Theorem latleeqm2

Description: "Less than or equal to" in terms of meet. (Contributed by NM, 7-Nov-2011)

	Ref	Expression
Hypotheses	latmle.b	$$⊢B={\mathrm{Base}}_K$$
	latmle.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	latmle.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
Assertion	latleeqm2	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\le }Y\leftrightarrow Y\underset{\dot{}}{\wedge }X=X\right)$$

Proof

Step	Hyp	Ref	Expression
1		latmle.b	$$⊢B={\mathrm{Base}}_K$$
2		latmle.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		latmle.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4	1 2 3	latleeqm1	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\le }Y\leftrightarrow X\underset{\dot{}}{\wedge }Y=X\right)$$
5	1 3	latmcom	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\wedge }Y=Y\underset{\dot{}}{\wedge }X$$
6	5	eqeq1d	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\wedge }Y=X\leftrightarrow Y\underset{\dot{}}{\wedge }X=X\right)$$
7	4 6	bitrd	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\le }Y\leftrightarrow Y\underset{\dot{}}{\wedge }X=X\right)$$