Theorem hlatjcom

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

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

Proof

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