Theorem gcdcomnni

Description: Commutative law for gcd. (Contributed by metakunt, 25-Apr-2024)

	Ref	Expression
Hypotheses	gcdcomnni.1	⊢ 𝑀 ∈ ℕ
	gcdcomnni.2	⊢ 𝑁 ∈ ℕ
Assertion	gcdcomnni	⊢ ( 𝑀 gcd 𝑁 ) = ( 𝑁 gcd 𝑀 )

Proof

Step	Hyp	Ref	Expression
1		gcdcomnni.1	⊢ 𝑀 ∈ ℕ
2		gcdcomnni.2	⊢ 𝑁 ∈ ℕ
3	1	nnzi	⊢ 𝑀 ∈ ℤ
4	2	nnzi	⊢ 𝑁 ∈ ℤ
5	3 4	pm3.2i	⊢ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ )
6		gcdcom	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 gcd 𝑁 ) = ( 𝑁 gcd 𝑀 ) )
7	5 6	ax-mp	⊢ ( 𝑀 gcd 𝑁 ) = ( 𝑁 gcd 𝑀 )