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Metamath Proof Explorer

Theorem dvaplusg

Description: Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013)

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 dvaplusg 
|- ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ S e. E ) ) -> ( R .+ S ) = ( f e. T |-> ( ( R ` f ) o. ( S ` 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 1 2 3 4 5 6 dvafplusg 
 |-  ( ( K e. V /\ W e. H ) -> .+ = ( s e. E , t e. E |-> ( g e. T |-> ( ( s ` g ) o. ( t ` g ) ) ) ) )
8 7 oveqd 
 |-  ( ( K e. V /\ W e. H ) -> ( R .+ S ) = ( R ( s e. E , t e. E |-> ( g e. T |-> ( ( s ` g ) o. ( t ` g ) ) ) ) S ) )
9 eqid 
 |-  ( s e. E , t e. E |-> ( g e. T |-> ( ( s ` g ) o. ( t ` g ) ) ) ) = ( s e. E , t e. E |-> ( g e. T |-> ( ( s ` g ) o. ( t ` g ) ) ) )
10 9 2 tendopl 
 |-  ( ( R e. E /\ S e. E ) -> ( R ( s e. E , t e. E |-> ( g e. T |-> ( ( s ` g ) o. ( t ` g ) ) ) ) S ) = ( f e. T |-> ( ( R ` f ) o. ( S ` f ) ) ) )
11 8 10 sylan9eq 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ S e. E ) ) -> ( R .+ S ) = ( f e. T |-> ( ( R ` f ) o. ( S ` f ) ) ) )