Theorem hocofi

Description: Mapping of composition of Hilbert space operators. (Contributed by NM, 14-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hoeq.1	⊢ 𝑆 : ℋ ⟶ ℋ
	hoeq.2	⊢ 𝑇 : ℋ ⟶ ℋ
Assertion	hocofi	⊢ ( 𝑆 ∘ 𝑇 ) : ℋ ⟶ ℋ

Proof

Step	Hyp	Ref	Expression
1		hoeq.1	⊢ 𝑆 : ℋ ⟶ ℋ
2		hoeq.2	⊢ 𝑇 : ℋ ⟶ ℋ
3		fco	⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑆 ∘ 𝑇 ) : ℋ ⟶ ℋ )
4	1 2 3	mp2an	⊢ ( 𝑆 ∘ 𝑇 ) : ℋ ⟶ ℋ