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Theorem cshwsdisj 14441
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.
StepHypRef Expression
1 orc 385 . . . . 5  |-  ( n  =  j  ->  (
n  =  j  \/  ( { ( W cyclShift  n ) }  i^i  { ( W cyclShift  j ) } )  =  (/) ) )
21a1d 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 755 . . . . . . . 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 763 . . . . . . . 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 764 . . . . . . . 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 2736 . . . . . . . . . 10  |-  ( n  =/=  j  <->  j  =/=  n )
76biimpi 194 . . . . . . . . 9  |-  ( n  =/=  j  ->  j  =/=  n )
87adantr 465 . . . . . . . 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 )
)
109cshwshashlem3 14440 . . . . . . . . 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 ) ) )
1110imp 429 . . . . . . . 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 ) )
123, 4, 5, 8, 11syl13anc 1230 . . . . . . 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 4089 . . . . . . 7  |-  ( ( W cyclShift  n )  =/=  ( W cyclShift  j )  ->  ( { ( W cyclShift  n ) }  i^i  { ( W cyclShift  j ) } )  =  (/) )
1412, 13syl 16 . . . . . 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 ) } )  =  (/) )
1514olcd 393 . . . . 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 ) } )  =  (/) ) )
1615ex 434 . . . 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 ) } )  =  (/) ) ) )
172, 16pm2.61ine 2780 . . 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 ) } )  =  (/) ) )
1817ralrimivva 2885 . 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 6292 . . . 4  |-  ( n  =  j  ->  ( W cyclShift  n )  =  ( W cyclShift  j ) )
2019sneqd 4039 . . 3  |-  ( n  =  j  ->  { ( W cyclShift  n ) }  =  { ( W cyclShift  j ) } )
2120disjor 4431 . 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 ) } )  =  (/) ) )
2218, 21sylibr 212 1  |-  ( (
ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W ` 
0 ) )  -> Disj  n  e.  ( 0..^ (
# `  W )
) { ( W cyclShift  n ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815    i^i cin 3475   (/)c0 3785   {csn 4027  Disj wdisj 4417   ` cfv 5588  (class class class)co 6284   0cc0 9492  ..^cfzo 11792   #chash 12373  Word cword 12500   cyclShift ccsh 12722   Primecprime 14076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-hash 12374  df-word 12508  df-concat 12510  df-substr 12512  df-reps 12515  df-csh 12723  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-dvds 13848  df-gcd 14004  df-prm 14077  df-phi 14155
This theorem is referenced by:  cshwshashnsame  14446
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