Theorem ococi

Description: Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of Beran p. 102. (Contributed by NM, 11-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypothesis	ococ.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	ococi	$$⊢\perp \left(\perp \left(A\right)\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		ococ.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2	1	chshii	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
3		shocsh	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to \perp \left(A\right)\in {\mathbf{S}}_{\mathscr{H}}$$
4	2 3	ax-mp	$$⊢\perp \left(A\right)\in {\mathbf{S}}_{\mathscr{H}}$$
5		shocsh	$$⊢\perp \left(A\right)\in {\mathbf{S}}_{\mathscr{H}}\to \perp \left(\perp \left(A\right)\right)\in {\mathbf{S}}_{\mathscr{H}}$$
6	4 5	ax-mp	$$⊢\perp \left(\perp \left(A\right)\right)\in {\mathbf{S}}_{\mathscr{H}}$$
7		shococss	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to A\subseteq \perp \left(\perp \left(A\right)\right)$$
8	2 7	ax-mp	$$⊢A\subseteq \perp \left(\perp \left(A\right)\right)$$
9		incom	$$⊢\perp \left(\perp \left(A\right)\right)\cap \perp \left(A\right)=\perp \left(A\right)\cap \perp \left(\perp \left(A\right)\right)$$
10		ocin	$$⊢\perp \left(A\right)\in {\mathbf{S}}_{\mathscr{H}}\to \perp \left(A\right)\cap \perp \left(\perp \left(A\right)\right)=0_{\mathscr{H}}$$
11	4 10	ax-mp	$$⊢\perp \left(A\right)\cap \perp \left(\perp \left(A\right)\right)=0_{\mathscr{H}}$$
12	9 11	eqtri	$$⊢\perp \left(\perp \left(A\right)\right)\cap \perp \left(A\right)=0_{\mathscr{H}}$$
13	1 6 8 12	omlsii	$$⊢A=\perp \left(\perp \left(A\right)\right)$$
14	13	eqcomi	$$⊢\perp \left(\perp \left(A\right)\right)=A$$