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Metamath Proof Explorer

Theorem hoscli

Description: Closure of Hilbert space operator sum. (Contributed by NM, 12-Nov-2000) (New usage is discouraged.)

Ref Expression
Hypotheses hoeq.1 ⊢ 𝑆 : ℋ ⟶ ℋ
hoeq.2 ⊢ 𝑇 : ℋ ⟶ ℋ
Assertion hoscli ( 𝐴 ∈ ℋ → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) ∈ ℋ )

Proof

Step Hyp Ref Expression
1 hoeq.1 ⊢ 𝑆 : ℋ ⟶ ℋ
2 hoeq.2 ⊢ 𝑇 : ℋ ⟶ ℋ
3 hoscl ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) ∈ ℋ )
4 1 2 3 mpanl12 ⊢ ( 𝐴 ∈ ℋ → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) ∈ ℋ )