Theorem pjinormi

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, 2-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Hypothesis	pjadjt.1	⊢ 𝐻 ∈ Cℋ
Assertion	pjinormi	⊢ ( 𝐴 ∈ ℋ → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ·ih 𝐴 ) = ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ↑ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		pjadjt.1	⊢ 𝐻 ∈ Cℋ
2		fveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( ( projℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
3		id	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) )
4	2 3	oveq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ·ih 𝐴 ) = ( ( ( projℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
5		2fveq3	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
6	5	oveq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ↑ 2 ) = ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ↑ 2 ) )
7	4 6	eqeq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ·ih 𝐴 ) = ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ↑ 2 ) ↔ ( ( ( projℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ↑ 2 ) ) )
8		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
9	1 8	pjinormii	⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ↑ 2 )
10	7 9	dedth	⊢ ( 𝐴 ∈ ℋ → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ·ih 𝐴 ) = ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ↑ 2 ) )