Theorem ople0

Description: An element less than or equal to zero equals zero. ( chle0 analog.) (Contributed by NM, 21-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	ople0	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to \left(X\underset{\dot{}}{\le }\underset{\dot{}}{0}\leftrightarrow X=\underset{\dot{}}{0}\right)$$

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	1 2 3	op0le	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to \underset{\dot{}}{0}\underset{\dot{}}{\le }X$$
5	4	biantrud	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to \left(X\underset{\dot{}}{\le }\underset{\dot{}}{0}\leftrightarrow \left(X\underset{\dot{}}{\le }\underset{\dot{}}{0}\wedge \underset{\dot{}}{0}\underset{\dot{}}{\le }X\right)\right)$$
6		opposet	$$⊢K\in \mathrm{OP}\to K\in \mathrm{Poset}$$
7	6	adantr	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to K\in \mathrm{Poset}$$
8		simpr	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to X\in B$$
9	1 3	op0cl	$$⊢K\in \mathrm{OP}\to \underset{\dot{}}{0}\in B$$
10	9	adantr	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to \underset{\dot{}}{0}\in B$$
11	1 2	posasymb	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge \underset{\dot{}}{0}\in B\right)\to \left(\left(X\underset{\dot{}}{\le }\underset{\dot{}}{0}\wedge \underset{\dot{}}{0}\underset{\dot{}}{\le }X\right)\leftrightarrow X=\underset{\dot{}}{0}\right)$$
12	7 8 10 11	syl3anc	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to \left(\left(X\underset{\dot{}}{\le }\underset{\dot{}}{0}\wedge \underset{\dot{}}{0}\underset{\dot{}}{\le }X\right)\leftrightarrow X=\underset{\dot{}}{0}\right)$$
13	5 12	bitrd	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to \left(X\underset{\dot{}}{\le }\underset{\dot{}}{0}\leftrightarrow X=\underset{\dot{}}{0}\right)$$