Theorem latmlej21

Description: Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012)

	Ref	Expression
Hypotheses	latledi.b	$$⊢B={\mathrm{Base}}_K$$
	latledi.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	latledi.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	latledi.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
Assertion	latmlej21	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(Y\underset{\dot{}}{\wedge }X\right)\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		latledi.b	$$⊢B={\mathrm{Base}}_K$$
2		latledi.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		latledi.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		latledi.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5	1 4	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	3adant3r3	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to X\underset{\dot{}}{\wedge }Y=Y\underset{\dot{}}{\wedge }X$$
7	1 2 3 4	latmlej11	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{\wedge }Y\right)\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Z\right)$$
8	6 7	eqbrtrrd	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(Y\underset{\dot{}}{\wedge }X\right)\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Z\right)$$