Theorem clatglble

Description: The greatest lower bound is the least element. (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	clatglble	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\wedge X\in S\right)\to G\left(S\right)\underset{\dot{}}{\le }X$$

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		simp1	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\wedge X\in S\right)\to K\in \mathrm{CLat}$$
5	1 3	clatglbcl2	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\right)\to S\in \mathrm{dom}G$$
6	5	3adant3	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\wedge X\in S\right)\to S\in \mathrm{dom}G$$
7		simp3	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\wedge X\in S\right)\to X\in S$$
8	1 2 3 4 6 7	glble	$$⊢\left(K\in \mathrm{CLat}\wedge S\subseteq B\wedge X\in S\right)\to G\left(S\right)\underset{\dot{}}{\le }X$$