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Metamath Proof Explorer

Theorem hoaddcl

Description: The sum of Hilbert space operators is an operator. (Contributed by NM, 21-Feb-2006) (Revised by Mario Carneiro, 16-Nov-2013) (New usage is discouraged.)

Ref Expression
Assertion hoaddcl ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑆 +op 𝑇 ) : ℋ ⟶ ℋ )

Proof

Step Hyp Ref Expression
1 ffvelcdm ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑥 ∈ ℋ ) → ( 𝑆 ‘ 𝑥 ) ∈ ℋ )
2 1 adantlr ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝑥 ∈ ℋ ) → ( 𝑆 ‘ 𝑥 ) ∈ ℋ )
3 ffvelcdm ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑥 ∈ ℋ ) → ( 𝑇 ‘ 𝑥 ) ∈ ℋ )
4 3 adantll ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝑥 ∈ ℋ ) → ( 𝑇 ‘ 𝑥 ) ∈ ℋ )
5 hvaddcl ⊢ ( ( ( 𝑆 ‘ 𝑥 ) ∈ ℋ ∧ ( 𝑇 ‘ 𝑥 ) ∈ ℋ ) → ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑇 ‘ 𝑥 ) ) ∈ ℋ )
6 2 4 5 syl2anc ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝑥 ∈ ℋ ) → ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑇 ‘ 𝑥 ) ) ∈ ℋ )
7 6 fmpttd ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑥 ∈ ℋ ↦ ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑇 ‘ 𝑥 ) ) ) : ℋ ⟶ ℋ )
8 hosmval ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑆 +op 𝑇 ) = ( 𝑥 ∈ ℋ ↦ ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑇 ‘ 𝑥 ) ) ) )
9 8 feq1d ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ( 𝑆 +op 𝑇 ) : ℋ ⟶ ℋ ↔ ( 𝑥 ∈ ℋ ↦ ( ( 𝑆 ‘ 𝑥 ) +ℎ ( 𝑇 ‘ 𝑥 ) ) ) : ℋ ⟶ ℋ ) )
10 7 9 mpbird ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑆 +op 𝑇 ) : ℋ ⟶ ℋ )