Theorem zmulcld

Description: Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	zred.1	$$⊢\phi \to A\in \mathbb{Z}$$
	zaddcld.1	$$⊢\phi \to B\in \mathbb{Z}$$
Assertion	zmulcld	$$⊢\phi \to AB\in \mathbb{Z}$$

Proof

Step	Hyp	Ref	Expression
1		zred.1	$$⊢\phi \to A\in \mathbb{Z}$$
2		zaddcld.1	$$⊢\phi \to B\in \mathbb{Z}$$
3		zmulcl	$$⊢\left(A\in \mathbb{Z}\wedge B\in \mathbb{Z}\right)\to AB\in \mathbb{Z}$$
4	1 2 3	syl2anc	$$⊢\phi \to AB\in \mathbb{Z}$$