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Theorem mumul 23319
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 1000 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  ->  B  e.  NN )
2 mucl 23279 . . . . . 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 10979 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  B
)  e.  CC )
54mul02d 9789 . . 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 6310 . . 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 23317 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  A )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  0 )
983adantl3 1154 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  A )  =  0 )  -> 
( mmu `  ( A  x.  B )
)  =  0 )
105, 7, 93eqtr4rd 2519 . 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 999 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  ->  A  e.  NN )
12 mucl 23279 . . . . . 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 10979 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  A
)  e.  CC )
1514mul01d 9790 . . 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 6311 . . 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 10556 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  CC )
19 nncn 10556 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  CC )
20 mulcom 9590 . . . . . . . 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 5876 . . . . . 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 23317 . . . . . 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 2508 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  ( mmu `  B )  =  0 )  ->  ( mmu `  ( A  x.  B
) )  =  0 )
27263adantl3 1154 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( mmu `  B )  =  0 )  -> 
( mmu `  ( A  x.  B )
)  =  0 )
2815, 17, 273eqtr4rd 2519 . 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 999 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  A  e.  NN )
30 simpl2 1000 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  B  e.  NN )
3129, 30nnmulcld 10595 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( A  x.  B )  e.  NN )
32 mumullem2 23318 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  ( ( mmu `  A )  =/=  0  /\  ( mmu `  B
)  =/=  0 ) )  ->  ( mmu `  ( A  x.  B
) )  =/=  0
)
33 muval2 23272 . . . 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 10651 . . . . . 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 12062 . . . . . . 7  |-  ( 1 ... B )  e. 
Fin
38 prmnn 14095 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  NN )
3938ssriv 3513 . . . . . . . . 9  |-  Prime  C_  NN
40 rabss2 3588 . . . . . . . . 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 23244 . . . . . . . . 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 3520 . . . . . . 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 7752 . . . . . . 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 12408 . . . . . 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 12062 . . . . . . 7  |-  ( 1 ... A )  e. 
Fin
50 rabss2 3588 . . . . . . . . 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 23244 . . . . . . . . 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 3520 . . . . . . 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 7752 . . . . . . 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 12408 . . . . . 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 12292 . . . 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 999 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  A  e.  NN )
6261nnzd 10977 . . . . . . . . . . 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 1000 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  ( A  gcd  B )  =  1 )  /\  p  e.  Prime )  ->  B  e.  NN )
6564nnzd 10977 . . . . . . . . . . 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 14124 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
p  ||  ( A  x.  B )  <->  ( p  ||  A  \/  p  ||  B ) ) )
6860, 63, 66, 67syl3anc 1228 . . . . . . . . 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 3109 . . . . . . . 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 3774 . . . . . . . 8  |-  ( { p  e.  Prime  |  p 
||  A }  u.  { p  e.  Prime  |  p 
||  B } )  =  { p  e. 
Prime  |  ( p  ||  A  \/  p  ||  B ) }
7169, 70syl6eqr 2526 . . . . . . 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 5876 . . . . . 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 3775 . . . . . . . 8  |-  ( { p  e.  Prime  |  p 
||  A }  i^i  { p  e.  Prime  |  p 
||  B } )  =  { p  e. 
Prime  |  ( p  ||  A  /\  p  ||  B ) }
74 nprmdvds1 14127 . . . . . . . . . . . 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 14096 . . . . . . . . . . . . . 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 14056 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( p  ||  A  /\  p  ||  B )  ->  p  ||  ( A  gcd  B ) ) )
7977, 63, 66, 78syl3anc 1228 . . . . . . . . . . . 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 1037 . . . . . . . . . . . . 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 4465 . . . . . . . . . . . 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 2881 . . . . . . . . 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 3812 . . . . . . . . 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 2520 . . . . . . 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 12430 . . . . . . 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 1228 . . . . . 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 2508 . . . . 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 6311 . . . 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 23272 . . . . . 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 23272 . . . . . 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 6313 . . . 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 2519 . . 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 2511 . 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 2786 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   {crab 2821    u. cun 3479    i^i cin 3480    C_ wss 3481   (/)c0 3790   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Fincfn 7528   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509   -ucneg 9818   NNcn 10548   NN0cn0 10807   ZZcz 10876   ...cfz 11684   ^cexp 12146   #chash 12385    || cdivides 13863    gcd cgcd 14019   Primecprime 14092   mmucmu 23232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-fz 11685  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-dvds 13864  df-gcd 14020  df-prm 14093  df-pc 14236  df-mu 23238
This theorem is referenced by: (None)
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