Theorem chocvali

Description: Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of A is the set of vectors that are orthogonal to all vectors in A . (Contributed by NM, 8-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypothesis	chocval.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chocvali	$$⊢\perp \left(A\right)=\left\{x\in \mathscr{H}|\forall y\in Ax{\cdot }_{\mathrm{ih}}y=0\right\}$$

Proof

Step	Hyp	Ref	Expression
1		chocval.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2	1	chssii	$$⊢A\subseteq \mathscr{H}$$
3		ocval	$$⊢A\subseteq \mathscr{H}\to \perp \left(A\right)=\left\{x\in \mathscr{H}|\forall y\in Ax{\cdot }_{\mathrm{ih}}y=0\right\}$$
4	2 3	ax-mp	$$⊢\perp \left(A\right)=\left\{x\in \mathscr{H}|\forall y\in Ax{\cdot }_{\mathrm{ih}}y=0\right\}$$