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Metamath Proof Explorer

Theorem hoscl

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

Ref Expression
Assertion hoscl ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) ∈ ℋ )

Proof

Step Hyp Ref Expression
1 hosval ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑇 ‘ 𝐴 ) ) )
2 1 3expa ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑇 ‘ 𝐴 ) ) )
3 ffvelcdm ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝑆 ‘ 𝐴 ) ∈ ℋ )
4 ffvelcdm ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝑇 ‘ 𝐴 ) ∈ ℋ )
5 3 4 anim12i ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) ∧ ( 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) ) → ( ( 𝑆 ‘ 𝐴 ) ∈ ℋ ∧ ( 𝑇 ‘ 𝐴 ) ∈ ℋ ) )
6 5 anandirs ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 ‘ 𝐴 ) ∈ ℋ ∧ ( 𝑇 ‘ 𝐴 ) ∈ ℋ ) )
7 hvaddcl ⊢ ( ( ( 𝑆 ‘ 𝐴 ) ∈ ℋ ∧ ( 𝑇 ‘ 𝐴 ) ∈ ℋ ) → ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑇 ‘ 𝐴 ) ) ∈ ℋ )
8 6 7 syl ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑇 ‘ 𝐴 ) ) ∈ ℋ )
9 2 8 eqeltrd ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 +op 𝑇 ) ‘ 𝐴 ) ∈ ℋ )