Theorem latleeqj2

Description: "Less than or equal to" in terms of join. ( chlejb2 analog.) (Contributed by NM, 14-Nov-2011)

	Ref	Expression
Hypotheses	latlej.b	$$⊢B={\mathrm{Base}}_K$$
	latlej.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	latlej.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
Assertion	latleeqj2	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\le }Y\leftrightarrow Y\underset{\dot{}}{\vee }X=Y\right)$$

Proof

Step	Hyp	Ref	Expression
1		latlej.b	$$⊢B={\mathrm{Base}}_K$$
2		latlej.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		latlej.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4	1 2 3	latleeqj1	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\le }Y\leftrightarrow X\underset{\dot{}}{\vee }Y=Y\right)$$
5	1 3	latjcom	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\vee }Y=Y\underset{\dot{}}{\vee }X$$
6	5	eqeq1d	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\vee }Y=Y\leftrightarrow Y\underset{\dot{}}{\vee }X=Y\right)$$
7	4 6	bitrd	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\le }Y\leftrightarrow Y\underset{\dot{}}{\vee }X=Y\right)$$