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Theorem cshwsdisj 14238
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 2718 . . . . . . . . . 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 14237 . . . . . . . . 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 1221 . . . . . . 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 4040 . . . . . . 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 2762 . . 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 2908 . 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 6203 . . . 4  |-  ( n  =  j  ->  ( W cyclShift  n )  =  ( W cyclShift  j ) )
2019sneqd 3992 . . 3  |-  ( n  =  j  ->  { ( W cyclShift  n ) }  =  { ( W cyclShift  j ) } )
2120disjor 4379 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2645   A.wral 2796   E.wrex 2797    i^i cin 3430   (/)c0 3740   {csn 3980  Disj wdisj 4365   ` cfv 5521  (class class class)co 6195   0cc0 9388  ..^cfzo 11660   #chash 12215  Word cword 12334   cyclShift ccsh 12538   Primecprime 13876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-disj 4366  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-fz 11550  df-fzo 11661  df-fl 11754  df-mod 11821  df-seq 11919  df-exp 11978  df-hash 12216  df-word 12342  df-concat 12344  df-substr 12346  df-reps 12349  df-csh 12539  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-dvds 13649  df-gcd 13804  df-prm 13877  df-phi 13954
This theorem is referenced by:  cshwshashnsame  14243
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