Theorem pjpji

Description: Decomposition of a vector into projections. (Contributed by NM, 6-Nov-1999) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		pjpj.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjpj.2	$$⊢A\in \mathscr{H}$$
3	1 2	pjpj0i	$$⊢A={\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right){+}_{ℎ}{\mathrm{proj}}_{ℎ}\left(\perp \left(H\right)\right)\left(A\right)$$