Theorem cshwsdisj 15805

Description: The singletons resulting by cyclically shifting a given word of length being a prime number and not consisting of identical symbols is a disjoint collection. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)

Hypothesis

Ref	Expression
cshwshash.0	|- ( ph -> ( W e. Word V /\ ( # ` W ) e. Prime ) )

Assertion

Ref	Expression
cshwsdisj	|- ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> Disj n e. ( 0..^ ( # ` W ) ) { ( W cyclShift n ) } )

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

Allowed substitution hint:   V( n)

Proof of Theorem cshwsdisj

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 400	. . . . 5 |- ( n = j -> ( n = j \/ ( { ( W cyclShift n ) } i^i { ( W cyclShift j ) } ) = (/) ) )
2	1	a1d 25	. . . 4 |- ( n = j -> ( ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) /\ ( n e. ( 0..^ ( # ` W ) ) /\ j e. ( 0..^ ( # ` W ) ) ) ) -> ( n = j \/ ( { ( W cyclShift n ) } i^i { ( W cyclShift j ) } ) = (/) ) ) )
3		simprl 794	. . . . . . . 8 |- ( ( n =/= j /\ ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) /\ ( n e. ( 0..^ ( # ` W ) ) /\ j e. ( 0..^ ( # ` W ) ) ) ) ) -> ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) )
4		simprrl 804	. . . . . . . 8 |- ( ( n =/= j /\ ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) /\ ( n e. ( 0..^ ( # ` W ) ) /\ j e. ( 0..^ ( # ` W ) ) ) ) ) -> n e. ( 0..^ ( # ` W ) ) )
5		simprrr 805	. . . . . . . 8 |- ( ( n =/= j /\ ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) /\ ( n e. ( 0..^ ( # ` W ) ) /\ j e. ( 0..^ ( # ` W ) ) ) ) ) -> j e. ( 0..^ ( # ` W ) ) )
6		necom 2847	. . . . . . . . . 10 |- ( n =/= j <-> j =/= n )
7	6	biimpi 206	. . . . . . . . 9 |- ( n =/= j -> j =/= n )
8	7	adantr 481	. . . . . . . 8 |- ( ( n =/= j /\ ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) /\ ( n e. ( 0..^ ( # ` W ) ) /\ j e. ( 0..^ ( # ` W ) ) ) ) ) -> j =/= n )
9		cshwshash.0	. . . . . . . . . 10 |- ( ph -> ( W e. Word V /\ ( # ` W ) e. Prime ) )
10	9	cshwshashlem3 15804	. . . . . . . . 9 |- ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> ( ( n e. ( 0..^ ( # ` W ) ) /\ j e. ( 0..^ ( # ` W ) ) /\ j =/= n ) -> ( W cyclShift n ) =/= ( W cyclShift j ) ) )
11	10	imp 445	. . . . . . . 8 |- ( ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) /\ ( n e. ( 0..^ ( # ` W ) ) /\ j e. ( 0..^ ( # ` W ) ) /\ j =/= n ) ) -> ( W cyclShift n ) =/= ( W cyclShift j ) )
12	3, 4, 5, 8, 11	syl13anc 1328	. . . . . . 7 |- ( ( n =/= j /\ ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) /\ ( n e. ( 0..^ ( # ` W ) ) /\ j e. ( 0..^ ( # ` W ) ) ) ) ) -> ( W cyclShift n ) =/= ( W cyclShift j ) )
13		disjsn2 4247	. . . . . . 7 |- ( ( W cyclShift n ) =/= ( W cyclShift j ) -> ( { ( W cyclShift n ) } i^i { ( W cyclShift j ) } ) = (/) )
14	12, 13	syl 17	. . . . . 6 |- ( ( n =/= j /\ ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) /\ ( n e. ( 0..^ ( # ` W ) ) /\ j e. ( 0..^ ( # ` W ) ) ) ) ) -> ( { ( W cyclShift n ) } i^i { ( W cyclShift j ) } ) = (/) )
15	14	olcd 408	. . . . 5 |- ( ( n =/= j /\ ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) /\ ( n e. ( 0..^ ( # ` W ) ) /\ j e. ( 0..^ ( # ` W ) ) ) ) ) -> ( n = j \/ ( { ( W cyclShift n ) } i^i { ( W cyclShift j ) } ) = (/) ) )
16	15	ex 450	. . . 4 |- ( n =/= j -> ( ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) /\ ( n e. ( 0..^ ( # ` W ) ) /\ j e. ( 0..^ ( # ` W ) ) ) ) -> ( n = j \/ ( { ( W cyclShift n ) } i^i { ( W cyclShift j ) } ) = (/) ) ) )
17	2, 16	pm2.61ine 2877	. . 3 |- ( ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) /\ ( n e. ( 0..^ ( # ` W ) ) /\ j e. ( 0..^ ( # ` W ) ) ) ) -> ( n = j \/ ( { ( W cyclShift n ) } i^i { ( W cyclShift j ) } ) = (/) ) )
18	17	ralrimivva 2971	. 2 |- ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> A. n e. ( 0..^ ( # ` W ) ) A. j e. ( 0..^ ( # ` W ) ) ( n = j \/ ( { ( W cyclShift n ) } i^i { ( W cyclShift j ) } ) = (/) ) )
19		oveq2 6658	. . . 4 |- ( n = j -> ( W cyclShift n ) = ( W cyclShift j ) )
20	19	sneqd 4189	. . 3 |- ( n = j -> { ( W cyclShift n ) } = { ( W cyclShift j ) } )
21	20	disjor 4634	. 2 |- (Disj n e. ( 0..^ ( # ` W ) ) { ( W cyclShift n ) } <-> A. n e. ( 0..^ ( # ` W ) ) A. j e. ( 0..^ ( # ` W ) ) ( n = j \/ ( { ( W cyclShift n ) } i^i { ( W cyclShift j ) } ) = (/) ) )
22	18, 21	sylibr 224	1 |- ( ( ph /\ E. i e. ( 0..^ ( # ` W ) ) ( W ` i ) =/= ( W ` 0 ) ) -> Disj n e. ( 0..^ ( # ` W ) ) { ( W cyclShift n ) } )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   (/)c0 3915   {csn 4177  Disj wdisj 4620   ` cfv 5888  (class class class)co 6650   0cc0 9936  ..^cfzo 12465   #chash 13117  Word cword 13291   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-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-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-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-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-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-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471

This theorem is referenced by:  cshwshashnsame  15810