Theorem latjcl

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

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

Proof

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