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Metamath Proof Explorer

Theorem erngbase-rN

Description: The base set of the division ring on trace-preserving endomorphisms is the set of all trace-preserving endomorphisms (for a fiducial co-atom W ). (Contributed by NM, 9-Jun-2013) (New usage is discouraged.)

Ref Expression
Hypotheses erngset.h-r 
|- H = ( LHyp ` K )
erngset.t-r 
|- T = ( ( LTrn ` K ) ` W )
erngset.e-r 
|- E = ( ( TEndo ` K ) ` W )
erngset.d-r 
|- D = ( ( EDRingR ` K ) ` W )
erng.c-r 
|- C = ( Base ` D )
Assertion erngbase-rN 
|- ( ( K e. V /\ W e. H ) -> C = E )

Proof

Step Hyp Ref Expression
1 erngset.h-r 
 |-  H = ( LHyp ` K )
2 erngset.t-r 
 |-  T = ( ( LTrn ` K ) ` W )
3 erngset.e-r 
 |-  E = ( ( TEndo ` K ) ` W )
4 erngset.d-r 
 |-  D = ( ( EDRingR ` K ) ` W )
5 erng.c-r 
 |-  C = ( Base ` D )
6 1 2 3 4 erngset-rN 
 |-  ( ( K e. V /\ W e. H ) -> D = { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } )
7 6 fveq2d 
 |-  ( ( K e. V /\ W e. H ) -> ( Base ` D ) = ( Base ` { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } ) )
8 3 fvexi 
 |-  E e. _V
9 eqid 
 |-  { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } = { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. }
10 9 rngbase 
 |-  ( E e. _V -> E = ( Base ` { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } ) )
11 8 10 ax-mp 
 |-  E = ( Base ` { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } )
12 7 5 11 3eqtr4g 
 |-  ( ( K e. V /\ W e. H ) -> C = E )