Theorem tendo0cl

Description: The additive identity is a trace-preserving endormorphism. (Contributed by NM, 12-Jun-2013)

	Ref	Expression
Hypotheses	tendo0.b	|- B = ( Base ` K )
	tendo0.h	|- H = ( LHyp ` K )
	tendo0.t	|- T = ( ( LTrn ` K ) ` W )
	tendo0.e	|- E = ( ( TEndo ` K ) ` W )
	tendo0.o	|- O = ( f e. T |-> ( _I |` B ) )
Assertion	tendo0cl	|- ( ( K e. HL /\ W e. H ) -> O e. E )

Proof

Step	Hyp	Ref	Expression
1		tendo0.b	|- B = ( Base ` K )
2		tendo0.h	|- H = ( LHyp ` K )
3		tendo0.t	|- T = ( ( LTrn ` K ) ` W )
4		tendo0.e	|- E = ( ( TEndo ` K ) ` W )
5		tendo0.o	|- O = ( f e. T |-> ( _I |` B ) )
6		eqid	|- ( le ` K ) = ( le ` K )
7		eqid	|- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
8		id	|- ( ( K e. HL /\ W e. H ) -> ( K e. HL /\ W e. H ) )
9	1 2 3	idltrn	|- ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. T )
10	9	adantr	|- ( ( ( K e. HL /\ W e. H ) /\ g e. T ) -> ( _I |` B ) e. T )
11	5	tendo0cbv	|- O = ( g e. T |-> ( _I |` B ) )
12	10 11	fmptd	|- ( ( K e. HL /\ W e. H ) -> O : T --> T )
13	1 2 3 4 5	tendo0co2	|- ( ( ( K e. HL /\ W e. H ) /\ g e. T /\ h e. T ) -> ( O ` ( g o. h ) ) = ( ( O ` g ) o. ( O ` h ) ) )
14	1 2 3 4 5 6 7	tendo0tp	|- ( ( ( K e. HL /\ W e. H ) /\ g e. T ) -> ( ( ( trL ` K ) ` W ) ` ( O ` g ) ) ( le ` K ) ( ( ( trL ` K ) ` W ) ` g ) )
15	6 2 3 7 4 8 12 13 14	istendod	|- ( ( K e. HL /\ W e. H ) -> O e. E )