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Metamath Proof Explorer

Theorem dvabase

Description: The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom W ). (Contributed by NM, 9-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

Ref Expression
Hypotheses dvabase.h 
|- H = ( LHyp ` K )
dvabase.e 
|- E = ( ( TEndo ` K ) ` W )
dvabase.u 
|- U = ( ( DVecA ` K ) ` W )
dvabase.f 
|- F = ( Scalar ` U )
dvabase.c 
|- C = ( Base ` F )
Assertion dvabase 
|- ( ( K e. X /\ W e. H ) -> C = E )

Proof

Step Hyp Ref Expression
1 dvabase.h 
 |-  H = ( LHyp ` K )
2 dvabase.e 
 |-  E = ( ( TEndo ` K ) ` W )
3 dvabase.u 
 |-  U = ( ( DVecA ` K ) ` W )
4 dvabase.f 
 |-  F = ( Scalar ` U )
5 dvabase.c 
 |-  C = ( Base ` F )
6 eqid 
 |-  ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
7 1 6 3 4 dvasca 
 |-  ( ( K e. X /\ W e. H ) -> F = ( ( EDRing ` K ) ` W ) )
8 7 fveq2d 
 |-  ( ( K e. X /\ W e. H ) -> ( Base ` F ) = ( Base ` ( ( EDRing ` K ) ` W ) ) )
9 5 8 eqtrid 
 |-  ( ( K e. X /\ W e. H ) -> C = ( Base ` ( ( EDRing ` K ) ` W ) ) )
10 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
11 eqid 
 |-  ( Base ` ( ( EDRing ` K ) ` W ) ) = ( Base ` ( ( EDRing ` K ) ` W ) )
12 1 10 2 6 11 erngbase 
 |-  ( ( K e. X /\ W e. H ) -> ( Base ` ( ( EDRing ` K ) ` W ) ) = E )
13 9 12 eqtrd 
 |-  ( ( K e. X /\ W e. H ) -> C = E )