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	$$⊢S:\mathscr{H}⟶\mathscr{H}$$
	hoeq.2	$$⊢T:\mathscr{H}⟶\mathscr{H}$$
Assertion	hocofni	$$⊢\left(S\circ T\right)Fn\mathscr{H}$$

Proof

Step	Hyp	Ref	Expression
1		hoeq.1	$$⊢S:\mathscr{H}⟶\mathscr{H}$$
2		hoeq.2	$$⊢T:\mathscr{H}⟶\mathscr{H}$$
3	1 2	hocofi	$$⊢\left(S\circ T\right):\mathscr{H}⟶\mathscr{H}$$
4		ffn	$$⊢\left(S\circ T\right):\mathscr{H}⟶\mathscr{H}\to \left(S\circ T\right)Fn\mathscr{H}$$
5	3 4	ax-mp	$$⊢\left(S\circ T\right)Fn\mathscr{H}$$