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Metamath Proof Explorer

Theorem dvasca

Description: The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom W ). (Contributed by NM, 22-Jun-2014)

Ref Expression
Hypotheses dvasca.h 
|- H = ( LHyp ` K )
dvasca.d 
|- D = ( ( EDRing ` K ) ` W )
dvasca.u 
|- U = ( ( DVecA ` K ) ` W )
dvasca.f 
|- F = ( Scalar ` U )
Assertion dvasca 
|- ( ( K e. X /\ W e. H ) -> F = D )

Proof

Step Hyp Ref Expression
1 dvasca.h 
 |-  H = ( LHyp ` K )
2 dvasca.d 
 |-  D = ( ( EDRing ` K ) ` W )
3 dvasca.u 
 |-  U = ( ( DVecA ` K ) ` W )
4 dvasca.f 
 |-  F = ( Scalar ` U )
5 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
6 eqid 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
7 1 5 6 2 3 dvaset 
 |-  ( ( K e. X /\ W e. H ) -> U = ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( LTrn ` K ) ` W ) |-> ( s ` f ) ) >. } ) )
8 7 fveq2d 
 |-  ( ( K e. X /\ W e. H ) -> ( Scalar ` U ) = ( Scalar ` ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( LTrn ` K ) ` W ) |-> ( s ` f ) ) >. } ) ) )
9 2 fvexi 
 |-  D e. _V
10 eqid 
 |-  ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( LTrn ` K ) ` W ) |-> ( s ` f ) ) >. } ) = ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( LTrn ` K ) ` W ) |-> ( s ` f ) ) >. } )
11 10 lmodsca 
 |-  ( D e. _V -> D = ( Scalar ` ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( LTrn ` K ) ` W ) |-> ( s ` f ) ) >. } ) ) )
12 9 11 ax-mp 
 |-  D = ( Scalar ` ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` W ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` W ) , g e. ( ( LTrn ` K ) ` W ) |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` W ) , f e. ( ( LTrn ` K ) ` W ) |-> ( s ` f ) ) >. } ) )
13 8 4 12 3eqtr4g 
 |-  ( ( K e. X /\ W e. H ) -> F = D )