Theorem psubcliN

Description: Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	psubclset.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	psubclset.p	$$⊢\underset{\dot{}}{\perp }={\perp }_{𝑃}\left(K\right)$$
	psubclset.c	$$⊢C=\mathrm{PSubCl}\left(K\right)$$
Assertion	psubcliN	$$⊢\left(K\in D\wedge X\in C\right)\to \left(X\subseteq A\wedge \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(X\right)\right)=X\right)$$

Proof

Step	Hyp	Ref	Expression
1		psubclset.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
2		psubclset.p	$$⊢\underset{\dot{}}{\perp }={\perp }_{𝑃}\left(K\right)$$
3		psubclset.c	$$⊢C=\mathrm{PSubCl}\left(K\right)$$
4	1 2 3	ispsubclN	$$⊢K\in D\to \left(X\in C\leftrightarrow \left(X\subseteq A\wedge \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(X\right)\right)=X\right)\right)$$
5	4	biimpa	$$⊢\left(K\in D\wedge X\in C\right)\to \left(X\subseteq A\wedge \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(X\right)\right)=X\right)$$