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	syl5eq	|- ( ( 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 ) ) ) ) )