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Theorem mumul 20419
Description: The Möbius function is a multiplicative function. This is one of the primary interests of the Möbius function as an arithmetic function. (Contributed by Mario Carneiro, 3-Oct-2014.)
Assertion
Ref Expression
mumul  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  (
mmu `  ( A  x.  B ) )  =  ( ( mmu `  A )  x.  (
mmu `  B )
) )

Proof of Theorem mumul
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 simpl2 959 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  ->  B  e.  NN )
2 mucl 20379 . . . . . 6  |-  ( B  e.  NN  ->  (
mmu `  B )  e.  ZZ )
31, 2syl 15 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  B
)  e.  ZZ )
43zcnd 10118 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  B
)  e.  CC )
54mul02d 9010 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( 0  x.  (
mmu `  B )
)  =  0 )
6 simpr 447 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  A
)  =  0 )
76oveq1d 5873 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( ( mmu `  A )  x.  (
mmu `  B )
)  =  ( 0  x.  ( mmu `  B ) ) )
8 mumullem1 20417 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  A )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  0 )
983adantl3 1113 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  ( A  x.  B )
)  =  0 )
105, 7, 93eqtr4rd 2326 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  ( A  x.  B )
)  =  ( ( mmu `  A )  x.  ( mmu `  B ) ) )
11 simpl1 958 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  ->  A  e.  NN )
12 mucl 20379 . . . . . 6  |-  ( A  e.  NN  ->  (
mmu `  A )  e.  ZZ )
1311, 12syl 15 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  A
)  e.  ZZ )
1413zcnd 10118 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  A
)  e.  CC )
1514mul01d 9011 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( ( mmu `  A )  x.  0 )  =  0 )
16 simpr 447 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  B
)  =  0 )
1716oveq2d 5874 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( ( mmu `  A )  x.  (
mmu `  B )
)  =  ( ( mmu `  A )  x.  0 ) )
18 nncn 9754 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  CC )
19 nncn 9754 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
20 mulcom 8823 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
2118, 19, 20syl2an 463 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
2221fveq2d 5529 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( mmu `  ( A  x.  B )
)  =  ( mmu `  ( B  x.  A
) ) )
2322adantr 451 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  ( mmu `  ( B  x.  A ) ) )
24 mumullem1 20417 . . . . . 6  |-  ( ( ( B  e.  NN  /\  A  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( B  x.  A
) )  =  0 )
2524ancom1s 780 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( B  x.  A
) )  =  0 )
2623, 25eqtrd 2315 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  0 )
27263adantl3 1113 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  ( A  x.  B )
)  =  0 )
2815, 17, 273eqtr4rd 2326 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  ( A  x.  B )
)  =  ( ( mmu `  A )  x.  ( mmu `  B ) ) )
29 simpl1 958 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  A  e.  NN )
30 simpl2 959 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  B  e.  NN )
3129, 30nnmulcld 9793 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( A  x.  B )  e.  NN )
32 mumullem2 20418 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  ( A  x.  B
) )  =/=  0
)
33 muval2 20372 . . . 4  |-  ( ( ( A  x.  B
)  e.  NN  /\  ( mmu `  ( A  x.  B ) )  =/=  0 )  -> 
( mmu `  ( A  x.  B )
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  ( A  x.  B ) } ) ) )
3431, 32, 33syl2anc 642 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  ( A  x.  B
) )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  ( A  x.  B ) } ) ) )
35 neg1cn 9813 . . . . . 6  |-  -u 1  e.  CC
3635a1i 10 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  -u 1  e.  CC )
37 fzfi 11034 . . . . . . 7  |-  ( 1 ... B )  e. 
Fin
38 prmnn 12761 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  NN )
3938ssriv 3184 . . . . . . . . 9  |-  Prime  C_  NN
40 rabss2 3256 . . . . . . . . 9  |-  ( Prime  C_  NN  ->  { p  e.  Prime  |  p  ||  B }  C_  { p  e.  NN  |  p  ||  B } )
4139, 40ax-mp 8 . . . . . . . 8  |-  { p  e.  Prime  |  p  ||  B }  C_  { p  e.  NN  |  p  ||  B }
42 sgmss 20344 . . . . . . . . 9  |-  ( B  e.  NN  ->  { p  e.  NN  |  p  ||  B }  C_  ( 1 ... B ) )
4330, 42syl 15 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  { p  e.  NN  |  p  ||  B }  C_  ( 1 ... B ) )
4441, 43syl5ss 3190 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  { p  e.  Prime  |  p  ||  B }  C_  ( 1 ... B ) )
45 ssfi 7083 . . . . . . 7  |-  ( ( ( 1 ... B
)  e.  Fin  /\  { p  e.  Prime  |  p 
||  B }  C_  ( 1 ... B
) )  ->  { p  e.  Prime  |  p  ||  B }  e.  Fin )
4637, 44, 45sylancr 644 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  { p  e.  Prime  |  p  ||  B }  e.  Fin )
47 hashcl 11350 . . . . . 6  |-  ( { p  e.  Prime  |  p 
||  B }  e.  Fin  ->  ( # `  {
p  e.  Prime  |  p 
||  B } )  e.  NN0 )
4846, 47syl 15 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( # `  {
p  e.  Prime  |  p 
||  B } )  e.  NN0 )
49 fzfi 11034 . . . . . . 7  |-  ( 1 ... A )  e. 
