Metamath Proof Explorer


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 )