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	⊢ ( 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