Description: The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011) (Proof shortened by SN, 15-Sep-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | gcdabs | |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | zabscl | |- ( N e. ZZ -> ( abs ` N ) e. ZZ ) |
|
| 2 | gcdabs1 | |- ( ( M e. ZZ /\ ( abs ` N ) e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd ( abs ` N ) ) ) |
|
| 3 | 1 2 | sylan2 | |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd ( abs ` N ) ) ) |
| 4 | gcdabs2 | |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd ( abs ` N ) ) = ( M gcd N ) ) |
|
| 5 | 3 4 | eqtrd | |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) ) |