Theorem hococli

Description: Closure 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	hococli	⊢ ( 𝐴 ∈ ℋ → ( ( 𝑆 ∘ 𝑇 ) ‘ 𝐴 ) ∈ ℋ )

Proof

Step	Hyp	Ref	Expression
1		hoeq.1	⊢ 𝑆 : ℋ ⟶ ℋ
2		hoeq.2	⊢ 𝑇 : ℋ ⟶ ℋ
3	1 2	hocoi	⊢ ( 𝐴 ∈ ℋ → ( ( 𝑆 ∘ 𝑇 ) ‘ 𝐴 ) = ( 𝑆 ‘ ( 𝑇 ‘ 𝐴 ) ) )
4	2	ffvelrni	⊢ ( 𝐴 ∈ ℋ → ( 𝑇 ‘ 𝐴 ) ∈ ℋ )
5	1	ffvelrni	⊢ ( ( 𝑇 ‘ 𝐴 ) ∈ ℋ → ( 𝑆 ‘ ( 𝑇 ‘ 𝐴 ) ) ∈ ℋ )
6	4 5	syl	⊢ ( 𝐴 ∈ ℋ → ( 𝑆 ‘ ( 𝑇 ‘ 𝐴 ) ) ∈ ℋ )
7	3 6	eqeltrd	⊢ ( 𝐴 ∈ ℋ → ( ( 𝑆 ∘ 𝑇 ) ‘ 𝐴 ) ∈ ℋ )