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Metamath Proof Explorer

Theorem dvafmulr

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

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

Proof

Step Hyp Ref Expression
1 dvafmul.h 
 |-  H = ( LHyp ` K )
2 dvafmul.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 dvafmul.e 
 |-  E = ( ( TEndo ` K ) ` W )
4 dvafmul.u 
 |-  U = ( ( DVecA ` K ) ` W )
5 dvafmul.f 
 |-  F = ( Scalar ` U )
6 dvafmul.p 
 |-  .x. = ( .r ` F )
7 eqid 
 |-  ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
8 1 7 4 5 dvasca 
 |-  ( ( K e. V /\ W e. H ) -> F = ( ( EDRing ` K ) ` W ) )
9 8 fveq2d 
 |-  ( ( K e. V /\ W e. H ) -> ( .r ` F ) = ( .r ` ( ( EDRing ` K ) ` W ) ) )
10 6 9 eqtrid 
 |-  ( ( K e. V /\ W e. H ) -> .x. = ( .r ` ( ( EDRing ` K ) ` W ) ) )
11 eqid 
 |-  ( .r ` ( ( EDRing ` K ) ` W ) ) = ( .r ` ( ( EDRing ` K ) ` W ) )
12 1 2 3 7 11 erngfmul 
 |-  ( ( K e. V /\ W e. H ) -> ( .r ` ( ( EDRing ` K ) ` W ) ) = ( s e. E , t e. E |-> ( s o. t ) ) )
13 10 12 eqtrd 
 |-  ( ( K e. V /\ W e. H ) -> .x. = ( s e. E , t e. E |-> ( s o. t ) ) )