Theorem latmcl

Description: Closure of meet operation in a lattice. ( incom analog.) (Contributed by NM, 14-Sep-2011)

	Ref	Expression
Hypotheses	latmcl.b	$$⊢B={\mathrm{Base}}_K$$
	latmcl.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
Assertion	latmcl	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\wedge }Y\in B$$

Proof

Step	Hyp	Ref	Expression
1		latmcl.b	$$⊢B={\mathrm{Base}}_K$$
2		latmcl.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
3		eqid	$$⊢\mathrm{join}\left(K\right)=\mathrm{join}\left(K\right)$$
4	1 3 2	latlem	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(X\mathrm{join}\left(K\right)Y\in B\wedge X\underset{\dot{}}{\wedge }Y\in B\right)$$
5	4	simprd	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\wedge }Y\in B$$