Theorem oplecon1b

Description: Contraposition law for strict ordering in orthoposets. ( chsscon1 analog.) (Contributed by NM, 6-Nov-2011)

	Ref	Expression
Hypotheses	opcon3.b	$$⊢B={\mathrm{Base}}_K$$
	opcon3.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	opcon3.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{oc}\left(K\right)$$
Assertion	oplecon1b	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\wedge Y\in B\right)\to \left(\underset{\dot{}}{\perp }\left(X\right)\underset{\dot{}}{\le }Y\leftrightarrow \underset{\dot{}}{\perp }\left(Y\right)\underset{\dot{}}{\le }X\right)$$

Proof

Step	Hyp	Ref	Expression
1		opcon3.b	$$⊢B={\mathrm{Base}}_K$$
2		opcon3.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		opcon3.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{oc}\left(K\right)$$
4	1 3	opoccl	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to \underset{\dot{}}{\perp }\left(X\right)\in B$$
5	4	3adant3	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\wedge Y\in B\right)\to \underset{\dot{}}{\perp }\left(X\right)\in B$$
6	1 2 3	oplecon3b	$$⊢\left(K\in \mathrm{OP}\wedge \underset{\dot{}}{\perp }\left(X\right)\in B\wedge Y\in B\right)\to \left(\underset{\dot{}}{\perp }\left(X\right)\underset{\dot{}}{\le }Y\leftrightarrow \underset{\dot{}}{\perp }\left(Y\right)\underset{\dot{}}{\le }\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(X\right)\right)\right)$$
7	5 6	syld3an2	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\wedge Y\in B\right)\to \left(\underset{\dot{}}{\perp }\left(X\right)\underset{\dot{}}{\le }Y\leftrightarrow \underset{\dot{}}{\perp }\left(Y\right)\underset{\dot{}}{\le }\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(X\right)\right)\right)$$
8	1 3	opococ	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\right)\to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(X\right)\right)=X$$
9	8	3adant3	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\wedge Y\in B\right)\to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(X\right)\right)=X$$
10	9	breq2d	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\wedge Y\in B\right)\to \left(\underset{\dot{}}{\perp }\left(Y\right)\underset{\dot{}}{\le }\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(X\right)\right)\leftrightarrow \underset{\dot{}}{\perp }\left(Y\right)\underset{\dot{}}{\le }X\right)$$
11	7 10	bitrd	$$⊢\left(K\in \mathrm{OP}\wedge X\in B\wedge Y\in B\right)\to \left(\underset{\dot{}}{\perp }\left(X\right)\underset{\dot{}}{\le }Y\leftrightarrow \underset{\dot{}}{\perp }\left(Y\right)\underset{\dot{}}{\le }X\right)$$