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Metamath Proof Explorer

Theorem dvhfplusr

Description: Ring addition operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

Ref Expression
Hypotheses dvhfplusr.h 
|- H = ( LHyp ` K )
dvhfplusr.t 
|- T = ( ( LTrn ` K ) ` W )
dvhfplusr.e 
|- E = ( ( TEndo ` K ) ` W )
dvhfplusr.u 
|- U = ( ( DVecH ` K ) ` W )
dvhfplusr.f 
|- F = ( Scalar ` U )
dvhfplusr.p 
|- .+ = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
dvhfplusr.s 
|- .+b = ( +g ` F )
Assertion dvhfplusr 
|- ( ( K e. V /\ W e. H ) -> .+b = .+ )

Proof

Step Hyp Ref Expression
1 dvhfplusr.h 
 |-  H = ( LHyp ` K )
2 dvhfplusr.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 dvhfplusr.e 
 |-  E = ( ( TEndo ` K ) ` W )
4 dvhfplusr.u 
 |-  U = ( ( DVecH ` K ) ` W )
5 dvhfplusr.f 
 |-  F = ( Scalar ` U )
6 dvhfplusr.p 
 |-  .+ = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
7 dvhfplusr.s 
 |-  .+b = ( +g ` F )
8 eqid 
 |-  ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
9 1 8 4 5 dvhsca 
 |-  ( ( K e. V /\ W e. H ) -> F = ( ( EDRing ` K ) ` W ) )
10 9 fveq2d 
 |-  ( ( K e. V /\ W e. H ) -> ( +g ` F ) = ( +g ` ( ( EDRing ` K ) ` W ) ) )
11 eqid 
 |-  ( +g ` ( ( EDRing ` K ) ` W ) ) = ( +g ` ( ( EDRing ` K ) ` W ) )
12 1 2 3 8 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 ) -> ( +g ` F ) = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) )
14 13 7 6 3eqtr4g 
 |-  ( ( K e. V /\ W e. H ) -> .+b = .+ )