Theorem pjocvec

Description: The set of vectors belonging to the orthocomplemented subspace of a projection. Second part of Theorem 27.3 of Halmos p. 45. (Contributed by NM, 24-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	pjocvec	|- ( H e. CH -> ( _|_ ` H ) = { x e. ~H | ( ( projh ` H ) ` x ) = 0h } )

Proof

Step	Hyp	Ref	Expression
1		choccl	|- ( H e. CH -> ( _|_ ` H ) e. CH )
2		chss	|- ( ( _|_ ` H ) e. CH -> ( _|_ ` H ) C_ ~H )
3	1 2	syl	|- ( H e. CH -> ( _|_ ` H ) C_ ~H )
4		sseqin2	|- ( ( _|_ ` H ) C_ ~H <-> ( ~H i^i ( _|_ ` H ) ) = ( _|_ ` H ) )
5	3 4	sylib	|- ( H e. CH -> ( ~H i^i ( _|_ ` H ) ) = ( _|_ ` H ) )
6		pjoc2	|- ( ( H e. CH /\ x e. ~H ) -> ( x e. ( _|_ ` H ) <-> ( ( projh ` H ) ` x ) = 0h ) )
7	6	rabbi2dva	|- ( H e. CH -> ( ~H i^i ( _|_ ` H ) ) = { x e. ~H | ( ( projh ` H ) ` x ) = 0h } )
8	5 7	eqtr3d	|- ( H e. CH -> ( _|_ ` H ) = { x e. ~H | ( ( projh ` H ) ` x ) = 0h } )