Theorem chocini

Description: Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of Kalmbach p. 65. (Contributed by NM, 11-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypothesis	ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chocini	$$⊢A\cap \perp \left(A\right)=0_{\mathscr{H}}$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2	1	chshii	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
3		ocin	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to A\cap \perp \left(A\right)=0_{\mathscr{H}}$$
4	2 3	ax-mp	$$⊢A\cap \perp \left(A\right)=0_{\mathscr{H}}$$