Theorem tendo1mul

Description: Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013)

	Ref	Expression
Hypotheses	tendof.h	|- H = ( LHyp ` K )
	tendof.t	|- T = ( ( LTrn ` K ) ` W )
	tendof.e	|- E = ( ( TEndo ` K ) ` W )
Assertion	tendo1mul	|- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( ( _I |` T ) o. U ) = U )

Proof

Step	Hyp	Ref	Expression
1		tendof.h	|- H = ( LHyp ` K )
2		tendof.t	|- T = ( ( LTrn ` K ) ` W )
3		tendof.e	|- E = ( ( TEndo ` K ) ` W )
4	1 2 3	tendof	|- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> U : T --> T )
5		fcoi2	|- ( U : T --> T -> ( ( _I |` T ) o. U ) = U )
6	4 5	syl	|- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( ( _I |` T ) o. U ) = U )