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Metamath Proof Explorer

Theorem dvaplusgv

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 dvaplusgv 
|- ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ S e. E /\ G e. T ) ) -> ( ( R .+ S ) ` G ) = ( ( R ` G ) o. ( S ` G ) ) )

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 dvaplusg 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ S e. E ) ) -> ( R .+ S ) = ( f e. T |-> ( ( R ` f ) o. ( S ` f ) ) ) )
8 7 fveq1d 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ S e. E ) ) -> ( ( R .+ S ) ` G ) = ( ( f e. T |-> ( ( R ` f ) o. ( S ` f ) ) ) ` G ) )
9 8 3adantr3 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ S e. E /\ G e. T ) ) -> ( ( R .+ S ) ` G ) = ( ( f e. T |-> ( ( R ` f ) o. ( S ` f ) ) ) ` G ) )
10 simpr3 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ S e. E /\ G e. T ) ) -> G e. T )
11 fveq2 
 |-  ( f = G -> ( R ` f ) = ( R ` G ) )
12 fveq2 
 |-  ( f = G -> ( S ` f ) = ( S ` G ) )
13 11 12 coeq12d 
 |-  ( f = G -> ( ( R ` f ) o. ( S ` f ) ) = ( ( R ` G ) o. ( S ` G ) ) )
14 eqid 
 |-  ( f e. T |-> ( ( R ` f ) o. ( S ` f ) ) ) = ( f e. T |-> ( ( R ` f ) o. ( S ` f ) ) )
15 fvex 
 |-  ( R ` G ) e. _V
16 fvex 
 |-  ( S ` G ) e. _V
17 15 16 coex 
 |-  ( ( R ` G ) o. ( S ` G ) ) e. _V
18 13 14 17 fvmpt 
 |-  ( G e. T -> ( ( f e. T |-> ( ( R ` f ) o. ( S ` f ) ) ) ` G ) = ( ( R ` G ) o. ( S ` G ) ) )
19 10 18 syl 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ S e. E /\ G e. T ) ) -> ( ( f e. T |-> ( ( R ` f ) o. ( S ` f ) ) ) ` G ) = ( ( R ` G ) o. ( S ` G ) ) )
20 9 19 eqtrd 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( R e. E /\ S e. E /\ G e. T ) ) -> ( ( R .+ S ) ` G ) = ( ( R ` G ) o. ( S ` G ) ) )