Theorem clatglbcl

Description: Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 14-Sep-2011)

	Ref	Expression
Hypotheses	clatglbcl.b	$$⊢B={\mathrm{Base}}_K$$
	clatglbcl.g	$$⊢G=\mathrm{glb}\left(K\right)$$
Assertion	clatglbcl	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\right)\to G\left(S\right)\in B$$

Proof

Step	Hyp	Ref	Expression
1		clatglbcl.b	$$⊢B={\mathrm{Base}}_K$$
2		clatglbcl.g	$$⊢G=\mathrm{glb}\left(K\right)$$
3		eqid	$$⊢\mathrm{lub}\left(K\right)=\mathrm{lub}\left(K\right)$$
4	1 3 2	clatlem	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\right)\to \left(\mathrm{lub}\left(K\right)\left(S\right)\in B\wedge G\left(S\right)\in B\right)$$
5	4	simprd	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\right)\to G\left(S\right)\in B$$