Theorem cshwshash 15811

Description: If a word has a length being a prime number, the size of the set of (different!) words resulting by cyclically shifting the original word equals the length of the original word or 1. (Contributed by AV, 19-May-2018.) (Revised by AV, 10-Nov-2018.)

Hypothesis

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

Assertion

Ref	Expression
cshwshash	|- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( ( # ` M ) = ( # ` W ) \/ ( # ` M ) = 1 ) )

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

Allowed substitution hints:   M( w, n)

Proof of Theorem cshwshash

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		repswsymballbi 13527	. . . . 5 |- ( W e. Word V -> ( W = ( ( W ` 0 ) repeatS ( # ` W ) ) <-> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) )
2	1	adantr 481	. . . 4 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( W = ( ( W ` 0 ) repeatS ( # ` W ) ) <-> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) )
3		prmnn 15388	. . . . . . . . 9 |- ( ( # ` W ) e. Prime -> ( # ` W ) e. NN )
4	3	nnge1d 11063	. . . . . . . 8 |- ( ( # ` W ) e. Prime -> 1 <_ ( # ` W ) )
5		wrdsymb1 13342	. . . . . . . 8 |- ( ( W e. Word V /\ 1 <_ ( # ` W ) ) -> ( W ` 0 ) e. V )
6	4, 5	sylan2 491	. . . . . . 7 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( W ` 0 ) e. V )
7	6	adantr 481	. . . . . 6 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ W = ( ( W ` 0 ) repeatS ( # ` W ) ) ) -> ( W ` 0 ) e. V )
8	3	ad2antlr 763	. . . . . 6 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ W = ( ( W ` 0 ) repeatS ( # ` W ) ) ) -> ( # ` W ) e. NN )
9		simpr 477	. . . . . 6 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ W = ( ( W ` 0 ) repeatS ( # ` W ) ) ) -> W = ( ( W ` 0 ) repeatS ( # ` W ) ) )
10		cshwrepswhash1.m	. . . . . . 7 |- M = { w e. Word V | E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w }
11	10	cshwrepswhash1 15809	. . . . . 6 |- ( ( ( W ` 0 ) e. V /\ ( # ` W ) e. NN /\ W = ( ( W ` 0 ) repeatS ( # ` W ) ) ) -> ( # ` M ) = 1 )
12	7, 8, 9, 11	syl3anc 1326	. . . . 5 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ W = ( ( W ` 0 ) repeatS ( # ` W ) ) ) -> ( # ` M ) = 1 )
13	12	ex 450	. . . 4 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( W = ( ( W ` 0 ) repeatS ( # ` W ) ) -> ( # ` M ) = 1 ) )
14	2, 13	sylbird 250	. . 3 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) -> ( # ` M ) = 1 ) )
15		olc 399	. . 3 |- ( ( # ` M ) = 1 -> ( ( # ` M ) = ( # ` W ) \/ ( # ` M ) = 1 ) )
16	14, 15	syl6com 37	. 2 |- ( A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) -> ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( ( # ` M ) = ( # ` W ) \/ ( # ` M ) = 1 ) ) )
17		rexnal 2995	. . . 4 |- ( E. i e. ( 0..^ ( # ` W ) ) -. ( W ` i ) = ( W ` 0 ) <-> -. A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) )
18		df-ne 2795	. . . . . 6 |- ( ( W ` i ) =/= ( W ` 0 ) <-> -. ( W ` i ) = ( W ` 0 ) )
19	18	bicomi 214	. . . . 5 |- ( -. ( W ` i ) = ( W ` 0 ) <-> ( W ` i ) =/= ( W ` 0 ) )
20	19	rexbii 3041	. . . 4 |- ( E. i e. ( 0..^ ( # ` W ) ) -. ( W ` i ) = ( W ` 0 ) <-> E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) )
21	17, 20	bitr3i 266	. . 3 |- ( -. A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) <-> E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) )
22	10	cshwshashnsame 15810	. . . 4 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) -> ( # ` M ) = ( # ` W ) ) )
23		orc 400	. . . 4 |- ( ( # ` M ) = ( # ` W ) -> ( ( # ` M ) = ( # ` W ) \/ ( # ` M ) = 1 ) )
24	22, 23	syl6com 37	. . 3 |- ( E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) -> ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( ( # ` M ) = ( # ` W ) \/ ( # ` M ) = 1 ) ) )
25	21, 24	sylbi 207	. 2 |- ( -. A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) -> ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( ( # ` M ) = ( # ` W ) \/ ( # ` M ) = 1 ) ) )
26	16, 25	pm2.61i 176	1 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( ( # ` M ) = ( # ` W ) \/ ( # ` M ) = 1 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   <_ cle 10075   NNcn 11020  ..^cfzo 12465   #chash 13117  Word cword 13291   repeatS creps 13298   cyclShift ccsh 13534   Primecprime 15385

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-inf2 8538  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-fal 1489  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-disj 4621  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-se 5074  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-isom 5897  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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  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-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-reps 13306  df-csh 13535  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471

This theorem is referenced by:  hashecclwwlksn1  26954