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Theorem muval2 20372
Description: The value of the Möbius function at a squarefree number. (Contributed by Mario Carneiro, 3-Oct-2014.)
Assertion
Ref Expression
muval2  |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0 )  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
Distinct variable group:    A, p

Proof of Theorem muval2
StepHypRef Expression
1 df-ne 2448 . . 3  |-  ( ( mmu `  A )  =/=  0  <->  -.  (
mmu `  A )  =  0 )
2 ifeqor 3602 . . . . 5  |-  ( if ( E. p  e. 
Prime  ( p ^ 2 )  ||  A , 
0 ,  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )  =  0  \/  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) )
3 muval 20370 . . . . . . 7  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
43eqeq1d 2291 . . . . . 6  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =  0  <->  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  0 ) )
53eqeq1d 2291 . . . . . 6  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  <->  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
64, 5orbi12d 690 . . . . 5  |-  ( A  e.  NN  ->  (
( ( mmu `  A )  =  0  \/  ( mmu `  A )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) )  <-> 
( if ( E. p  e.  Prime  (
p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  0  \/  if ( E. p  e.  Prime  (
p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) ) )
72, 6mpbiri 224 . . . 4  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =  0  \/  ( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) ) )
87ord 366 . . 3  |-  ( A  e.  NN  ->  ( -.  ( mmu `  A
)  =  0  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) ) )
91, 8syl5bi 208 . 2  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =/=  0  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) ) )
109imp 418 1  |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0 )  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   ifcif 3565   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738   -ucneg 9038   NNcn 9746   2c2 9795   ^cexp 11104   #chash 11337    || cdivides 12531   Primecprime 12758   mmucmu 20332
This theorem is referenced by:  mumul  20419  musum  20431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-i2m1 8805
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-mu 20338
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