Theorem pjcocli

Description: Closure of composition of projections. (Contributed by NM, 29-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjco.1	|- G e. CH
	pjco.2	|- H e. CH
Assertion	pjcocli	|- ( A e. ~H -> ( ( ( projh ` G ) o. ( projh ` H ) ) ` A ) e. G )

Proof

Step	Hyp	Ref	Expression
1		pjco.1	|- G e. CH
2		pjco.2	|- H e. CH
3	1 2	pjcoi	|- ( A e. ~H -> ( ( ( projh ` G ) o. ( projh ` H ) ) ` A ) = ( ( projh ` G ) ` ( ( projh ` H ) ` A ) ) )
4	2	pjhcli	|- ( A e. ~H -> ( ( projh ` H ) ` A ) e. ~H )
5	1	pjcli	|- ( ( ( projh ` H ) ` A ) e. ~H -> ( ( projh ` G ) ` ( ( projh ` H ) ` A ) ) e. G )
6	4 5	syl	|- ( A e. ~H -> ( ( projh ` G ) ` ( ( projh ` H ) ` A ) ) e. G )
7	3 6	eqeltrd	|- ( A e. ~H -> ( ( ( projh ` G ) o. ( projh ` H ) ) ` A ) e. G )