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Metamath Proof Explorer

Theorem istendod

Description: Deduce the predicate "is 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 )
istendod.1 
|- ( ph -> ( K e. V /\ W e. H ) )
istendod.2 
|- ( ph -> S : T --> T )
istendod.3 
|- ( ( ph /\ f e. T /\ g e. T ) -> ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) )
istendod.4 
|- ( ( ph /\ f e. T ) -> ( R ` ( S ` f ) ) .<_ ( R ` f ) )
Assertion istendod 
|- ( ph -> S e. E )

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 istendod.1 
 |-  ( ph -> ( K e. V /\ W e. H ) )
7 istendod.2 
 |-  ( ph -> S : T --> T )
8 istendod.3 
 |-  ( ( ph /\ f e. T /\ g e. T ) -> ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) )
9 istendod.4 
 |-  ( ( ph /\ f e. T ) -> ( R ` ( S ` f ) ) .<_ ( R ` f ) )
10 8 3expb 
 |-  ( ( ph /\ ( f e. T /\ g e. T ) ) -> ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) )
11 10 ralrimivva 
 |-  ( ph -> A. f e. T A. g e. T ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) )
12 9 ralrimiva 
 |-  ( ph -> A. f e. T ( R ` ( S ` f ) ) .<_ ( R ` f ) )
13 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 ) ) ) )
14 6 13 syl 
 |-  ( ph -> ( 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 ) ) ) )
15 7 11 12 14 mpbir3and 
 |-  ( ph -> S e. E )