Theorem pjoccl

Description: The part of a vector that belongs to the orthocomplemented space. (Contributed by NM, 11-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	pjoccl	|- ( ( H e. CH /\ A e. ~H ) -> ( A -h ( ( projh ` H ) ` A ) ) e. ( _|_ ` H ) )

Proof

Step	Hyp	Ref	Expression
1		pjop	|- ( ( H e. CH /\ A e. ~H ) -> ( ( projh ` ( _|_ ` H ) ) ` A ) = ( A -h ( ( projh ` H ) ` A ) ) )
2		choccl	|- ( H e. CH -> ( _|_ ` H ) e. CH )
3		axpjcl	|- ( ( ( _|_ ` H ) e. CH /\ A e. ~H ) -> ( ( projh ` ( _|_ ` H ) ) ` A ) e. ( _|_ ` H ) )
4	2 3	sylan	|- ( ( H e. CH /\ A e. ~H ) -> ( ( projh ` ( _|_ ` H ) ) ` A ) e. ( _|_ ` H ) )
5	1 4	eqeltrrd	|- ( ( H e. CH /\ A e. ~H ) -> ( A -h ( ( projh ` H ) ` A ) ) e. ( _|_ ` H ) )