Description: Commutative law for gcd. (Contributed by metakunt, 25-Apr-2024)
Ref | Expression | ||
---|---|---|---|
Hypotheses | gcdcomnni.1 | |- M e. NN |
|
gcdcomnni.2 | |- N e. NN |
||
Assertion | gcdcomnni | |- ( M gcd N ) = ( N gcd M ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcdcomnni.1 | |- M e. NN |
|
2 | gcdcomnni.2 | |- N e. NN |
|
3 | 1 | nnzi | |- M e. ZZ |
4 | 2 | nnzi | |- N e. ZZ |
5 | 3 4 | pm3.2i | |- ( M e. ZZ /\ N e. ZZ ) |
6 | gcdcom | |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( N gcd M ) ) |
|
7 | 5 6 | ax-mp | |- ( M gcd N ) = ( N gcd M ) |