Theorem pclidN

Description: The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pclid.s	$$⊢S=\mathrm{PSubSp}\left(K\right)$$
	pclid.c	$$⊢U=\mathrm{PCl}\left(K\right)$$
Assertion	pclidN	$$⊢\left(K\in V\wedge X\in S\right)\to U\left(X\right)=X$$

Proof

Step	Hyp	Ref	Expression
1		pclid.s	$$⊢S=\mathrm{PSubSp}\left(K\right)$$
2		pclid.c	$$⊢U=\mathrm{PCl}\left(K\right)$$
3		eqid	$$⊢\mathrm{Atoms}\left(K\right)=\mathrm{Atoms}\left(K\right)$$
4	3 1	psubssat	$$⊢\left(K\in V\wedge X\in S\right)\to X\subseteq \mathrm{Atoms}\left(K\right)$$
5	3 1 2	pclvalN	$$⊢\left(K\in V\wedge X\subseteq \mathrm{Atoms}\left(K\right)\right)\to U\left(X\right)=\bigcap \left\{y\in S|X\subseteq y\right\}$$
6	4 5	syldan	$$⊢\left(K\in V\wedge X\in S\right)\to U\left(X\right)=\bigcap \left\{y\in S|X\subseteq y\right\}$$
7		intmin	$$⊢X\in S\to \bigcap \left\{y\in S|X\subseteq y\right\}=X$$
8	7	adantl	$$⊢\left(K\in V\wedge X\in S\right)\to \bigcap \left\{y\in S|X\subseteq y\right\}=X$$
9	6 8	eqtrd	$$⊢\left(K\in V\wedge X\in S\right)\to U\left(X\right)=X$$