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Theorem muval2 20388
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 2461 . . 3  |-  ( ( mmu `  A )  =/=  0  <->  -.  (
mmu `  A )  =  0 )
2 ifeqor 3615 . . . . 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 20386 . . . . . . 7  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
43eqeq1d 2304 . . . . . 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 2304 . . . . . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560   ifcif 3578   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754   -ucneg 9054   NNcn 9762   2c2 9811   ^cexp 11120   #chash 11353    || cdivides 12547   Primecprime 12774   mmucmu 20348
This theorem is referenced by:  mumul  20435  musum  20447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-mulcl 8815  ax-i2m1 8821
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-mu 20354
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