Theorem cshwshashnsame 15810

Description: If a word (not consisting of identical symbols) 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. (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
cshwshashnsame	|- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) -> ( # ` M ) = ( # ` W ) ) )

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

Allowed substitution hints:   M( w, i, n)

Proof of Theorem cshwshashnsame

Step	Hyp	Ref	Expression
1		cshwrepswhash1.m	. . . . . 6 |- M = { w e. Word V | E. n e. ( 0..^ ( # ` W ) ) ( W cyclShift n ) = w }
2	1	cshwsiun 15806	. . . . 5 |- ( W e. Word V -> M = U_ n e. ( 0..^ ( # ` W ) ) { ( W cyclShift n ) } )
3	2	ad2antrr 762	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> M = U_ n e. ( 0..^ ( # ` W ) ) { ( W cyclShift n ) } )
4	3	fveq2d 6195	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> ( # ` M ) = ( # ` U_ n e. ( 0..^ ( # ` W ) ) { ( W cyclShift n ) } ) )
5		fzofi 12773	. . . . 5 |- ( 0..^ ( # ` W ) ) e. Fin
6	5	a1i 11	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> ( 0..^ ( # ` W ) ) e. Fin )
7		snfi 8038	. . . . 5 |- { ( W cyclShift n ) } e. Fin
8	7	a1i 11	. . . 4 |- ( ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) /\ n e. ( 0..^ ( # ` W ) ) ) -> { ( W cyclShift n ) } e. Fin )
9		id 22	. . . . 5 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( W e. Word V /\ ( # ` W ) e. Prime ) )
10	9	cshwsdisj 15805	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> Disj n e. ( 0..^ ( # ` W ) ) { ( W cyclShift n ) } )
11	6, 8, 10	hashiun 14554	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> ( # ` U_ n e. ( 0..^ ( # ` W ) ) { ( W cyclShift n ) } ) = sum_ n e. ( 0..^ ( # ` W ) ) ( # ` { ( W cyclShift n ) } ) )
12		ovex 6678	. . . . . 6 |- ( W cyclShift n ) e. _V
13		hashsng 13159	. . . . . 6 |- ( ( W cyclShift n ) e. _V -> ( # ` { ( W cyclShift n ) } ) = 1 )
14	12, 13	mp1i 13	. . . . 5 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> ( # ` { ( W cyclShift n ) } ) = 1 )
15	14	sumeq2sdv 14435	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> sum_ n e. ( 0..^ ( # ` W ) ) ( # ` { ( W cyclShift n ) } ) = sum_ n e. ( 0..^ ( # ` W ) ) 1 )
16		1cnd 10056	. . . . . . 7 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> 1 e. CC )
17		fsumconst 14522	. . . . . . 7 |- ( ( ( 0..^ ( # ` W ) ) e. Fin /\ 1 e. CC ) -> sum_ n e. ( 0..^ ( # ` W ) ) 1 = ( ( # ` ( 0..^ ( # ` W ) ) ) x. 1 ) )
18	5, 16, 17	sylancr 695	. . . . . 6 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> sum_ n e. ( 0..^ ( # ` W ) ) 1 = ( ( # ` ( 0..^ ( # ` W ) ) ) x. 1 ) )
19		lencl 13324	. . . . . . . . 9 |- ( W e. Word V -> ( # ` W ) e. NN0 )
20	19	adantr 481	. . . . . . . 8 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( # ` W ) e. NN0 )
21		hashfzo0 13217	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( # ` ( 0..^ ( # ` W ) ) ) = ( # ` W ) )
22	20, 21	syl 17	. . . . . . 7 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( # ` ( 0..^ ( # ` W ) ) ) = ( # ` W ) )
23	22	oveq1d 6665	. . . . . 6 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( ( # ` ( 0..^ ( # ` W ) ) ) x. 1 ) = ( ( # ` W ) x. 1 ) )
24		prmnn 15388	. . . . . . . . 9 |- ( ( # ` W ) e. Prime -> ( # ` W ) e. NN )
25	24	nnred 11035	. . . . . . . 8 |- ( ( # ` W ) e. Prime -> ( # ` W ) e. RR )
26	25	adantl 482	. . . . . . 7 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( # ` W ) e. RR )
27		ax-1rid 10006	. . . . . . 7 |- ( ( # ` W ) e. RR -> ( ( # ` W ) x. 1 ) = ( # ` W ) )
28	26, 27	syl 17	. . . . . 6 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( ( # ` W ) x. 1 ) = ( # ` W ) )
29	18, 23, 28	3eqtrd 2660	. . . . 5 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> sum_ n e. ( 0..^ ( # ` W ) ) 1 = ( # ` W ) )
30	29	adantr 481	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> sum_ n e. ( 0..^ ( # ` W ) ) 1 = ( # ` W ) )
31	15, 30	eqtrd 2656	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> sum_ n e. ( 0..^ ( # ` W ) ) ( # ` { ( W cyclShift n ) } ) = ( # ` W ) )
32	4, 11, 31	3eqtrd 2660	. 2 |- ( ( ( W e. Word V /\ ( # ` W ) e. Prime ) /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> ( # ` M ) = ( # ` W ) )
33	32	ex 450	1 |- ( ( W e. Word V /\ ( # ` W ) e. Prime ) -> ( E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) -> ( # ` M ) = ( # ` W ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   {csn 4177   U_ciun 4520   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Word cword 13291   cyclShift ccsh 13534   sum_csu 14416   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:  cshwshash  15811  umgrhashecclwwlk  26955