Theorem nn0mulcli

Description: Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002)

	Ref	Expression
Hypotheses	nn0addcli.1	$$⊢M\in {\mathbb{N}}_0$$
	nn0addcli.2	$$⊢N\in {\mathbb{N}}_0$$
Assertion	nn0mulcli	$$⊢\mathrm{<mi\; mathcolor="\#808">M</mi>}\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\in {\mathbb{N}}_0$$

Proof

Step	Hyp	Ref	Expression
1		nn0addcli.1	$$⊢M\in {\mathbb{N}}_0$$
2		nn0addcli.2	$$⊢N\in {\mathbb{N}}_0$$
3		nn0mulcl	$$⊢\left(M\in {\mathbb{N}}_0\wedge N\in {\mathbb{N}}_0\right)\to \mathrm{<mi\; mathcolor="\#808">M</mi>}\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\in {\mathbb{N}}_0$$
4	1 2 3	mp2an	$$⊢\mathrm{<mi\; mathcolor="\#808">M</mi>}\cdot \mathrm{<mi\; mathcolor="\#808">N</mi>}\in {\mathbb{N}}_0$$