Theorem nn0mulcld

Description: Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	nn0red.1	$$⊢\phi \to A\in {\mathbb{N}}_0$$
	nn0addcld.2	$$⊢\phi \to B\in {\mathbb{N}}_0$$
Assertion	nn0mulcld	$$⊢\phi \to AB\in {\mathbb{N}}_0$$

Proof

Step	Hyp	Ref	Expression
1		nn0red.1	$$⊢\phi \to A\in {\mathbb{N}}_0$$
2		nn0addcld.2	$$⊢\phi \to B\in {\mathbb{N}}_0$$
3		nn0mulcl	$$⊢\left(A\in {\mathbb{N}}_0\wedge B\in {\mathbb{N}}_0\right)\to AB\in {\mathbb{N}}_0$$
4	1 2 3	syl2anc	$$⊢\phi \to AB\in {\mathbb{N}}_0$$