Theorem axpjpj

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

	Ref	Expression
Assertion	axpjpj	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \mathscr{H}\right)\to A={\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right){+}_{ℎ}{\mathrm{proj}}_{ℎ}\left(\perp \left(H\right)\right)\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \mathscr{H}\right)\to H\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjhth	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to H{+}_{\mathscr{H}}\perp \left(H\right)=\mathscr{H}$$
3	2	eleq2d	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to \left(A\in \left(H{+}_{\mathscr{H}}\perp \left(H\right)\right)\leftrightarrow A\in \mathscr{H}\right)$$
4	3	biimpar	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \mathscr{H}\right)\to A\in \left(H{+}_{\mathscr{H}}\perp \left(H\right)\right)$$
5	1 4	pjpjpre	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \mathscr{H}\right)\to A={\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right){+}_{ℎ}{\mathrm{proj}}_{ℎ}\left(\perp \left(H\right)\right)\left(A\right)$$