Metamath Proof Explorer


Theorem dvhbase

Description: The ring base set of the constructed full vector space H. (Contributed by NM, 29-Oct-2013) (Revised by Mario Carneiro, 22-Jun-2014)

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

Proof

Step Hyp Ref Expression
1 dvhbase.h
 |-  H = ( LHyp ` K )
2 dvhbase.e
 |-  E = ( ( TEndo ` K ) ` W )
3 dvhbase.u
 |-  U = ( ( DVecH ` K ) ` W )
4 dvhbase.f
 |-  F = ( Scalar ` U )
5 dvhbase.c
 |-  C = ( Base ` F )
6 eqid
 |-  ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
7 1 6 3 4 dvhsca
 |-  ( ( 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 syl5eq
 |-  ( ( 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 )