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Metamath Proof Explorer

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 ∈ C_ℋ
		pjidm.2	⊢ 𝐴 ∈ ℋ
	Assertion	pjinormii	⊢ ( ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ·_ih 𝐴 ) = ( ( norm_ℎ ‘ ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ↑ 2 )

Proof

Step	Hyp	Ref	Expression
1		pjidm.1	⊢ 𝐻 ∈ C_ℋ
2		pjidm.2	⊢ 𝐴 ∈ ℋ
3	1 2	pjhclii	⊢ ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ
4	3	normsqi	⊢ ( ( norm_ℎ ‘ ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ↑ 2 ) = ( ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ·_ih ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) )
5	1 3 2	pjadjii	⊢ ( ( ( proj_ℎ ‘ 𝐻 ) ‘ ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ·_ih 𝐴 ) = ( ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ·_ih ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) )
6	1 2	pjidmi	⊢ ( ( proj_ℎ ‘ 𝐻 ) ‘ ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 )
7	6	oveq1i	⊢ ( ( ( proj_ℎ ‘ 𝐻 ) ‘ ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ·_ih 𝐴 ) = ( ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ·_ih 𝐴 )
8	4 5 7	3eqtr2ri	⊢ ( ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ·_ih 𝐴 ) = ( ( norm_ℎ ‘ ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ↑ 2 )