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 ∈ Cℋ
	pjidm.2	⊢ 𝐴 ∈ ℋ
Assertion	pjinormii	⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ·ih 𝐴 ) = ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ↑ 2 )

Proof

Step	Hyp	Ref	Expression
1		pjidm.1	⊢ 𝐻 ∈ Cℋ
2		pjidm.2	⊢ 𝐴 ∈ ℋ
3	1 2	pjhclii	⊢ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 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 )