Theorem psubclssatN

Description: A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	psubclssat.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	psubclssat.c	$$⊢C=\mathrm{PSubCl}\left(K\right)$$
Assertion	psubclssatN	$$⊢\left(K\in D\wedge X\in C\right)\to X\subseteq A$$

Proof

Step	Hyp	Ref	Expression
1		psubclssat.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
2		psubclssat.c	$$⊢C=\mathrm{PSubCl}\left(K\right)$$
3		eqid	$$⊢{\perp }_{𝑃}\left(K\right)={\perp }_{𝑃}\left(K\right)$$
4	1 3 2	psubcliN	$$⊢\left(K\in D\wedge X\in C\right)\to \left(X\subseteq A\wedge {\perp }_{𝑃}\left(K\right)\left({\perp }_{𝑃}\left(K\right)\left(X\right)\right)=X\right)$$
5	4	simpld	$$⊢\left(K\in D\wedge X\in C\right)\to X\subseteq A$$