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Theorem cshnz 12533
Description: A cyclical shift is the empty set if the number of shifts is not an integer. (Contributed by Alexander van der Vekens, 21-May-2018.) (Revised by AV, 17-Nov-2018.)
Assertion
Ref Expression
cshnz  |-  ( -.  N  e.  ZZ  ->  ( W cyclShift  N )  =  (/) )

Proof of Theorem cshnz
Dummy variables  f 
l  n  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-csh 12530 . . 3  |- cyclShift  =  ( w  e.  { f  |  E. l  e. 
NN0  f  Fn  (
0..^ l ) } ,  n  e.  ZZ  |->  if ( w  =  (/) ,  (/) ,  ( ( w substr  <. ( n  mod  ( # `
 w ) ) ,  ( # `  w
) >. ) concat  ( w substr  <.
0 ,  ( n  mod  ( # `  w
) ) >. )
) ) )
2 0ex 4522 . . . 4  |-  (/)  e.  _V
3 ovex 6217 . . . 4  |-  ( ( w substr  <. ( n  mod  ( # `  w ) ) ,  ( # `  w ) >. ) concat  ( w substr  <. 0 ,  ( n  mod  ( # `  w ) ) >.
) )  e.  _V
42, 3ifex 3958 . . 3  |-  if ( w  =  (/) ,  (/) ,  ( ( w substr  <. (
n  mod  ( # `  w
) ) ,  (
# `  w ) >. ) concat  ( w substr  <. 0 ,  ( n  mod  ( # `  w ) ) >. ) ) )  e.  _V
51, 4dmmpt2 6746 . 2  |-  dom cyclShift  =  ( { f  |  E. l  e.  NN0  f  Fn  ( 0..^ l ) }  X.  ZZ )
6 id 22 . . 3  |-  ( -.  N  e.  ZZ  ->  -.  N  e.  ZZ )
76intnand 907 . 2  |-  ( -.  N  e.  ZZ  ->  -.  ( W  e.  {
f  |  E. l  e.  NN0  f  Fn  (
0..^ l ) }  /\  N  e.  ZZ ) )
8 ndmovg 6348 . 2  |-  ( ( dom cyclShift  =  ( {
f  |  E. l  e.  NN0  f  Fn  (
0..^ l ) }  X.  ZZ )  /\  -.  ( W  e.  {
f  |  E. l  e.  NN0  f  Fn  (
0..^ l ) }  /\  N  e.  ZZ ) )  ->  ( W cyclShift  N )  =  (/) )
95, 7, 8sylancr 663 1  |-  ( -.  N  e.  ZZ  ->  ( W cyclShift  N )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2436   E.wrex 2796   (/)c0 3737   ifcif 3891   <.cop 3983    X. cxp 4938   dom cdm 4940    Fn wfn 5513   ` cfv 5518  (class class class)co 6192   0cc0 9385   NN0cn0 10682   ZZcz 10749  ..^cfzo 11651    mod cmo 11811   #chash 12206   concat cconcat 12327   substr csubstr 12329   cyclShift ccsh 12529
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 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-csh 12530
This theorem is referenced by:  0csh0  12534  cshwcl  12539
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