Theorem clatglb

Description: Properties of greatest lower bound of a complete lattice. (Contributed by NM, 5-Dec-2011)

	Ref	Expression
Hypotheses	clatglb.b	$$⊢B={\mathrm{Base}}_K$$
	clatglb.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	clatglb.g	$$⊢G=\mathrm{glb}\left(K\right)$$
Assertion	clatglb	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\right)\to \left(\forall y\in SG\left(S\right)\underset{\dot{}}{\le }y\wedge \forall z\in B\left(\forall y\in Sz\underset{\dot{}}{\le }y\to z\underset{\dot{}}{\le }G\left(S\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		clatglb.b	$$⊢B={\mathrm{Base}}_K$$
2		clatglb.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		clatglb.g	$$⊢G=\mathrm{glb}\left(K\right)$$
4		simpl	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\right)\to K\in \mathrm{CLat}$$
5	1 3	clatglbcl2	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\right)\to S\in \mathrm{dom}G$$
6	1 2 3 4 5	glbprop	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\right)\to \left(\forall y\in SG\left(S\right)\underset{\dot{}}{\le }y\wedge \forall z\in B\left(\forall y\in Sz\underset{\dot{}}{\le }y\to z\underset{\dot{}}{\le }G\left(S\right)\right)\right)$$