Theorem latpos

Description: A lattice is a poset. (Contributed by NM, 17-Sep-2011)

	Ref	Expression
Assertion	latpos	$$⊢K\in \mathrm{Lat}\to K\in \mathrm{Poset}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
2		eqid	$$⊢\mathrm{join}\left(K\right)=\mathrm{join}\left(K\right)$$
3		eqid	$$⊢\mathrm{meet}\left(K\right)=\mathrm{meet}\left(K\right)$$
4	1 2 3	islat	$$⊢K\in \mathrm{Lat}\leftrightarrow \left(K\in \mathrm{Poset}\wedge \left(\mathrm{dom}\mathrm{join}\left(K\right)={\mathrm{Base}}_K\times {\mathrm{Base}}_K\wedge \mathrm{dom}\mathrm{meet}\left(K\right)={\mathrm{Base}}_K\times {\mathrm{Base}}_K\right)\right)$$
5	4	simplbi	$$⊢K\in \mathrm{Lat}\to K\in \mathrm{Poset}$$