Theorem pjpoi

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

	Ref	Expression
Hypotheses	pjop.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
	pjop.2	$$⊢A\in \mathscr{H}$$
Assertion	pjpoi	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)=A{-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(\perp \left(H\right)\right)\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		pjop.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjop.2	$$⊢A\in \mathscr{H}$$
3		pjpo	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \mathscr{H}\right)\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)=A{-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(\perp \left(H\right)\right)\left(A\right)$$
4	1 2 3	mp2an	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)=A{-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(\perp \left(H\right)\right)\left(A\right)$$