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 e. CH
Assertion	chocvali	|- ( _|_ ` A ) = { x e. ~H | A. y e. A ( x .ih y ) = 0 }

Proof

Step	Hyp	Ref	Expression
1		chocval.1	|- A e. CH
2	1	chssii	|- A C_ ~H
3		ocval	|- ( A C_ ~H -> ( _|_ ` A ) = { x e. ~H | A. y e. A ( x .ih y ) = 0 } )
4	2 3	ax-mp	|- ( _|_ ` A ) = { x e. ~H | A. y e. A ( x .ih y ) = 0 }