Theorem nnmulcli

Description: Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014)

	Ref	Expression
Hypotheses	nnmulcli.1	$$⊢A\in \mathbb{N}$$
	nnmulcli.2	$$⊢B\in \mathbb{N}$$
Assertion	nnmulcli	$$⊢AB\in \mathbb{N}$$

Proof

Step	Hyp	Ref	Expression
1		nnmulcli.1	$$⊢A\in \mathbb{N}$$
2		nnmulcli.2	$$⊢B\in \mathbb{N}$$
3		nnmulcl	$$⊢\left(A\in \mathbb{N}\wedge B\in \mathbb{N}\right)\to AB\in \mathbb{N}$$
4	1 2 3	mp2an	$$⊢AB\in \mathbb{N}$$