MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cshwsdisj Structured version   Unicode version

Theorem cshwsdisj 15032
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 386 . . . . 5  |-  ( n  =  j  ->  (
n  =  j  \/  ( { ( W cyclShift  n ) }  i^i  { ( W cyclShift  j ) } )  =  (/) ) )
21a1d 26 . . . 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 762 . . . . . . . 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 772 . . . . . . . 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 773 . . . . . . . 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 2700 . . . . . . . . . 10  |-  ( n  =/=  j  <->  j  =/=  n )
76biimpi 197 . . . . . . . . 9  |-  ( n  =/=  j  ->  j  =/=  n )
87adantr 466 . . . . . . . 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 15031 . . . . . . . . 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 430 . . . . . . . 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 1266 . . . . . . 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 4064 . . . . . . 7  |-  ( ( W cyclShift  n )  =/=  ( W cyclShift  j )  ->  ( { ( W cyclShift  n ) }  i^i  { ( W cyclShift  j ) } )  =  (/) )
1412, 13syl 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 ) } )  =  (/) )
1514olcd 394 . . . . 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 435 . . . 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 2744 . . 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 2853 . 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 6313 . . . 4  |-  ( n  =  j  ->  ( W cyclShift  n )  =  ( W cyclShift  j ) )
2019sneqd 4014 . . 3  |-  ( n  =  j  ->  { ( W cyclShift  n ) }  =  { ( W cyclShift  j ) } )
2120disjor 4411 . 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 215 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 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783    i^i cin 3441   (/)c0 3767   {csn 4002  Disj wdisj 4397   ` cfv 5601  (class class class)co 6305   0cc0 9538  ..^cfzo 11913   #chash 12512  Word cword 12643   cyclShift ccsh 12875   Primecprime 14593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-word 12651  df-concat 12653  df-substr 12655  df-reps 12658  df-csh 12876  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-dvds 14284  df-gcd 14443  df-prm 14594  df-phi 14683
This theorem is referenced by:  cshwshashnsame  15037
  Copyright terms: Public domain W3C validator