Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-lvec Structured version   Visualization version   GIF version

Definition df-lvec 19043
 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 multiplication is commutative i.e. a field. (Contributed by NM, 11-Nov-2013.)
Assertion
Ref Expression
df-lvec LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}

Detailed syntax breakdown of Definition df-lvec
StepHypRef Expression
1 clvec 19042 . 2 class LVec
2 vf . . . . . 6 setvar 𝑓
32cv 1479 . . . . 5 class 𝑓
4 csca 15884 . . . . 5 class Scalar
53, 4cfv 5857 . . . 4 class (Scalar‘𝑓)
6 cdr 18687 . . . 4 class DivRing
75, 6wcel 1987 . . 3 wff (Scalar‘𝑓) ∈ DivRing
8 clmod 18803 . . 3 class LMod
97, 2, 8crab 2912 . 2 class {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
101, 9wceq 1480 1 wff LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
 Colors of variables: wff setvar class This definition is referenced by:  islvec  19044  bj-vecssmod  32815
 Copyright terms: Public domain W3C validator