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Metamath Proof Explorer

Definition df-lvec

Description: Define the class of all left vector spaces. A left vector space over a division ring is an Abelian group (vectors) together with a division ring (scalars) and a left scalar product connecting them. Some authors call this a "left module over a division ring", reserving "vector space" for those where the division ring is commutative, i.e., is a field. (Contributed by NM, 11-Nov-2013)

Ref Expression
Assertion df-lvec 
|- LVec = { f e. LMod | ( Scalar ` f ) e. DivRing }

Detailed syntax breakdown

Step Hyp Ref Expression
0 clvec 
 |-  LVec
1 vf 
 |-  f
2 clmod 
 |-  LMod
3 csca 
 |-  Scalar
4 1 cv 
 |-  f
5 4 3 cfv 
 |-  ( Scalar ` f )
6 cdr 
 |-  DivRing
7 5 6 wcel 
 |-  ( Scalar ` f ) e. DivRing
8 7 1 2 crab 
 |-  { f e. LMod | ( Scalar ` f ) e. DivRing }
9 0 8 wceq 
 |-  LVec = { f e. LMod | ( Scalar ` f ) e. DivRing }