Metamath Proof Explorer


Theorem hocofni

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 ( 𝑆𝑇 ) Fn ℋ

Proof

Step Hyp Ref Expression
1 hoeq.1 𝑆 : ℋ ⟶ ℋ
2 hoeq.2 𝑇 : ℋ ⟶ ℋ
3 1 2 hocofi ( 𝑆𝑇 ) : ℋ ⟶ ℋ
4 ffn ( ( 𝑆𝑇 ) : ℋ ⟶ ℋ → ( 𝑆𝑇 ) Fn ℋ )
5 3 4 ax-mp ( 𝑆𝑇 ) Fn ℋ