Theorem nnaddcld

Description: Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	nnge1d.1	$$⊢\phi \to A\in \mathbb{N}$$
	nnmulcld.2	$$⊢\phi \to B\in \mathbb{N}$$
Assertion	nnaddcld	$$⊢\phi \to A+B\in \mathbb{N}$$

Proof

Step	Hyp	Ref	Expression
1		nnge1d.1	$$⊢\phi \to A\in \mathbb{N}$$
2		nnmulcld.2	$$⊢\phi \to B\in \mathbb{N}$$
3		nnaddcl	$$⊢\left(A\in \mathbb{N}\wedge B\in \mathbb{N}\right)\to A+B\in \mathbb{N}$$
4	1 2 3	syl2anc	$$⊢\phi \to A+B\in \mathbb{N}$$