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Theorem muval2 20919
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 2603 . . 3  |-  ( ( mmu `  A )  =/=  0  <->  -.  (
mmu `  A )  =  0 )
2 ifeqor 3778 . . . . 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 20917 . . . . . . 7  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
43eqeq1d 2446 . . . . . 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 2446 . . . . . 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 692 . . . . 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 226 . . . 4  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =  0  \/  ( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) ) )
87ord 368 . . 3  |-  ( A  e.  NN  ->  ( -.  ( mmu `  A
)  =  0  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) ) )
91, 8syl5bi 210 . 2  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =/=  0  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) ) )
109imp 420 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 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   {crab 2711   ifcif 3741   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   0cc0 8992   1c1 8993   -ucneg 9294   NNcn 10002   2c2 10051   ^cexp 11384   #chash 11620    || cdivides 12854   Primecprime 13081   mmucmu 20879
This theorem is referenced by:  mumul  20966  musum  20978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-mulcl 9054  ax-i2m1 9060
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-mu 20885
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