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Theorem mumul 20956
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 961 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  ->  B  e.  NN )
2 mucl 20916 . . . . . 6  |-  ( B  e.  NN  ->  (
mmu `  B )  e.  ZZ )
31, 2syl 16 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  B
)  e.  ZZ )
43zcnd 10368 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  B
)  e.  CC )
54mul02d 9256 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( 0  x.  (
mmu `  B )
)  =  0 )
6 simpr 448 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  A
)  =  0 )
76oveq1d 6088 . . 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 20954 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  A )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  0 )
983adantl3 1115 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  ( A  x.  B )
)  =  0 )
105, 7, 93eqtr4rd 2478 . 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 960 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  ->  A  e.  NN )
12 mucl 20916 . . . . . 6  |-  ( A  e.  NN  ->  (
mmu `  A )  e.  ZZ )
1311, 12syl 16 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  A
)  e.  ZZ )
1413zcnd 10368 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  A
)  e.  CC )
1514mul01d 9257 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( ( mmu `  A )  x.  0 )  =  0 )
16 simpr 448 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  B
)  =  0 )
1716oveq2d 6089 . . 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 10000 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  CC )
19 nncn 10000 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
20 mulcom 9068 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
2118, 19, 20syl2an 464 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
2221fveq2d 5724 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( mmu `  ( A  x.  B )
)  =  ( mmu `  ( B  x.  A
) ) )
2322adantr 452 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  ( mmu `  ( B  x.  A ) ) )
24 mumullem1 20954 . . . . . 6  |-  ( ( ( B  e.  NN  /\  A  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( B  x.  A
) )  =  0 )
2524ancom1s 781 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( B  x.  A
) )  =  0 )
2623, 25eqtrd 2467 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  0 )
27263adantl3 1115 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  ( A  x.  B )
)  =  0 )
2815, 17, 273eqtr4rd 2478 . 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 960 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  A  e.  NN )
30 simpl2 961 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  B  e.  NN )
3129, 30nnmulcld 10039 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( A  x.  B )  e.  NN )
32 mumullem2 20955 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  ( A  x.  B
) )  =/=  0
)
33 muval2 20909 . . . 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 643 . . 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 10059 . . . . . 6  |-  -u 1  e.  CC
3635a1i 11 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  -u 1  e.  CC )
37 fzfi 11303 . . . . . . 7  |-  ( 1 ... B )  e. 
Fin
38 prmnn 13074 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  NN )
3938ssriv 3344 . . . . . . . . 9  |-  Prime  C_  NN
40 rabss2 3418 . . . . . . . . 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 20881 . . . . . . . . 9  |-  ( B  e.  NN  ->  { p  e.  NN  |  p  ||  B }  C_  ( 1 ... B ) )
4330, 42syl 16 . . . . . . . 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 3351 . . . . . . 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 7321 . . . . . . 7  |-  ( ( ( 1 ... B
)  e.  Fin  /\  { p  e.  Prime  |  p 
||  B }  C_  ( 1 ... B
) )  ->  { p  e.  Prime  |  p  ||  B }  e.  Fin )
4637, 44, 45sylancr 645 . . . . . 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 11631 . . . . . 6  |-  ( { p  e.  Prime  |  p 
||  B }  e.  Fin  ->  ( # `  {
p  e.  Prime  |  p 
||  B } )  e.  NN0 )
4846, 47syl 16 . . . . 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 11303 . . . . . . 7  |-  ( 1 ... A )  e. 
Fin
50 rabss2 3418 . . . . . . . . 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 20881 . . . . . . . . 9  |-  ( A  e.  NN  ->  { p  e.  NN  |  p  ||  A }  C_  ( 1 ... A ) )
5329, 52syl 16 . . . . . . . 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 3351 . . . . . . 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 7321 . . . . . . 7  |-  ( ( ( 1 ... A
)  e.  Fin  /\  { p  e.  Prime  |  p 
||  A }  C_  ( 1 ... A
) )  ->  { p  e.  Prime  |  p  ||  A }  e.  Fin )
5649, 54, 55sylancr 645 . . . . . 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 11631 . . . . . 6  |-  ( { p  e.  Prime  |  p 
||  A }  e.  Fin  ->  ( # `  {
p  e.  Prime  |  p 
||  A } )  e.  NN0 )
5856, 57syl 16 . . . . 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 11517 . . . 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 448 . . . . . . . . . 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 960 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  NN )
6261nnzd 10366 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  ZZ )
6362adantlr 696 . . . . . . . . . 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 961 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  NN )
6564nnzd 10366 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  ZZ )
6665adantlr 696 . . . . . . . . . 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 13100 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
p  ||  ( A  x.  B )  <->  ( p  ||  A  \/  p  ||  B ) ) )
6860, 63, 66, 67syl3anc 1184 . . . . . . . . 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 2939 . . . . . . . 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 3604 . . . . . . . 8  |-  ( { p  e.  Prime  |  p 
||  A }  u.  { p  e.  Prime  |  p 
||  B } )  =  { p  e. 
Prime  |  ( p  ||  A  \/  p  ||  B ) }
7169, 70syl6eqr 2485 . . . . . . 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 5724 . . . . . 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 3605 . . . . . . . 8  |-  ( { p  e.  Prime  |  p 
||  A }  i^i  { p  e.  Prime  |  p 
||  B } )  =  { p  e. 
Prime  |  ( p  ||  A  /\  p  ||  B ) }
74 nprmdvds1 13103 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
7574adantl 453 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  -.  p  ||  1 )
76 prmz 13075 . . . . . . . . . . . . . 14  |-  ( p  e.  Prime  ->  p  e.  ZZ )
7776adantl 453 . . . . . . . . . . . . 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 13035 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( p  ||  A  /\  p  ||  B )  ->  p  ||  ( A  gcd  B ) ) )
7977, 63, 66, 78syl3anc 1184 . . . . . . . . . . . 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 998 . . . . . . . . . . . . 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 4216 . . . . . . . . . . . 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 206 . . . . . . . . . . 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 170 . . . . . . . . . 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 2781 . . . . . . . . 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 3641 . . . . . . . . 9  |-  ( { p  e.  Prime  |  ( p  ||  A  /\  p  ||  B ) }  =  (/)  <->  A. p  e.  Prime  -.  ( p  ||  A  /\  p  ||  B ) )
8684, 85sylibr 204 . . . . . . . 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 2479 . . . . . . 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 11648 . . . . . . 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 1184 . . . . . 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 2467 . . . . 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 6089 . . . 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 733 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  A )  =/=  0
)
93 muval2 20909 . . . . . 6  |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0 )  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
9429, 92, 93syl2anc 643 . . . . 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 734 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  B )  =/=  0
)
96 muval2 20909 . . . . . 6  |-  ( ( B  e.  NN  /\  ( mmu `  B )  =/=  0 )  -> 
( mmu `  B
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  B }
) ) )
9730, 95, 96syl2anc 643 . . . . 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 6091 . . . 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 2478 . . 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 2470 . 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 2677 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   {crab 2701    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Fincfn 7101   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987   -ucneg 9284   NNcn 9992   NN0cn0 10213   ZZcz 10274   ...cfz 11035   ^cexp 11374   #chash 11610    || cdivides 12844    gcd cgcd 12998   Primecprime 13071   mmucmu 20869
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203  df-mu 20875
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