Theorem op0le

Description: Orthoposet zero is less than or equal to any element. ( ch0le analog.) (Contributed by NM, 12-Oct-2011)

	Ref	Expression
Hypotheses	op0le.b	$$⊢B={\mathrm{Base}}_K$$
	op0le.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	op0le.z	$$⊢\underset{\dot{}}{0}=\mathrm{0.}\left(K\right)$$
Assertion	op0le	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to \underset{\dot{}}{0}\underset{\dot{}}{\le }X$$

Proof

Step	Hyp	Ref	Expression
1		op0le.b	$$⊢B={\mathrm{Base}}_K$$
2		op0le.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		op0le.z	$$⊢\underset{\dot{}}{0}=\mathrm{0.}\left(K\right)$$
4		eqid	$$⊢\mathrm{glb}\left(K\right)=\mathrm{glb}\left(K\right)$$
5		simpl	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to K\in \mathrm{OP}$$
6		simpr	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to X\in B$$
7		eqid	$$⊢\mathrm{lub}\left(K\right)=\mathrm{lub}\left(K\right)$$
8	1 7 4	op01dm	$$⊢K\in \mathrm{OP}\to \left(B\in \mathrm{dom}\mathrm{lub}\left(K\right)\wedge B\in \mathrm{dom}\mathrm{glb}\left(K\right)\right)$$
9	8	simprd	$$⊢K\in \mathrm{OP}\to B\in \mathrm{dom}\mathrm{glb}\left(K\right)$$
10	9	adantr	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to B\in \mathrm{dom}\mathrm{glb}\left(K\right)$$
11	1 4 2 3 5 6 10	p0le	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to \underset{\dot{}}{0}\underset{\dot{}}{\le }X$$