Theorem csslss

Description: A closed subspace of a pre-Hilbert space is a linear subspace. (Contributed by Mario Carneiro, 13-Oct-2015)

	Ref	Expression
Hypotheses	csslss.c	$$⊢C=\mathrm{ClSubSp}\left(W\right)$$
	csslss.l	$$⊢L=\mathrm{LSubSp}\left(W\right)$$
Assertion	csslss	$$⊢\left(W\in \mathrm{PreHil}\wedge S\in C\right)\to S\in L$$

Proof

Step	Hyp	Ref	Expression
1		csslss.c	$$⊢C=\mathrm{ClSubSp}\left(W\right)$$
2		csslss.l	$$⊢L=\mathrm{LSubSp}\left(W\right)$$
3		eqid	$$⊢\mathrm{ocv}\left(W\right)=\mathrm{ocv}\left(W\right)$$
4	3 1	cssi	$$⊢S\in C\to S=\mathrm{ocv}\left(W\right)\left(\mathrm{ocv}\left(W\right)\left(S\right)\right)$$
5	4	adantl	$$⊢\left(W\in \mathrm{PreHil}\wedge S\in C\right)\to S=\mathrm{ocv}\left(W\right)\left(\mathrm{ocv}\left(W\right)\left(S\right)\right)$$
6		eqid	$$⊢{\mathrm{Base}}_W={\mathrm{Base}}_W$$
7	6 3	ocvss	$$⊢\mathrm{ocv}\left(W\right)\left(S\right)\subseteq {\mathrm{Base}}_W$$
8	7	a1i	$$⊢S\in C\to \mathrm{ocv}\left(W\right)\left(S\right)\subseteq {\mathrm{Base}}_W$$
9	6 3 2	ocvlss	$$⊢\left(W\in \mathrm{PreHil}\wedge \mathrm{ocv}\left(W\right)\left(S\right)\subseteq {\mathrm{Base}}_W\right)\to \mathrm{ocv}\left(W\right)\left(\mathrm{ocv}\left(W\right)\left(S\right)\right)\in L$$
10	8 9	sylan2	$$⊢\left(W\in \mathrm{PreHil}\wedge S\in C\right)\to \mathrm{ocv}\left(W\right)\left(\mathrm{ocv}\left(W\right)\left(S\right)\right)\in L$$
11	5 10	eqeltrd	$$⊢\left(W\in \mathrm{PreHil}\wedge S\in C\right)\to S\in L$$