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Metamath Proof Explorer
Description: Multiplicative property of a linear Hilbert space operator.
(Contributed by NM, 11-May-2005) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypothesis |
lnopl.1 |
⊢ 𝑇 ∈ LinOp |
|
Assertion |
lnopmuli |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
lnopl.1 |
⊢ 𝑇 ∈ LinOp |
| 2 |
|
lnopmul |
⊢ ( ( 𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ) |
| 3 |
1 2
|
mp3an1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ) |