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Metamath Proof Explorer

Theorem dvafvsca

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

Ref Expression
Hypotheses dvafvsca.h 
|- H = ( LHyp ` K )
dvafvsca.t 
|- T = ( ( LTrn ` K ) ` W )
dvafvsca.e 
|- E = ( ( TEndo ` K ) ` W )
dvafvsca.u 
|- U = ( ( DVecA ` K ) ` W )
dvafvsca.s 
|- .x. = ( .s ` U )
Assertion dvafvsca 
|- ( ( K e. V /\ W e. H ) -> .x. = ( s e. E , f e. T |-> ( s ` f ) ) )

Proof

Step Hyp Ref Expression
1 dvafvsca.h 
 |-  H = ( LHyp ` K )
2 dvafvsca.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 dvafvsca.e 
 |-  E = ( ( TEndo ` K ) ` W )
4 dvafvsca.u 
 |-  U = ( ( DVecA ` K ) ` W )
5 dvafvsca.s 
 |-  .x. = ( .s ` U )
6 eqid 
 |-  ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
7 1 2 3 6 4 dvaset 
 |-  ( ( K e. V /\ 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. E , f e. T |-> ( s ` f ) ) >. } ) )
8 7 fveq2d 
 |-  ( ( K e. V /\ W e. H ) -> ( .s ` U ) = ( .s ` ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) ) )
9 3 fvexi 
 |-  E e. _V
10 2 fvexi 
 |-  T e. _V
11 9 10 mpoex 
 |-  ( s e. E , f e. T |-> ( s ` f ) ) e. _V
12 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. E , 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. E , f e. T |-> ( s ` f ) ) >. } )
13 12 lmodvsca 
 |-  ( ( s e. E , f e. T |-> ( s ` f ) ) e. _V -> ( s e. E , f e. T |-> ( s ` f ) ) = ( .s ` ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) ) )
14 11 13 ax-mp 
 |-  ( s e. E , f e. T |-> ( s ` f ) ) = ( .s ` ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) )
15 8 5 14 3eqtr4g 
 |-  ( ( K e. V /\ W e. H ) -> .x. = ( s e. E , f e. T |-> ( s ` f ) ) )