Theorem gcdmultipled

Description: The greatest common divisor of a nonnegative integer M and a multiple of it is M itself. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	gcdmultipled.1	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
	gcdmultipled.2	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
Assertion	gcdmultipled	⊢ ( 𝜑 → ( 𝑀 gcd ( 𝑁 · 𝑀 ) ) = 𝑀 )

Proof

Step	Hyp	Ref	Expression
1		gcdmultipled.1	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
2		gcdmultipled.2	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
3	1	nn0zd	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
4		0zd	⊢ ( 𝜑 → 0 ∈ ℤ )
5		gcdaddm	⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ ) → ( 𝑀 gcd 0 ) = ( 𝑀 gcd ( 0 + ( 𝑁 · 𝑀 ) ) ) )
6	2 3 4 5	syl3anc	⊢ ( 𝜑 → ( 𝑀 gcd 0 ) = ( 𝑀 gcd ( 0 + ( 𝑁 · 𝑀 ) ) ) )
7		nn0gcdid0	⊢ ( 𝑀 ∈ ℕ0 → ( 𝑀 gcd 0 ) = 𝑀 )
8	1 7	syl	⊢ ( 𝜑 → ( 𝑀 gcd 0 ) = 𝑀 )
9	2 3	zmulcld	⊢ ( 𝜑 → ( 𝑁 · 𝑀 ) ∈ ℤ )
10	9	zcnd	⊢ ( 𝜑 → ( 𝑁 · 𝑀 ) ∈ ℂ )
11	10	addid2d	⊢ ( 𝜑 → ( 0 + ( 𝑁 · 𝑀 ) ) = ( 𝑁 · 𝑀 ) )
12	11	oveq2d	⊢ ( 𝜑 → ( 𝑀 gcd ( 0 + ( 𝑁 · 𝑀 ) ) ) = ( 𝑀 gcd ( 𝑁 · 𝑀 ) ) )
13	6 8 12	3eqtr3rd	⊢ ( 𝜑 → ( 𝑀 gcd ( 𝑁 · 𝑀 ) ) = 𝑀 )