This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 )