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 syl6bi
 |-  ( ( 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 )