Metamath Proof Explorer
Description: Functionality of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypotheses |
hoeq.1 |
⊢ 𝑆 : ℋ ⟶ ℋ |
|
|
hoeq.2 |
⊢ 𝑇 : ℋ ⟶ ℋ |
|
Assertion |
hocofni |
⊢ ( 𝑆 ∘ 𝑇 ) Fn ℋ |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
hoeq.1 |
⊢ 𝑆 : ℋ ⟶ ℋ |
2 |
|
hoeq.2 |
⊢ 𝑇 : ℋ ⟶ ℋ |
3 |
1 2
|
hocofi |
⊢ ( 𝑆 ∘ 𝑇 ) : ℋ ⟶ ℋ |
4 |
|
ffn |
⊢ ( ( 𝑆 ∘ 𝑇 ) : ℋ ⟶ ℋ → ( 𝑆 ∘ 𝑇 ) Fn ℋ ) |
5 |
3 4
|
ax-mp |
⊢ ( 𝑆 ∘ 𝑇 ) Fn ℋ |