Theorem latref

Description: A lattice ordering is reflexive. ( ssid analog.) (Contributed by NM, 8-Oct-2011)

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

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	posref	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to X\underset{\dot{}}{\le }X$$
5	3 4	sylan	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\right)\to X\underset{\dot{}}{\le }X$$