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

Proof

Step	Hyp	Ref	Expression
1		hoeq.1	$$⊢S:\mathscr{H}⟶\mathscr{H}$$
2		hoeq.2	$$⊢T:\mathscr{H}⟶\mathscr{H}$$
3		fco	$$⊢\left(S:\mathscr{H}⟶\mathscr{H}\wedge T:\mathscr{H}⟶\mathscr{H}\right)\to \left(S\circ T\right):\mathscr{H}⟶\mathscr{H}$$
4	1 2 3	mp2an	$$⊢\left(S\circ T\right):\mathscr{H}⟶\mathscr{H}$$