Theorem pjhcl

Description: Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		chss	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to H\subseteq \mathscr{H}$$
2	1	adantr	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \mathscr{H}\right)\to H\subseteq \mathscr{H}$$
3		axpjcl	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \mathscr{H}\right)\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in H$$
4	2 3	sseldd	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \mathscr{H}\right)\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in \mathscr{H}$$