Theorem pjpj0i

Description: Decomposition of a vector into projections. (Contributed by NM, 26-Oct-1999) (Revised by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

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

Proof

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