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Theorem mumul 22524
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 992 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  ->  B  e.  NN )
2 mucl 22484 . . . . . 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 10753 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  B
)  e.  CC )
54mul02d 9572 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( 0  x.  (
mmu `  B )
)  =  0 )
6 simpr 461 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  A
)  =  0 )
76oveq1d 6111 . . 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 22522 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  A )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  0 )
983adantl3 1146 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  ( A  x.  B )
)  =  0 )
105, 7, 93eqtr4rd 2486 . 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 991 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  ->  A  e.  NN )
12 mucl 22484 . . . . . 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 10753 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  A
)  e.  CC )
1514mul01d 9573 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( ( mmu `  A )  x.  0 )  =  0 )
16 simpr 461 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  B
)  =  0 )
1716oveq2d 6112 . . 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 10335 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  CC )
19 nncn 10335 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
20 mulcom 9373 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
2118, 19, 20syl2an 477 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
2221fveq2d 5700 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( mmu `  ( A  x.  B )
)  =  ( mmu `  ( B  x.  A
) ) )
2322adantr 465 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  ( mmu `  ( B  x.  A ) ) )
24 mumullem1 22522 . . . . . 6  |-  ( ( ( B  e.  NN  /\  A  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( B  x.  A
) )  =  0 )
2524ancom1s 803 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( B  x.  A
) )  =  0 )
2623, 25eqtrd 2475 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  0 )
27263adantl3 1146 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  ( A  x.  B )
)  =  0 )
2815, 17, 273eqtr4rd 2486 . 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 991 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  A  e.  NN )
30 simpl2 992 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  B  e.  NN )
3129, 30nnmulcld 10374 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( A  x.  B )  e.  NN )
32 mumullem2 22523 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  ( A  x.  B
) )  =/=  0
)
33 muval2 22477 . . . 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 661 . . 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 10430 . . . . . 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 11799 . . . . . . 7  |-  ( 1 ... B )  e. 
Fin
38 prmnn 13771 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  NN )
3938ssriv 3365 . . . . . . . . 9  |-  Prime  C_  NN
40 rabss2 3440 . . . . . . . . 9  |-  ( Prime  C_  NN  ->  { p  e.  Prime  |  p  ||  B }  C_  { p  e.  NN  |  p  ||  B } )
4139, 40ax-mp 5 . . . . . . . 8  |-  { p  e.  Prime  |  p  ||  B }  C_  { p  e.  NN  |  p  ||  B }
42 sgmss 22449 . . . . . . . . 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 3372 . . . . . . 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 7538 . . . . . . 7  |-  ( ( ( 1 ... B
)  e.  Fin  /\  { p  e.  Prime  |  p 
||  B }  C_  ( 1 ... B
) )  ->  { p  e.  Prime  |  p  ||  B }  e.  Fin )
4637, 44, 45sylancr 663 . . . . . 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 12131 . . . . . 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 11799 . . . . . . 7  |-  ( 1 ... A )  e. 
Fin
50 rabss2 3440 . . . . . . . . 9  |-  ( Prime  C_  NN  ->  { p  e.  Prime  |  p  ||  A }  C_  { p  e.  NN  |  p  ||  A } )
5139, 50ax-mp 5 . . . . . . . 8  |-  { p  e.  Prime  |  p  ||  A }  C_  { p  e.  NN  |  p  ||  A }
52 sgmss 22449 . . . . . . . . 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 3372 . . . . . . 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 7538 . . . . . . 7  |-  ( ( ( 1 ... A
)  e.  Fin  /\  { p  e.  Prime  |  p 
||  A }  C_  ( 1 ... A
) )  ->  { p  e.  Prime  |  p  ||  A }  e.  Fin )
5649, 54, 55sylancr 663 . . . . . 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 12131 . . . . . 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 12015 . . . 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 461 . . . . . . . . . 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 991 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  NN )
6261nnzd 10751 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  ZZ )
6362adantlr 714 . . . . . . . . . 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 992 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  NN )
6564nnzd 10751 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  ZZ )
6665adantlr 714 . . . . . . . . . 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 13799 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
p  ||  ( A  x.  B )  <->  ( p  ||  A  \/  p  ||  B ) ) )
6860, 63, 66, 67syl3anc 1218 . . . . . . . . 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 2968 . . . . . . . 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 3626 . . . . . . . 8  |-  ( { p  e.  Prime  |  p 
||  A }  u.  { p  e.  Prime  |  p 
||  B } )  =  { p  e. 
Prime  |  ( p  ||  A  \/  p  ||  B ) }
7169, 70syl6eqr 2493 . . . . . . 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 5700 . . . . . 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 3627 . . . . . . . 8  |-  ( { p  e.  Prime  |  p 
||  A }  i^i  { p  e.  Prime  |  p 
||  B } )  =  { p  e. 
Prime  |  ( p  ||  A  /\  p  ||  B ) }
74 nprmdvds1 13802 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
7574adantl 466 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  -.  p  ||  1 )
76 prmz 13772 . . . . . . . . . . . . . 14  |-  ( p  e.  Prime  ->  p  e.  ZZ )
7776adantl 466 . . . . . . . . . . . . 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 13732 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( p  ||  A  /\  p  ||  B )  ->  p  ||  ( A  gcd  B ) ) )
7977, 63, 66, 78syl3anc 1218 . . . . . . . . . . . 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 1029 . . . . . . . . . . . . 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 4309 . . . . . . . . . . . 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 214 . . . . . . . . . . 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 177 . . . . . . . . . 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 2804 . . . . . . . . 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 3664 . . . . . . . . 9  |-  ( { p  e.  Prime  |  ( p  ||  A  /\  p  ||  B ) }  =  (/)  <->  A. p  e.  Prime  -.  ( p  ||  A  /\  p  ||  B ) )
8684, 85sylibr 212 . . . . . . . 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 2487 . . . . . . 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 12150 . . . . . . 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 1218 . . . . . 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 2475 . . . . 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 6112 . . . 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 755 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  A )  =/=  0
)
93 muval2 22477 . . . . . 6  |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0 )  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
9429, 92, 93syl2anc 661 . . . . 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 756 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  B )  =/=  0
)
96 muval2 22477 . . . . . 6  |-  ( ( B  e.  NN  /\  ( mmu `  B )  =/=  0 )  -> 
( mmu `  B
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  B }
) ) )
9730, 95, 96syl2anc 661 . . . . 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 6114 . . . 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 2486 . . 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 2478 . 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 2695 1  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  ->  (
mmu `  ( A  x.  B ) )  =  ( ( mmu `  A )  x.  (
mmu `  B )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   {crab 2724    u. cun 3331    i^i cin 3332    C_ wss 3333   (/)c0 3642   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Fincfn 7315   CCcc 9285   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292   -ucneg 9601   NNcn 10327   NN0cn0 10584   ZZcz 10651   ...cfz 11442   ^cexp 11870   #chash 12108    || cdivides 13540    gcd cgcd 13695   Primecprime 13768   mmucmu 22437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-fz 11443  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-dvds 13541  df-gcd 13696  df-prm 13769  df-pc 13909  df-mu 22443
This theorem is referenced by: (None)
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