Theorem psubcli2N

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

	Ref	Expression
Hypotheses	psubcli2.p	$$⊢\underset{\dot{}}{\perp }={\perp }_{𝑃}\left(K\right)$$
	psubcli2.c	$$⊢C=\mathrm{PSubCl}\left(K\right)$$
Assertion	psubcli2N	$$⊢\left(K\in D\wedge X\in C\right)\to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(X\right)\right)=X$$

Proof

Step	Hyp	Ref	Expression
1		psubcli2.p	$$⊢\underset{\dot{}}{\perp }={\perp }_{𝑃}\left(K\right)$$
2		psubcli2.c	$$⊢C=\mathrm{PSubCl}\left(K\right)$$
3		eqid	$$⊢\mathrm{Atoms}\left(K\right)=\mathrm{Atoms}\left(K\right)$$
4	3 1 2	ispsubclN	$$⊢K\in D\to \left(X\in C\leftrightarrow \left(X\subseteq \mathrm{Atoms}\left(K\right)\wedge \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(X\right)\right)=X\right)\right)$$
5	4	simplbda	$$⊢\left(K\in D\wedge X\in C\right)\to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(X\right)\right)=X$$