Theorem hlatjidm

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

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

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 1	latjidm	$$⊢\left(K\in \mathrm{Lat}\wedge X\in {\mathrm{Base}}_K\right)\to X\underset{\dot{}}{\vee }X=X$$
7	3 5 6	syl2an	$$⊢\left(K\in \mathrm{HL}\wedge X\in A\right)\to X\underset{\dot{}}{\vee }X=X$$