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Metamath Proof Explorer

Theorem pjhcli

Description: Closure of a projection in Hilbert space. (Contributed by NM, 7-Oct-2000) (New usage is discouraged.)

Ref Expression
Hypothesis pjcl.1 ⊢ 𝐻 ∈ Cℋ
Assertion pjhcli ( 𝐴 ∈ ℋ → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ )

Proof

Step Hyp Ref Expression
1 pjcl.1 ⊢ 𝐻 ∈ Cℋ
2 pjhcl ⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ )
3 1 2 mpan ⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ )