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 e. ~H -> ( ( S o. T ) ` A ) e. ~H )

Proof

Step	Hyp	Ref	Expression
1		hoeq.1	|- S : ~H --> ~H
2		hoeq.2	|- T : ~H --> ~H
3	1 2	hocoi	|- ( A e. ~H -> ( ( S o. T ) ` A ) = ( S ` ( T ` A ) ) )
4	2	ffvelcdmi	|- ( A e. ~H -> ( T ` A ) e. ~H )
5	1	ffvelcdmi	|- ( ( T ` A ) e. ~H -> ( S ` ( T ` A ) ) e. ~H )
6	4 5	syl	|- ( A e. ~H -> ( S ` ( T ` A ) ) e. ~H )
7	3 6	eqeltrd	|- ( A e. ~H -> ( ( S o. T ) ` A ) e. ~H )