Theorem nnmcli

Description: _om is closed under multiplication. Inference form of nnmcl . (Contributed by Scott Fenton, 20-Apr-2012)

	Ref	Expression
Hypotheses	nncli.1	$$⊢A\in \mathrm{\omega }$$
	nncli.2	$$⊢B\in \mathrm{\omega }$$
Assertion	nnmcli	$$⊢A{\cdot }_{𝑜}B\in \mathrm{\omega }$$

Proof

Step	Hyp	Ref	Expression
1		nncli.1	$$⊢A\in \mathrm{\omega }$$
2		nncli.2	$$⊢B\in \mathrm{\omega }$$
3		nnmcl	$$⊢\left(A\in \mathrm{\omega }\wedge B\in \mathrm{\omega }\right)\to A{\cdot }_{𝑜}B\in \mathrm{\omega }$$
4	1 2 3	mp2an	$$⊢A{\cdot }_{𝑜}B\in \mathrm{\omega }$$