( 0..^ ( # ` W ) ) ) -> A e. V )
35	1	3ad2ant2 1083	. . . . . . . . . . . . 13 |- ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) -> N e. NN0 )
36	35	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) /\ u e. Word V ) /\ n e. ( 0..^ ( # ` W ) ) ) -> N e. NN0 )
37		elfzoelz 12470	. . . . . . . . . . . . 13 |- ( n e. ( 0..^ ( # ` W ) ) -> n e. ZZ )
38	37	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) /\ u e. Word V ) /\ n e. ( 0..^ ( # ` W ) ) ) -> n e. ZZ )
39		repswcshw 13558	. . . . . . . . . . . 12 |- ( ( A e. V /\ N e. NN0 /\ n e. ZZ ) -> ( ( A repeatS N ) cyclShift n ) = ( A repeatS N ) )
40	34, 36, 38, 39	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) /\ u e. Word V ) /\ n e. ( 0..^ ( # ` W ) ) ) -> ( ( A repeatS N ) cyclShift n ) = ( A repeatS N ) )
41	32, 40	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) /\ u e. Word V ) /\ n e. ( 0..^ ( # ` W ) ) ) -> ( W cyclShift n ) = ( A repeatS N ) )
42	41	eqeq1d 2624	. . . . . . . . 9 |- ( ( ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) /\ u e. Word V ) /\ n e. ( 0..^ ( # ` W ) ) ) -> ( ( W cyclShift n ) = u <-> ( A repeatS N ) = u ) )
43	42	biimpd 219	. . . . . . . 8 |- ( ( ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) /\ u e. Word V ) /\ n e. ( 0..^ ( # ` W ) ) ) -> ( ( W cyclShift n ) = u -> ( A repeatS N ) = u ) )
44	43	rexlimdva 3031	. . . . . . 7 |- ( ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) /\ u e. Word V ) -> ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = u -> ( A repeatS N ) = u ) )
45	44	ralrimiva 2966	. . . . . 6 |- ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) -> A. u e. Word V ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = u -> ( A repeatS N ) = u ) )
46		eqeq1 2626	. . . . . . . . 9 |- ( w = ( A repeatS N ) -> ( w = u <-> ( A repeatS N ) = u ) )
47	46	imbi2d 330	. . . . . . . 8 |- ( w = ( A repeatS N ) -> ( ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = u -> w = u ) <-> ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = u -> ( A repeatS N ) = u ) ) )
48	47	ralbidv 2986	. . . . . . 7 |- ( w = ( A repeatS N ) -> ( A. u e. Word V ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = u -> w = u ) <-> A. u e. Word V ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = u -> ( A repeatS N ) = u ) ) )
49	48	rspcev 3309	. . . . . 6 |- ( ( ( A repeatS N ) e. Word V /\ A. u e. Word V ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = u -> ( A repeatS N ) = u ) ) -> E. w e. Word V A. u e. Word V ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = u -> w = u ) )
50	30, 45, 49	syl2anc 693	. . . . 5 |- ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) -> E. w e. Word V A. u e. Word V ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = u -> w = u ) )
51		eqeq2 2633	. . . . . . 7 |- ( w = u -> ( ( W cyclShift n ) = w <-> ( W cyclShift n ) = u ) )
52	51	rexbidv 3052	. . . . . 6 |- ( w = u -> ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w <-> E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = u ) )
53	52	reu7 3401	. . . . 5 |- ( E! w e. Word V E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w <-> ( E. w e. Word V E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w /\ E. w e. Word V A. u e. Word V ( E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = u -> w = u ) ) )
54	27, 50, 53	sylanbrc 698	. . . 4 |- ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) -> E! w e. Word V E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w )
55		reusn 4262	. . . 4 |- ( E! w e. Word V E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w <-> E. r { w e. Word V | E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w } = { r } )
56	54, 55	sylib 208	. . 3 |- ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) -> E. r { w e. Word V | E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w } = { r } )
57		cshwrepswhash1.m	. . . . 5 |- M = { w e. Word V | E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w }
58	57	eqeq1i 2627	. . . 4 |- ( M = { r } <-> { w e. Word V | E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w } = { r } )
59	58	exbii 1774	. . 3 |- ( E. r M = { r } <-> E. r { w e. Word V | E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w } = { r } )
60	56, 59	sylibr 224	. 2 |- ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) -> E. r M = { r } )
61	57	cshwsex 15807	. . . . . 6 |- ( W e. Word V -> M e. _V )
62	61	3ad2ant1 1082	. . . . 5 |- ( ( W e. Word V /\ ( # ` W ) = N /\ A. i e. ( 0..^ N ) ( W ` i ) = A ) -> M e. _V )
63	3, 62	syl6bi 243	. . . 4 |- ( ( A e. V /\ N e. NN ) -> ( W = ( A repeatS N ) -> M e. _V ) )
64	63	3impia 1261	. . 3 |- ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) -> M e. _V )
65		hash1snb 13207	. . 3 |- ( M e. _V -> ( ( # ` M ) = 1 <-> E. r M = { r } ) )
66	64, 65	syl 17	. 2 |- ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) -> ( ( # ` M ) = 1 <-> E. r M = { r } ) )
67	60, 66	mpbird 247	1 |- ( ( A e. V /\ N e. NN /\ W = ( A repeatS N ) ) -> ( # ` M ) = 1 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   {crab 2916   _Vcvv 3200   {csn 4177   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   NNcn 11020   NN0cn0 11292   ZZcz 11377  ..^cfzo 12465   #chash 13117  Word cword 13291   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-cda 8990  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:  cshwshash  15811