Theorem gcdaddmzz2nncomi

Description: Adding a multiple of one operand of the gcd operator to the other does not alter the result. (Contributed by metakunt, 25-Apr-2024)

	Ref	Expression
Hypotheses	gcdaddmzz2nncomi.1	$$⊢M\in \mathbb{N}$$
	gcdaddmzz2nncomi.2	$$⊢N\in \mathbb{N}$$
	gcdaddmzz2nncomi.3	$$⊢K\in \mathbb{Z}$$
Assertion	gcdaddmzz2nncomi	$$⊢M\gcd N=M\gcd \left(K\cdot \mathrm{<mi\; mathcolor="\#808">M</mi>}+N\right)$$

Proof

Step	Hyp	Ref	Expression
1		gcdaddmzz2nncomi.1	$$⊢M\in \mathbb{N}$$
2		gcdaddmzz2nncomi.2	$$⊢N\in \mathbb{N}$$
3		gcdaddmzz2nncomi.3	$$⊢K\in \mathbb{Z}$$
4	1 2 3	gcdaddmzz2nni	$$⊢M\gcd N=M\gcd \left(N+K\cdot \mathrm{<mi\; mathcolor="\#808">M</mi>}\right)$$
5	2	nncni	$$⊢N\in ℂ$$
6		zcn	$$⊢K\in \mathbb{Z}\to K\in ℂ$$
7	3 6	ax-mp	$$⊢K\in ℂ$$
8	1	nncni	$$⊢M\in ℂ$$
9	7 8	mulcli	$$⊢K\cdot \mathrm{<mi\; mathcolor="\#808">M</mi>}\in ℂ$$
10	5 9	addcomi	$$⊢N+K\cdot \mathrm{<mi\; mathcolor="\#808">M</mi>}=K\cdot \mathrm{<mi\; mathcolor="\#808">M</mi>}+N$$
11	10	oveq2i	$$⊢M\gcd \left(N+K\cdot \mathrm{<mi\; mathcolor="\#808">M</mi>}\right)=M\gcd \left(K\cdot \mathrm{<mi\; mathcolor="\#808">M</mi>}+N\right)$$
12	4 11	eqtri	$$⊢M\gcd N=M\gcd \left(K\cdot \mathrm{<mi\; mathcolor="\#808">M</mi>}+N\right)$$