Theorem css1

Description: The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015)

	Ref	Expression
Hypotheses	css1.v	$$⊢V={\mathrm{Base}}_W$$
	css1.c	$$⊢C=\mathrm{ClSubSp}\left(W\right)$$
Assertion	css1	$$⊢W\in \mathrm{PreHil}\to V\in C$$

Proof

Step	Hyp	Ref	Expression
1		css1.v	$$⊢V={\mathrm{Base}}_W$$
2		css1.c	$$⊢C=\mathrm{ClSubSp}\left(W\right)$$
3		eqid	$$⊢\mathrm{ocv}\left(W\right)=\mathrm{ocv}\left(W\right)$$
4	1 3	ocv0	$$⊢\mathrm{ocv}\left(W\right)\left(\varnothing \right)=V$$
5		0ss	$$⊢\varnothing \subseteq V$$
6	1 2 3	ocvcss	$$⊢\left(W\in \mathrm{PreHil}\wedge \varnothing \subseteq V\right)\to \mathrm{ocv}\left(W\right)\left(\varnothing \right)\in C$$
7	5 6	mpan2	$$⊢W\in \mathrm{PreHil}\to \mathrm{ocv}\left(W\right)\left(\varnothing \right)\in C$$
8	4 7	eqeltrrid	$$⊢W\in \mathrm{PreHil}\to V\in C$$