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Metamath Proof Explorer

Theorem dvafplusg

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 dvafplus.h 
|- H = ( LHyp ` K )
dvafplus.t 
|- T = ( ( LTrn ` K ) ` W )
dvafplus.e 
|- E = ( ( TEndo ` K ) ` W )
dvafplus.u 
|- U = ( ( DVecA ` K ) ` W )
dvafplus.f 
|- F = ( Scalar ` U )
dvafplus.p 
|- .+ = ( +g ` F )
Assertion dvafplusg 
|- ( ( K e. V /\ W e. H ) -> .+ = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) )

Proof

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