Theorem pjorthcoi

Description: Composition of projections of orthogonal subspaces. Part (i)->(iia) of Theorem 27.4 of Halmos p. 45. (Contributed by NM, 6-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjco.1	$$⊢G\in {\mathbf{C}}_{\mathscr{H}}$$
	pjco.2	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	pjorthcoi	$$⊢G\subseteq \perp \left(H\right)\to {\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)=0_{\mathrm{hop}}$$

Proof

Step	Hyp	Ref	Expression
1		pjco.1	$$⊢G\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjco.2	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
3	2	pjcli	$$⊢x\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\in H$$
4	1 2	chsscon2i	$$⊢G\subseteq \perp \left(H\right)\leftrightarrow H\subseteq \perp \left(G\right)$$
5		ssel	$$⊢H\subseteq \perp \left(G\right)\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\in H\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\in \perp \left(G\right)\right)$$
6	4 5	sylbi	$$⊢G\subseteq \perp \left(H\right)\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\in H\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\in \perp \left(G\right)\right)$$
7	3 6	syl5com	$$⊢x\in \mathscr{H}\to \left(G\subseteq \perp \left(H\right)\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\in \perp \left(G\right)\right)$$
8	2	pjhcli	$$⊢x\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\in \mathscr{H}$$
9		pjoc2	$$⊢\left(G\in {\mathbf{C}}_{\mathscr{H}}\wedge {\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\in \mathscr{H}\right)\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\in \perp \left(G\right)\leftrightarrow {\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\right)=0_{ℎ}\right)$$
10	1 8 9	sylancr	$$⊢x\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\in \perp \left(G\right)\leftrightarrow {\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\right)=0_{ℎ}\right)$$
11	7 10	sylibd	$$⊢x\in \mathscr{H}\to \left(G\subseteq \perp \left(H\right)\to {\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\right)=0_{ℎ}\right)$$
12	11	impcom	$$⊢\left(G\subseteq \perp \left(H\right)\wedge x\in \mathscr{H}\right)\to {\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\right)=0_{ℎ}$$
13	1 2	pjcoi	$$⊢x\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\right)$$
14	13	adantl	$$⊢\left(G\subseteq \perp \left(H\right)\wedge x\in \mathscr{H}\right)\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(x\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\right)$$
15		ho0val	$$⊢x\in \mathscr{H}\to 0_{\mathrm{hop}}\left(x\right)=0_{ℎ}$$
16	15	adantl	$$⊢\left(G\subseteq \perp \left(H\right)\wedge x\in \mathscr{H}\right)\to 0_{\mathrm{hop}}\left(x\right)=0_{ℎ}$$
17	12 14 16	3eqtr4d	$$⊢\left(G\subseteq \perp \left(H\right)\wedge x\in \mathscr{H}\right)\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(x\right)=0_{\mathrm{hop}}\left(x\right)$$
18	17	ralrimiva	$$⊢G\subseteq \perp \left(H\right)\to \forall x\in \mathscr{H}\left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(x\right)=0_{\mathrm{hop}}\left(x\right)$$
19	1	pjfi	$$⊢{\mathrm{proj}}_{ℎ}\left(G\right):\mathscr{H}⟶\mathscr{H}$$
20	2	pjfi	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right):\mathscr{H}⟶\mathscr{H}$$
21	19 20	hocofi	$$⊢\left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right):\mathscr{H}⟶\mathscr{H}$$
22		ho0f	$$⊢0_{\mathrm{hop}}:\mathscr{H}⟶\mathscr{H}$$
23	21 22	hoeqi	$$⊢\forall x\in \mathscr{H}\left({\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)\right)\left(x\right)=0_{\mathrm{hop}}\left(x\right)\leftrightarrow {\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)=0_{\mathrm{hop}}$$
24	18 23	sylib	$$⊢G\subseteq \perp \left(H\right)\to {\mathrm{proj}}_{ℎ}\left(G\right)\circ {\mathrm{proj}}_{ℎ}\left(H\right)=0_{\mathrm{hop}}$$