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Metamath Proof Explorer

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