Description: Alternate definition for odd numbers. (Contributed by AV, 1-Jul-2020)
Ref | Expression | ||
---|---|---|---|
Assertion | dfodd7 | |- Odd = { z e. ZZ | ( 2 gcd z ) = 1 } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isodd7 | |- ( x e. Odd <-> ( x e. ZZ /\ ( 2 gcd x ) = 1 ) ) |
|
2 | oveq2 | |- ( z = x -> ( 2 gcd z ) = ( 2 gcd x ) ) |
|
3 | 2 | eqeq1d | |- ( z = x -> ( ( 2 gcd z ) = 1 <-> ( 2 gcd x ) = 1 ) ) |
4 | 3 | elrab | |- ( x e. { z e. ZZ | ( 2 gcd z ) = 1 } <-> ( x e. ZZ /\ ( 2 gcd x ) = 1 ) ) |
5 | 1 4 | bitr4i | |- ( x e. Odd <-> x e. { z e. ZZ | ( 2 gcd z ) = 1 } ) |
6 | 5 | eqriv | |- Odd = { z e. ZZ | ( 2 gcd z ) = 1 } |