Theorem hlatjcl

Description: Closure of join operation. Frequently-used special case of latjcl for atoms. (Contributed by NM, 15-Jun-2012)

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

Proof

Step	Hyp	Ref	Expression
1		hlatjcl.b	$$⊢B={\mathrm{Base}}_K$$
2		hlatjcl.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		hlatjcl.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		hllat	$$⊢K\in \mathrm{HL}\to K\in \mathrm{Lat}$$
5	1 3	atbase	$$⊢X\in A\to X\in B$$
6	1 3	atbase	$$⊢Y\in A\to Y\in B$$
7	1 2	latjcl	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\vee }Y\in B$$
8	4 5 6 7	syl3an	$$⊢\left(K\in \mathrm{HL}\wedge X\in A\wedge Y\in A\right)\to X\underset{\dot{}}{\vee }Y\in B$$