Fin
50 rabss2 3256 . . . . . . . . 9  |-  ( Prime  C_  NN  ->  { p  e.  Prime  |  p  ||  A }  C_  { p  e.  NN  |  p  ||  A } )
5139, 50ax-mp 8 . . . . . . . 8  |-  { p  e.  Prime  |  p  ||  A }  C_  { p  e.  NN  |  p  ||  A }
52 sgmss 20344 . . . . . . . . 9  |-  ( A  e.  NN  ->  { p  e.  NN  |  p  ||  A }  C_  ( 1 ... A ) )
5329, 52syl 15 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  { p  e.  NN  |  p  ||  A }  C_  ( 1 ... A ) )
5451, 53syl5ss 3190 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  { p  e.  Prime  |  p  ||  A }  C_  ( 1 ... A ) )
55 ssfi 7083 . . . . . . 7  |-  ( ( ( 1 ... A
)  e.  Fin  /\  { p  e.  Prime  |  p 
||  A }  C_  ( 1 ... A
) )  ->  { p  e.  Prime  |  p  ||  A }  e.  Fin )
5649, 54, 55sylancr 644 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  { p  e.  Prime  |  p  ||  A }  e.  Fin )
57 hashcl 11350 . . . . . 6  |-  ( { p  e.  Prime  |  p 
||  A }  e.  Fin  ->  ( # `  {
p  e.  Prime  |  p 
||  A } )  e.  NN0 )
5856, 57syl 15 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( # `  {
p  e.  Prime  |  p 
||  A } )  e.  NN0 )
5936, 48, 58expaddd 11247 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( -u 1 ^ ( ( # `  { p  e.  Prime  |  p  ||  A }
)  +  ( # `  { p  e.  Prime  |  p  ||  B }
) ) )  =  ( ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) )  x.  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  B }
) ) ) )
60 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  p  e. 
Prime )
61 simpl1 958 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  NN )
6261nnzd 10116 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  ZZ )
6362adantlr 695 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  A  e.  ZZ )
64 simpl2 959 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  NN )
6564nnzd 10116 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  ZZ )
6665adantlr 695 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  B  e.  ZZ )
67 euclemma 12787 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
p  ||  ( A  x.  B )  <->  ( p  ||  A  \/  p  ||  B ) ) )
6860, 63, 66, 67syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  ( p 
||  ( A  x.  B )  <->  ( p  ||  A  \/  p  ||  B ) ) )
6968rabbidva 2779 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  { p  e.  Prime  |  p  ||  ( A  x.  B
) }  =  {
p  e.  Prime  |  ( p  ||  A  \/  p  ||  B ) } )
70 unrab 3439 . . . . . . . 8  |-  ( { p  e.  Prime  |  p 
||  A }  u.  { p  e.  Prime  |  p 
||  B } )  =  { p  e. 
Prime  |  ( p  ||  A  \/  p  ||  B ) }
7169, 70syl6eqr 2333 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  { p  e.  Prime  |  p  ||  ( A  x.  B
) }  =  ( { p  e.  Prime  |  p  ||  A }  u.  { p  e.  Prime  |  p  ||  B }
) )
7271fveq2d 5529 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( # `  {
p  e.  Prime  |  p 
||  ( A  x.  B ) } )  =  ( # `  ( { p  e.  Prime  |  p  ||  A }  u.  { p  e.  Prime  |  p  ||  B }
) ) )
73 inrab 3440 . . . . . . . 8  |-  ( { p  e.  Prime  |  p 
||  A }  i^i  { p  e.  Prime  |  p 
||  B } )  =  { p  e. 
