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 : ~H --> ~H
	hoeq.2	|- T : ~H --> ~H
Assertion	hocofni	|- ( S o. T ) Fn ~H

Proof

Step	Hyp	Ref	Expression
1		hoeq.1	|- S : ~H --> ~H
2		hoeq.2	|- T : ~H --> ~H
3	1 2	hocofi	|- ( S o. T ) : ~H --> ~H
4		ffn	|- ( ( S o. T ) : ~H --> ~H -> ( S o. T ) Fn ~H )
5	3 4	ax-mp	|- ( S o. T ) Fn ~H