Theorem pjoi0i

Description: The inner product of projections on orthogonal subspaces vanishes. (Contributed by NM, 1-Nov-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoi0.1	|- G e. CH
	pjoi0.2	|- H e. CH
	pjoi0.3	|- A e. ~H
Assertion	pjoi0i	|- ( G C_ ( _|_ ` H ) -> ( ( ( projh ` G ) ` A ) .ih ( ( projh ` H ) ` A ) ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		pjoi0.1	|- G e. CH
2		pjoi0.2	|- H e. CH
3		pjoi0.3	|- A e. ~H
4	1 2 3	3pm3.2i	|- ( G e. CH /\ H e. CH /\ A e. ~H )
5		pjoi0	|- ( ( ( G e. CH /\ H e. CH /\ A e. ~H ) /\ G C_ ( _|_ ` H ) ) -> ( ( ( projh ` G ) ` A ) .ih ( ( projh ` H ) ` A ) ) = 0 )
6	4 5	mpan	|- ( G C_ ( _|_ ` H ) -> ( ( ( projh ` G ) ` A ) .ih ( ( projh ` H ) ` A ) ) = 0 )