Theorem pjsumi

Description: The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		pjsumt.1	$$⊢G\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjsumt.2	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
3	1 2	osumi	$$⊢G\subseteq \perp \left(H\right)\to G{+}_{\mathscr{H}}H=G{\vee }_{\mathscr{H}}H$$
4	3	fveq2d	$$⊢G\subseteq \perp \left(H\right)\to {\mathrm{proj}}_{ℎ}\left(G{+}_{\mathscr{H}}H\right)={\mathrm{proj}}_{ℎ}\left(G{\vee }_{\mathscr{H}}H\right)$$
5	4	fveq1d	$$⊢G\subseteq \perp \left(H\right)\to {\mathrm{proj}}_{ℎ}\left(G{+}_{\mathscr{H}}H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G{\vee }_{\mathscr{H}}H\right)\left(A\right)$$
6	5	adantl	$$⊢\left(A\in \mathscr{H}\wedge G\subseteq \perp \left(H\right)\right)\to {\mathrm{proj}}_{ℎ}\left(G{+}_{\mathscr{H}}H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G{\vee }_{\mathscr{H}}H\right)\left(A\right)$$
7		pjcjt2	$$⊢\left(G\in {\mathbf{C}}_{\mathscr{H}}\wedge H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \mathscr{H}\right)\to \left(G\subseteq \perp \left(H\right)\to {\mathrm{proj}}_{ℎ}\left(G{\vee }_{\mathscr{H}}H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){+}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)$$
8	1 2 7	mp3an12	$$⊢A\in \mathscr{H}\to \left(G\subseteq \perp \left(H\right)\to {\mathrm{proj}}_{ℎ}\left(G{\vee }_{\mathscr{H}}H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){+}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)$$
9	8	imp	$$⊢\left(A\in \mathscr{H}\wedge G\subseteq \perp \left(H\right)\right)\to {\mathrm{proj}}_{ℎ}\left(G{\vee }_{\mathscr{H}}H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){+}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)$$
10	6 9	eqtrd	$$⊢\left(A\in \mathscr{H}\wedge G\subseteq \perp \left(H\right)\right)\to {\mathrm{proj}}_{ℎ}\left(G{+}_{\mathscr{H}}H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){+}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)$$
11	10	ex	$$⊢A\in \mathscr{H}\to \left(G\subseteq \perp \left(H\right)\to {\mathrm{proj}}_{ℎ}\left(G{+}_{\mathscr{H}}H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){+}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)$$