Theorem latnlej1r

Description: An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012)

	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	latnlej1r	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\wedge \neg X\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\vee }Z\right)\right)\to X\ne Z$$

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	latnlej	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\wedge \neg X\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\vee }Z\right)\right)\to \left(X\ne Y\wedge X\ne Z\right)$$
5	4	simprd	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\wedge \neg X\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\vee }Z\right)\right)\to X\ne Z$$