Theorem clatpos

Description: A complete lattice is a poset. (Contributed by NM, 8-Sep-2018)

	Ref	Expression
Assertion	clatpos	$$⊢K\in \mathrm{CLat}\to K\in \mathrm{Poset}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
2		eqid	$$⊢\mathrm{lub}\left(K\right)=\mathrm{lub}\left(K\right)$$
3		eqid	$$⊢\mathrm{glb}\left(K\right)=\mathrm{glb}\left(K\right)$$
4	1 2 3	isclat	$$⊢K\in \mathrm{CLat}\leftrightarrow \left(K\in \mathrm{Poset}\wedge \left(\mathrm{dom}\mathrm{lub}\left(K\right)=𝒫{\mathrm{Base}}_K\wedge \mathrm{dom}\mathrm{glb}\left(K\right)=𝒫{\mathrm{Base}}_K\right)\right)$$
5	4	simplbi	$$⊢K\in \mathrm{CLat}\to K\in \mathrm{Poset}$$