Theorem pjcohocli

Description: Closure of composition of projection and Hilbert space operator. (Contributed by NM, 3-Dec-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjcohocl.1	|- H e. CH
	pjcohocl.2	|- T : ~H --> ~H
Assertion	pjcohocli	|- ( A e. ~H -> ( ( ( projh ` H ) o. T ) ` A ) e. H )

Proof

Step	Hyp	Ref	Expression
1		pjcohocl.1	|- H e. CH
2		pjcohocl.2	|- T : ~H --> ~H
3	1	pjfi	|- ( projh ` H ) : ~H --> ~H
4	3 2	hocoi	|- ( A e. ~H -> ( ( ( projh ` H ) o. T ) ` A ) = ( ( projh ` H ) ` ( T ` A ) ) )
5	2	ffvelrni	|- ( A e. ~H -> ( T ` A ) e. ~H )
6	1	pjcli	|- ( ( T ` A ) e. ~H -> ( ( projh ` H ) ` ( T ` A ) ) e. H )
7	5 6	syl	|- ( A e. ~H -> ( ( projh ` H ) ` ( T ` A ) ) e. H )
8	4 7	eqeltrd	|- ( A e. ~H -> ( ( ( projh ` H ) o. T ) ` A ) e. H )