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Metamath Proof Explorer

Theorem tendof

Description: Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013)

Ref Expression
Hypotheses tendof.h 
|- H = ( LHyp ` K )
tendof.t 
|- T = ( ( LTrn ` K ) ` W )
tendof.e 
|- E = ( ( TEndo ` K ) ` W )
Assertion tendof 
|- ( ( ( K e. V /\ W e. H ) /\ S e. E ) -> S : T --> T )

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 eqid 
 |-  ( le ` K ) = ( le ` K )
5 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
6 4 1 2 5 3 istendo 
 |-  ( ( K e. V /\ W e. H ) -> ( S e. E <-> ( S : T --> T /\ A. f e. T A. g e. T ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) /\ A. f e. T ( ( ( trL ` K ) ` W ) ` ( S ` f ) ) ( le ` K ) ( ( ( trL ` K ) ` W ) ` f ) ) ) )
7 simp1 
 |-  ( ( S : T --> T /\ A. f e. T A. g e. T ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) /\ A. f e. T ( ( ( trL ` K ) ` W ) ` ( S ` f ) ) ( le ` K ) ( ( ( trL ` K ) ` W ) ` f ) ) -> S : T --> T )
8 6 7 biimtrdi 
 |-  ( ( K e. V /\ W e. H ) -> ( S e. E -> S : T --> T ) )
9 8 imp 
 |-  ( ( ( K e. V /\ W e. H ) /\ S e. E ) -> S : T --> T )