Theorem pjinormii

Description: The inner product of a projection and its argument is the square of the norm of the projection. Remark in Halmos p. 44. (Contributed by NM, 13-Aug-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjidm.1	|- H e. CH
	pjidm.2	|- A e. ~H
Assertion	pjinormii	|- ( ( ( projh ` H ) ` A ) .ih A ) = ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 )

Proof

Step	Hyp	Ref	Expression
1		pjidm.1	|- H e. CH
2		pjidm.2	|- A e. ~H
3	1 2	pjhclii	|- ( ( projh ` H ) ` A ) e. ~H
4	3	normsqi	|- ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) = ( ( ( projh ` H ) ` A ) .ih ( ( projh ` H ) ` A ) )
5	1 3 2	pjadjii	|- ( ( ( projh ` H ) ` ( ( projh ` H ) ` A ) ) .ih A ) = ( ( ( projh ` H ) ` A ) .ih ( ( projh ` H ) ` A ) )
6	1 2	pjidmi	|- ( ( projh ` H ) ` ( ( projh ` H ) ` A ) ) = ( ( projh ` H ) ` A )
7	6	oveq1i	|- ( ( ( projh ` H ) ` ( ( projh ` H ) ` A ) ) .ih A ) = ( ( ( projh ` H ) ` A ) .ih A )
8	4 5 7	3eqtr2ri	|- ( ( ( projh ` H ) ` A ) .ih A ) = ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 )