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 ∈ Cℋ
	pjcli.2	⊢ 𝐴 ∈ ℋ
Assertion	pjhclii	⊢ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ

Proof

Step	Hyp	Ref	Expression
1		pjcli.1	⊢ 𝐻 ∈ Cℋ
2		pjcli.2	⊢ 𝐴 ∈ ℋ
3	1	pjhcli	⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ )
4	2 3	ax-mp	⊢ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