Theorem pjopi

Description: Orthocomplement projection in terms of projection. (Contributed by NM, 31-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjop.1	⊢ 𝐻 ∈ Cℋ
	pjop.2	⊢ 𝐴 ∈ ℋ
Assertion	pjopi	⊢ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = ( 𝐴 −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		pjop.1	⊢ 𝐻 ∈ Cℋ
2		pjop.2	⊢ 𝐴 ∈ ℋ
3		pjop	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = ( 𝐴 −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) )
4	1 2 3	mp2an	⊢ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = ( 𝐴 −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) )