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Metamath Proof Explorer

Theorem dvavbase

Description: The vectors (vector base set) of the constructed partial vector space A are all translations (for a fiducial co-atom W ). (Contributed by NM, 9-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

Ref Expression
Hypotheses dvavbase.h 
|- H = ( LHyp ` K )
dvavbase.t 
|- T = ( ( LTrn ` K ) ` W )
dvavbase.u 
|- U = ( ( DVecA ` K ) ` W )
dvavbase.v 
|- V = ( Base ` U )
Assertion dvavbase 
|- ( ( K e. X /\ W e. H ) -> V = T )

Proof

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