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Theorem muval 20915
Description: The value of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
Assertion
Ref Expression
muval  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
Distinct variable group:    A, p

Proof of Theorem muval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 breq2 4216 . . . 4  |-  ( x  =  A  ->  (
( p ^ 2 )  ||  x  <->  ( p ^ 2 )  ||  A ) )
21rexbidv 2726 . . 3  |-  ( x  =  A  ->  ( E. p  e.  Prime  ( p ^ 2 ) 
||  x  <->  E. p  e.  Prime  ( p ^
2 )  ||  A
) )
3 breq2 4216 . . . . . 6  |-  ( x  =  A  ->  (
p  ||  x  <->  p  ||  A
) )
43rabbidv 2948 . . . . 5  |-  ( x  =  A  ->  { p  e.  Prime  |  p  ||  x }  =  {
p  e.  Prime  |  p 
||  A } )
54fveq2d 5732 . . . 4  |-  ( x  =  A  ->  ( # `
 { p  e. 
Prime  |  p  ||  x } )  =  (
# `  { p  e.  Prime  |  p  ||  A } ) )
65oveq2d 6097 . . 3  |-  ( x  =  A  ->  ( -u 1 ^ ( # `  { p  e.  Prime  |  p  ||  x }
) )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) )
72, 6ifbieq2d 3759 . 2  |-  ( x  =  A  ->  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  x ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  x } ) ) )  =  if ( E. p  e. 
Prime  ( p ^ 2 )  ||  A , 
0 ,  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) ) )
8 df-mu 20883 . 2  |-  mmu  =  ( x  e.  NN  |->  if ( E. p  e. 
Prime  ( p ^ 2 )  ||  x ,  0 ,  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  x }
) ) ) )
9 c0ex 9085 . . 3  |-  0  e.  _V
10 ovex 6106 . . 3  |-  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  e.  _V
119, 10ifex 3797 . 2  |-  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  e.  _V
127, 8, 11fvmpt 5806 1  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   E.wrex 2706   {crab 2709   ifcif 3739   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991   -ucneg 9292   NNcn 10000   2c2 10049   ^cexp 11382   #chash 11618    || cdivides 12852   Primecprime 13079   mmucmu 20877
This theorem is referenced by:  muval1  20916  muval2  20917  isnsqf  20918  mule1  20931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-mulcl 9052  ax-i2m1 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-mu 20883
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