Theorem pjorthi

Description: Projection components on orthocomplemented subspaces are orthogonal. (Contributed by NM, 26-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjorth.1	|- A e. ~H
	pjorth.2	|- B e. ~H
Assertion	pjorthi	|- ( H e. CH -> ( ( ( projh ` H ) ` A ) .ih ( ( projh ` ( _|_ ` H ) ) ` B ) ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		pjorth.1	|- A e. ~H
2		pjorth.2	|- B e. ~H
3		chsh	|- ( H e. CH -> H e. SH )
4		axpjcl	|- ( ( H e. CH /\ A e. ~H ) -> ( ( projh ` H ) ` A ) e. H )
5	1 4	mpan2	|- ( H e. CH -> ( ( projh ` H ) ` A ) e. H )
6		choccl	|- ( H e. CH -> ( _|_ ` H ) e. CH )
7		axpjcl	|- ( ( ( _|_ ` H ) e. CH /\ B e. ~H ) -> ( ( projh ` ( _|_ ` H ) ) ` B ) e. ( _|_ ` H ) )
8	6 2 7	sylancl	|- ( H e. CH -> ( ( projh ` ( _|_ ` H ) ) ` B ) e. ( _|_ ` H ) )
9	5 8	jca	|- ( H e. CH -> ( ( ( projh ` H ) ` A ) e. H /\ ( ( projh ` ( _|_ ` H ) ) ` B ) e. ( _|_ ` H ) ) )
10		shocorth	|- ( H e. SH -> ( ( ( ( projh ` H ) ` A ) e. H /\ ( ( projh ` ( _|_ ` H ) ) ` B ) e. ( _|_ ` H ) ) -> ( ( ( projh ` H ) ` A ) .ih ( ( projh ` ( _|_ ` H ) ) ` B ) ) = 0 ) )
11	3 9 10	sylc	|- ( H e. CH -> ( ( ( projh ` H ) ` A ) .ih ( ( projh ` ( _|_ ` H ) ) ` B ) ) = 0 )