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 )