Theorem clatglbcl2

Description: Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 13-Sep-2018)

	Ref	Expression
Hypotheses	clatglbcl.b	$$⊢B={\mathrm{Base}}_K$$
	clatglbcl.g	$$⊢G=\mathrm{glb}\left(K\right)$$
Assertion	clatglbcl2	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\right)\to S\in \mathrm{dom}G$$

Proof

Step	Hyp	Ref	Expression
1		clatglbcl.b	$$⊢B={\mathrm{Base}}_K$$
2		clatglbcl.g	$$⊢G=\mathrm{glb}\left(K\right)$$
3	1	fvexi	$$⊢B\in \mathrm{V}$$
4	3	elpw2	$$⊢S\in 𝒫B\leftrightarrow S\subseteq B$$
5	4	biimpri	$$⊢S\subseteq B\to S\in 𝒫B$$
6	5	adantl	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\right)\to S\in 𝒫B$$
7		eqid	$$⊢\mathrm{lub}\left(K\right)=\mathrm{lub}\left(K\right)$$
8	1 7 2	isclat	$$⊢K\in \mathrm{CLat}\leftrightarrow \left(K\in \mathrm{Poset}\wedge \left(\mathrm{dom}\mathrm{lub}\left(K\right)=𝒫B\wedge \mathrm{dom}G=𝒫B\right)\right)$$
9		simprr	$$⊢\left(K\in \mathrm{Poset}\wedge \left(\mathrm{dom}\mathrm{lub}\left(K\right)=𝒫B\wedge \mathrm{dom}G=𝒫B\right)\right)\to \mathrm{dom}G=𝒫B$$
10	8 9	sylbi	$$⊢K\in \mathrm{CLat}\to \mathrm{dom}G=𝒫B$$
11	10	adantr	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\right)\to \mathrm{dom}G=𝒫B$$
12	6 11	eleqtrrd	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\right)\to S\in \mathrm{dom}G$$