Step |
Hyp |
Ref |
Expression |
1 |
|
df-mu |
|- mmu = ( x e. NN |-> if ( E. p e. Prime ( p ^ 2 ) || x , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || x } ) ) ) ) |
2 |
|
0z |
|- 0 e. ZZ |
3 |
|
neg1z |
|- -u 1 e. ZZ |
4 |
|
prmdvdsfi |
|- ( x e. NN -> { p e. Prime | p || x } e. Fin ) |
5 |
|
hashcl |
|- ( { p e. Prime | p || x } e. Fin -> ( # ` { p e. Prime | p || x } ) e. NN0 ) |
6 |
4 5
|
syl |
|- ( x e. NN -> ( # ` { p e. Prime | p || x } ) e. NN0 ) |
7 |
|
zexpcl |
|- ( ( -u 1 e. ZZ /\ ( # ` { p e. Prime | p || x } ) e. NN0 ) -> ( -u 1 ^ ( # ` { p e. Prime | p || x } ) ) e. ZZ ) |
8 |
3 6 7
|
sylancr |
|- ( x e. NN -> ( -u 1 ^ ( # ` { p e. Prime | p || x } ) ) e. ZZ ) |
9 |
|
ifcl |
|- ( ( 0 e. ZZ /\ ( -u 1 ^ ( # ` { p e. Prime | p || x } ) ) e. ZZ ) -> if ( E. p e. Prime ( p ^ 2 ) || x , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || x } ) ) ) e. ZZ ) |
10 |
2 8 9
|
sylancr |
|- ( x e. NN -> if ( E. p e. Prime ( p ^ 2 ) || x , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || x } ) ) ) e. ZZ ) |
11 |
1 10
|
fmpti |
|- mmu : NN --> ZZ |