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 = .+ )