Theorem hosubfni

Description: Functionality of difference of Hilbert space operators. (Contributed by NM, 2-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hoeq.1	⊢ 𝑆 : ℋ ⟶ ℋ
	hoeq.2	⊢ 𝑇 : ℋ ⟶ ℋ
Assertion	hosubfni	⊢ ( 𝑆 −op 𝑇 ) Fn ℋ

Proof

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