Theorem pjhclii

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

	Ref	Expression
Hypotheses	pjcli.1	|- H e. CH
	pjcli.2	|- A e. ~H
Assertion	pjhclii	|- ( ( projh ` H ) ` A ) e. ~H

Proof

Step	Hyp	Ref	Expression
1		pjcli.1	|- H e. CH
2		pjcli.2	|- A e. ~H
3	1	pjhcli	|- ( A e. ~H -> ( ( projh ` H ) ` A ) e. ~H )
4	2 3	ax-mp	|- ( ( projh ` H ) ` A ) e. ~H