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Metamath Proof Explorer
Description: A linear Hilbert space operator is a Hilbert space operator.
(Contributed by NM, 18-Jan-2006) (New usage is discouraged.)
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|
Ref |
Expression |
|
Assertion |
lnopf |
⊢ ( 𝑇 ∈ LinOp → 𝑇 : ℋ ⟶ ℋ ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ellnop |
⊢ ( 𝑇 ∈ LinOp ↔ ( 𝑇 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℋ ∀ 𝑧 ∈ ℋ ( 𝑇 ‘ ( ( 𝑥 ·ℎ 𝑦 ) +ℎ 𝑧 ) ) = ( ( 𝑥 ·ℎ ( 𝑇 ‘ 𝑦 ) ) +ℎ ( 𝑇 ‘ 𝑧 ) ) ) ) |
| 2 |
1
|
simplbi |
⊢ ( 𝑇 ∈ LinOp → 𝑇 : ℋ ⟶ ℋ ) |