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Theorem mumul 23834
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 1001 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  ->  B  e.  NN )
2 mucl 23794 . . . . . 6  |-  ( B  e.  NN  ->  (
mmu `  B )  e.  ZZ )
31, 2syl 17 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  B
)  e.  ZZ )
43zcnd 11008 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  B
)  e.  CC )
54mul02d 9811 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( 0  x.  (
mmu `  B )
)  =  0 )
6 simpr 459 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  A
)  =  0 )
76oveq1d 6292 . . 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 23832 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  A )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  0 )
983adantl3 1155 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  ( A  x.  B )
)  =  0 )
105, 7, 93eqtr4rd 2454 . 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 1000 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  ->  A  e.  NN )
12 mucl 23794 . . . . . 6  |-  ( A  e.  NN  ->  (
mmu `  A )  e.  ZZ )
1311, 12syl 17 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  A
)  e.  ZZ )
1413zcnd 11008 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  A
)  e.  CC )
1514mul01d 9812 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( ( mmu `  A )  x.  0 )  =  0 )
16 simpr 459 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  B
)  =  0 )
1716oveq2d 6293 . . 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 10583 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  CC )
19 nncn 10583 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
20 mulcom 9607 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
2118, 19, 20syl2an 475 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
2221fveq2d 5852 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( mmu `  ( A  x.  B )
)  =  ( mmu `  ( B  x.  A
) ) )
2322adantr 463 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  ( mmu `  ( B  x.  A ) ) )
24 mumullem1 23832 . . . . . 6  |-  ( ( ( B  e.  NN  /\  A  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( B  x.  A
) )  =  0 )
2524ancom1s 806 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( B  x.  A
) )  =  0 )
2623, 25eqtrd 2443 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  0 )
27263adantl3 1155 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  ( A  x.  B )
)  =  0 )
2815, 17, 273eqtr4rd 2454 . 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 1000 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  A  e.  NN )
30 simpl2 1001 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  B  e.  NN )
3129, 30nnmulcld 10623 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( A  x.  B )  e.  NN )
32 mumullem2 23833 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  ( A  x.  B
) )  =/=  0
)
33 muval2 23787 . . . 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 659 . . 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 10679 . . . . . 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 12121 . . . . . . 7  |-  ( 1 ... B )  e. 
Fin
38 prmnn 14427 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  NN )
3938ssriv 3445 . . . . . . . . 9  |-  Prime  C_  NN
40 rabss2 3521 . . . . . . . . 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 23759 . . . . . . . . 9  |-  ( B  e.  NN  ->  { p  e.  NN  |  p  ||  B }  C_  ( 1 ... B ) )
4330, 42syl 17 . . . . . . . 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 3452 . . . . . . 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 7774 . . . . . . 7  |-  ( ( ( 1 ... B
)  e.  Fin  /\  { p  e.  Prime  |  p 
||  B }  C_  ( 1 ... B
) )  ->  { p  e.  Prime  |  p  ||  B }  e.  Fin )
4637, 44, 45sylancr 661 . . . . . 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 12473 . . . . . 6  |-  ( { p  e.  Prime  |  p 
||  B }  e.  Fin  ->  ( # `  {
p  e.  Prime  |  p 
||  B } )  e.  NN0 )
4846, 47syl 17 . . . . 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 12121 . . . . . . 7  |-  ( 1 ... A )  e. 
Fin
50 rabss2 3521 . . . . . . . . 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 23759 . . . . . . . . 9  |-  ( A  e.  NN  ->  { p  e.  NN  |  p  ||  A }  C_  ( 1 ... A ) )
5329, 52syl 17 . . . . . . . 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 3452 . . . . . . 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 7774 . . . . . . 7  |-  ( ( ( 1 ... A
)  e.  Fin  /\  { p  e.  Prime  |  p 
||  A }  C_  ( 1 ... A
) )  ->  { p  e.  Prime  |  p  ||  A }  e.  Fin )
5649, 54, 55sylancr 661 . . . . . 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 12473 . . . . . 6  |-  ( { p  e.  Prime  |  p 
||  A }  e.  Fin  ->  ( # `  {
p  e.  Prime  |  p 
||  A } )  e.  NN0 )
5856, 57syl 17 . . . . 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 12354 . . . 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 459 . . . . . . . . . 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 1000 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  NN )
6261nnzd 11006 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  ZZ )
6362adantlr 713 . . . . . . . . . 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 1001 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  NN )
6564nnzd 11006 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  ZZ )
6665adantlr 713 . . . . . . . . . 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 14456 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
p  ||  ( A  x.  B )  <->  ( p  ||  A  \/  p  ||  B ) ) )
6860, 63, 66, 67syl3anc 1230 . . . . . . . . 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 3049 . . . . . . . 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 3720 . . . . . . . 8  |-  ( { p  e.  Prime  |  p 
||  A }  u.  { p  e.  Prime  |  p 
||  B } )  =  { p  e. 
Prime  |  ( p  ||  A  \/  p  ||  B ) }
7169, 70syl6eqr 2461 . . . . . . 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 5852 . . . . . 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 3721 . . . . . . . 8  |-  ( { p  e.  Prime  |  p 
||  A }  i^i  { p  e.  Prime  |  p 
||  B } )  =  { p  e. 
Prime  |  ( p  ||  A  /\  p  ||  B ) }
74 nprmdvds1 14459 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
7574adantl 464 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  (
( mmu `  A
)  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  /\  p  e. 
Prime )  ->  -.  p  ||  1 )
76 prmz 14428 . . . . . . . . . . . . . 14  |-  ( p  e.  Prime  ->  p  e.  ZZ )
7776adantl 464 . . . . . . . . . . . . 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 14388 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( p  ||  A  /\  p  ||  B )  ->  p  ||  ( A  gcd  B ) ) )
7977, 63, 66, 78syl3anc 1230 . . . . . . . . . . . 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 1038 . . . . . . . . . . . . 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 4406 . . . . . . . . . . . 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 2817 . . . . . . . . 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 3760 . . . . . . . . 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 2455 . . . . . . 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 12496 . . . . . . 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 1230 . . . . . 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 2443 . . . . 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 6293 . . . 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 756 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  A )  =/=  0
)
93 muval2 23787 . . . . . 6  |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0 )  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
9429, 92, 93syl2anc 659 . . . . 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 758 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  B )  =/=  0
)
96 muval2 23787 . . . . . 6  |-  ( ( B  e.  NN  /\  ( mmu `  B )  =/=  0 )  -> 
( mmu `  B
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  B }
) ) )
9730, 95, 96syl2anc 659 . . . . 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 6295 . . . 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 2454 . . 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 2446 . 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 2722 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 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   {crab 2757    u. cun 3411    i^i cin 3412    C_ wss 3413   (/)c0 3737   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Fincfn 7553   CCcc 9519   0cc0 9521   1c1 9522    + caddc 9524    x. cmul 9526   -ucneg 9841   NNcn 10575   NN0cn0 10835   ZZcz 10904   ...cfz 11724   ^cexp 12208   #chash 12450    || cdvds 14193    gcd cgcd 14351   Primecprime 14424   mmucmu 23747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-q 11227  df-rp 11265  df-fz 11725  df-fl 11964  df-mod 12033  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-dvds 14194  df-gcd 14352  df-prm 14425  df-pc 14568  df-mu 23753
This theorem is referenced by: (None)
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