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Metamath Proof Explorer

Theorem tendotp

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

Ref Expression
Hypotheses tendoset.l 
|- .<_ = ( le ` K )
tendoset.h 
|- H = ( LHyp ` K )
tendoset.t 
|- T = ( ( LTrn ` K ) ` W )
tendoset.r 
|- R = ( ( trL ` K ) ` W )
tendoset.e 
|- E = ( ( TEndo ` K ) ` W )
Assertion tendotp 
|- ( ( ( K e. V /\ W e. H ) /\ S e. E /\ F e. T ) -> ( R ` ( S ` F ) ) .<_ ( R ` F ) )

Proof

Step Hyp Ref Expression
1 tendoset.l 
 |-  .<_ = ( le ` K )
2 tendoset.h 
 |-  H = ( LHyp ` K )
3 tendoset.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 tendoset.r 
 |-  R = ( ( trL ` K ) ` W )
5 tendoset.e 
 |-  E = ( ( TEndo ` K ) ` W )
6 1 2 3 4 5 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 ( R ` ( S ` f ) ) .<_ ( R ` f ) ) ) )
7 2fveq3 
 |-  ( f = F -> ( R ` ( S ` f ) ) = ( R ` ( S ` F ) ) )
8 fveq2 
 |-  ( f = F -> ( R ` f ) = ( R ` F ) )
9 7 8 breq12d 
 |-  ( f = F -> ( ( R ` ( S ` f ) ) .<_ ( R ` f ) <-> ( R ` ( S ` F ) ) .<_ ( R ` F ) ) )
10 9 rspccv 
 |-  ( A. f e. T ( R ` ( S ` f ) ) .<_ ( R ` f ) -> ( F e. T -> ( R ` ( S ` F ) ) .<_ ( R ` F ) ) )
11 10 3ad2ant3 
 |-  ( ( S : T --> T /\ A. f e. T A. g e. T ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) /\ A. f e. T ( R ` ( S ` f ) ) .<_ ( R ` f ) ) -> ( F e. T -> ( R ` ( S ` F ) ) .<_ ( R ` F ) ) )
12 6 11 biimtrdi 
 |-  ( ( K e. V /\ W e. H ) -> ( S e. E -> ( F e. T -> ( R ` ( S ` F ) ) .<_ ( R ` F ) ) ) )
13 12 3imp 
 |-  ( ( ( K e. V /\ W e. H ) /\ S e. E /\ F e. T ) -> ( R ` ( S ` F ) ) .<_ ( R ` F ) )