Theorem lcmfn0cl

Description: Closure of the _lcm function. (Contributed by AV, 21-Aug-2020)

	Ref	Expression
Assertion	lcmfn0cl	$$⊢\left(Z\subseteq \mathbb{Z}\wedge Z\in \mathrm{Fin}\wedge 0\notin Z\right)\to \underset{\_}{\mathrm{lcm}}\left(Z\right)\in \mathbb{N}$$

Proof

Step	Hyp	Ref	Expression
1		ssrab2	$$⊢\left\{n\in \mathbb{N}|\forall m\in Zm\parallel n\right\}\subseteq \mathbb{N}$$
2		lcmfcllem	$$⊢\left(Z\subseteq \mathbb{Z}\wedge Z\in \mathrm{Fin}\wedge 0\notin Z\right)\to \underset{\_}{\mathrm{lcm}}\left(Z\right)\in \left\{n\in \mathbb{N}|\forall m\in Zm\parallel n\right\}$$
3	1 2	sseldi	$$⊢\left(Z\subseteq \mathbb{Z}\wedge Z\in \mathrm{Fin}\wedge 0\notin Z\right)\to \underset{\_}{\mathrm{lcm}}\left(Z\right)\in \mathbb{N}$$