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Metamath Proof Explorer

Theorem dvafvadd

Description: The vector sum operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

Ref Expression
Hypotheses dvafvadd.h 
|- H = ( LHyp ` K )
dvafvadd.t 
|- T = ( ( LTrn ` K ) ` W )
dvafvadd.u 
|- U = ( ( DVecA ` K ) ` W )
dvafvadd.v 
|- .+ = ( +g ` U )
Assertion dvafvadd 
|- ( ( K e. X /\ W e. H ) -> .+ = ( f e. T , g e. T |-> ( f o. g ) ) )

Proof

Step Hyp Ref Expression
1 dvafvadd.h 
 |-  H = ( LHyp ` K )
2 dvafvadd.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 dvafvadd.u 
 |-  U = ( ( DVecA ` K ) ` W )
4 dvafvadd.v 
 |-  .+ = ( +g ` 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 ) -> ( +g ` U ) = ( +g ` ( { <. ( 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 9 9 mpoex 
 |-  ( f e. T , g e. T |-> ( f o. g ) ) e. _V
11 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 ) ) >. } )
12 11 lmodplusg 
 |-  ( ( f e. T , g e. T |-> ( f o. g ) ) e. _V -> ( f e. T , g e. T |-> ( f o. g ) ) = ( +g ` ( { <. ( 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 10 12 ax-mp 
 |-  ( f e. T , g e. T |-> ( f o. g ) ) = ( +g ` ( { <. ( 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 ) ) >. } ) )
14 8 4 13 3eqtr4g 
 |-  ( ( K e. X /\ W e. H ) -> .+ = ( f e. T , g e. T |-> ( f o. g ) ) )