Theorem lattr

Description: A lattice ordering is transitive. ( sstr analog.) (Contributed by NM, 17-Nov-2011)

	Ref	Expression
Hypotheses	latref.b	$$⊢B={\mathrm{Base}}_K$$
	latref.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
Assertion	lattr	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }Z\right)\to X\underset{\dot{}}{\le }Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		latref.b	$$⊢B={\mathrm{Base}}_K$$
2		latref.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		latpos	$$⊢K\in \mathrm{Lat}\to K\in \mathrm{Poset}$$
4	1 2	postr	$$⊢\left(K\in \mathrm{Poset}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }Z\right)\to X\underset{\dot{}}{\le }Z\right)$$
5	3 4	sylan	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }Z\right)\to X\underset{\dot{}}{\le }Z\right)$$