Theorem tendospcl

Description: Closure of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013)

	Ref	Expression
Hypotheses	tendosp.h	|- H = ( LHyp ` K )
	tendosp.t	|- T = ( ( LTrn ` K ) ` W )
	tendosp.e	|- E = ( ( TEndo ` K ) ` W )
Assertion	tendospcl	|- ( ( ( K e. V /\ W e. H ) /\ U e. E /\ F e. T ) -> ( U ` F ) e. T )

Proof

Step	Hyp	Ref	Expression
1		tendosp.h	|- H = ( LHyp ` K )
2		tendosp.t	|- T = ( ( LTrn ` K ) ` W )
3		tendosp.e	|- E = ( ( TEndo ` K ) ` W )
4	1 2 3	tendocl	|- ( ( ( K e. V /\ W e. H ) /\ U e. E /\ F e. T ) -> ( U ` F ) e. T )