Prime  |  ( p  ||  A  /\  p  ||  B ) }
74 nprmdvds1 12790 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
7574adantl 452 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  -.  p  ||  1 )
76 prmz 12762 . . . . . . . . . . . . . 14  |-  ( p  e.  Prime  ->  p  e.  ZZ )
7776adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  p  e.  ZZ )
78 dvdsgcd 12722 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( p  ||  A  /\  p  ||  B )  ->  p  ||  ( A  gcd  B ) ) )
7977, 63, 66, 78syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  ( ( p  ||  A  /\  p  ||  B )  ->  p  ||  ( A  gcd  B ) ) )
80 simpll3 996 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  ( A  gcd  B )  =  1 )
8180breq2d 4035 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  ( p 
||  ( A  gcd  B )  <->  p  ||  1 ) )
8279, 81sylibd 205 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  ( ( p  ||  A  /\  p  ||  B )  ->  p  ||  1 ) )
8375, 82mtod 168 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  -.  (
p  ||  A  /\  p  ||  B ) )
8483ralrimiva 2626 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  A. p  e.  Prime  -.  ( p  ||  A  /\  p  ||  B ) )
85 rabeq0 3476 . . . . . . . . 9  |-  ( { p  e.  Prime  |  ( p  ||  A  /\  p  ||  B ) }  =  (/)  <->  A. p  e.  Prime  -.  ( p  ||  A  /\  p  ||  B ) )
8684, 85sylibr 203 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  { p  e.  Prime  |  ( p 
||  A  /\  p  ||  B ) }  =  (/) )
8773, 86syl5eq 2327 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( {
p  e.  Prime  |  p 
||  A }  i^i  { p  e.  Prime  |  p 
||  B } )  =  (/) )
88 hashun 11364 . . . . . . 7  |-  ( ( { p  e.  Prime  |  p  ||  A }  e.  Fin  /\  { p  e.  Prime  |  p  ||  B }  e.  Fin  /\  ( { p  e. 
Prime  |  p  ||  A }  i^i  { p  e. 
Prime  |  p  ||  B } )  =  (/) )  ->  ( # `  ( { p  e.  Prime  |  p  ||  A }  u.  { p  e.  Prime  |  p  ||  B }
) )  =  ( ( # `  {
p  e.  Prime  |  p 
||  A } )  +  ( # `  {
p  e.  Prime  |  p 
||  B } ) ) )
8956, 46, 87, 88syl3anc 1182 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( # `  ( { p  e.  Prime  |  p  ||  A }  u.  { p  e.  Prime  |  p  ||  B }
) )  =  ( ( # `  {
p  e.  Prime  |  p 
||  A } )  +  ( # `  {
p  e.  Prime  |  p 
||  B } ) ) )
9072, 89eqtrd 2315 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( # `  {
p  e.  Prime  |  p 
||  ( A  x.  B ) } )  =  ( ( # `  { p  e.  Prime  |  p  ||  A }
)  +  ( # `  { p  e.  Prime  |  p  ||  B }
) ) )
9190oveq2d 5874 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  ( A  x.  B ) } ) )  =  ( -u
1 ^ ( (
# `  { p  e.  Prime  |  p  ||  A } )  +  (
# `  { p  e.  Prime  |  p  ||  B } ) ) ) )
92 simprl 732 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  A )  =/=  0
)
93 muval2 20372 . . . . . 6  |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0 )  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
9429, 92, 93syl2anc 642 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  A )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) )
95 simprr 733 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  B )  =/=  0
)
96 muval2 20372 . . . . . 6  |-  ( ( B  e.  NN  /\  ( mmu `  B )  =/=  0 )  -> 
( mmu `  B
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  B }
) ) )
9730, 95, 96syl2anc 642 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  B )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  B } ) ) )
9894, 97oveq12d 5876 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( (
mmu `  A )  x.  ( mmu `  B
) )  =  ( ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) )  x.  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  B }
) ) ) )
9959, 91, 983eqtr4rd 2326 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( (
mmu `  A )  x.  ( mmu `  B
) )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  ( A  x.  B ) } ) ) )
10034, 99eqtr4d 2318 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  ( A  x.  B
) )  =  ( ( mmu `  A
)  x.  ( mmu `  B ) ) )
10110, 28, 100pm2.61da2ne 2525 1  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  (
mmu `  ( A  x.  B ) )  =  ( ( mmu `  A )  x.  (
mmu `  B )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   -ucneg 9038   NNcn 9746   NN0cn0 9965   ZZcz 10024   ...cfz 10782   ^cexp 11104   #chash 11337    || cdivides 12531    gcd cgcd 12685   Primecprime 12758   mmucmu 20332
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-mu 20338
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