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Metamath Proof Explorer

Theorem hodcl

Description: Closure of the difference of two Hilbert space operators. (Contributed by NM, 15-Nov-2002) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 hodval ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 −op 𝑇 ) ‘ 𝐴 ) = ( ( 𝑆 ‘ 𝐴 ) −ℎ ( 𝑇 ‘ 𝐴 ) ) )
2 ffvelcdm ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝑆 ‘ 𝐴 ) ∈ ℋ )
3 2 3adant2 ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝑆 ‘ 𝐴 ) ∈ ℋ )
4 ffvelcdm ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝑇 ‘ 𝐴 ) ∈ ℋ )
5 4 3adant1 ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝑇 ‘ 𝐴 ) ∈ ℋ )
6 hvsubcl ⊢ ( ( ( 𝑆 ‘ 𝐴 ) ∈ ℋ ∧ ( 𝑇 ‘ 𝐴 ) ∈ ℋ ) → ( ( 𝑆 ‘ 𝐴 ) −ℎ ( 𝑇 ‘ 𝐴 ) ) ∈ ℋ )
7 3 5 6 syl2anc ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 ‘ 𝐴 ) −ℎ ( 𝑇 ‘ 𝐴 ) ) ∈ ℋ )
8 1 7 eqeltrd ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 −op 𝑇 ) ‘ 𝐴 ) ∈ ℋ )
9 8 3expa ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 −op 𝑇 ) ‘ 𝐴 ) ∈ ℋ )