Step |
Hyp |
Ref |
Expression |
1 |
|
df-ne |
|- ( ( mmu ` A ) =/= 0 <-> -. ( mmu ` A ) = 0 ) |
2 |
|
ifeqor |
|- ( if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) = 0 \/ if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) |
3 |
|
muval |
|- ( A e. NN -> ( mmu ` A ) = if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) ) |
4 |
3
|
eqeq1d |
|- ( A e. NN -> ( ( mmu ` A ) = 0 <-> if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) = 0 ) ) |
5 |
3
|
eqeq1d |
|- ( A e. NN -> ( ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) <-> if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) ) |
6 |
4 5
|
orbi12d |
|- ( A e. NN -> ( ( ( mmu ` A ) = 0 \/ ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) <-> ( if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) = 0 \/ if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) ) ) |
7 |
2 6
|
mpbiri |
|- ( A e. NN -> ( ( mmu ` A ) = 0 \/ ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) ) |
8 |
7
|
ord |
|- ( A e. NN -> ( -. ( mmu ` A ) = 0 -> ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) ) |
9 |
1 8
|
syl5bi |
|- ( A e. NN -> ( ( mmu ` A ) =/= 0 -> ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) ) |
10 |
9
|
imp |
|- ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) -> ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) |