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Theorem cshnz 12754
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 12751 . . 3  |- cyclShift  =  ( w  e.  { f  |  E. l  e. 
NN0  f  Fn  (
0..^ l ) } ,  n  e.  ZZ  |->  if ( w  =  (/) ,  (/) ,  ( ( w substr  <. ( n  mod  ( # `
 w ) ) ,  ( # `  w
) >. ) ++  ( w substr  <. 0 ,  ( n  mod  ( # `  w
) ) >. )
) ) )
2 0ex 4569 . . . 4  |-  (/)  e.  _V
3 ovex 6298 . . . 4  |-  ( ( w substr  <. ( n  mod  ( # `  w ) ) ,  ( # `  w ) >. ) ++  ( w substr  <. 0 ,  ( n  mod  ( # `
 w ) )
>. ) )  e.  _V
42, 3ifex 3997 . . 3  |-  if ( w  =  (/) ,  (/) ,  ( ( w substr  <. (
n  mod  ( # `  w
) ) ,  (
# `  w ) >. ) ++  ( w substr  <. 0 ,  ( n  mod  ( # `  w ) ) >. ) ) )  e.  _V
51, 4dmmpt2 6843 . 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 914 . 2  |-  ( -.  N  e.  ZZ  ->  -.  ( W  e.  {
f  |  E. l  e.  NN0  f  Fn  (
0..^ l ) }  /\  N  e.  ZZ ) )
8 ndmovg 6431 . 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 661 1  |-  ( -.  N  e.  ZZ  ->  ( W cyclShift  N )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439   E.wrex 2805   (/)c0 3783   ifcif 3929   <.cop 4022    X. cxp 4986   dom cdm 4988    Fn wfn 5565   ` cfv 5570  (class class class)co 6270   0cc0 9481   NN0cn0 10791   ZZcz 10860  ..^cfzo 11799    mod cmo 11978   #chash 12387   ++ cconcat 12520   substr csubstr 12522   cyclShift ccsh 12750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-csh 12751
This theorem is referenced by:  0csh0  12755  cshwcl  12760
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