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Definition df-lvec 18867
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 18866 . 2 class LVec
2 vf . . . . . 6 setvar 𝑓
32cv 1473 . . . . 5 class 𝑓
4 csca 15714 . . . . 5 class Scalar
53, 4cfv 5787 . . . 4 class (Scalar‘𝑓)
6 cdr 18513 . . . 4 class DivRing
75, 6wcel 1976 . . 3 wff (Scalar‘𝑓) ∈ DivRing
8 clmod 18629 . . 3 class LMod
97, 2, 8crab 2896 . 2 class {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
101, 9wceq 1474 1 wff LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
Colors of variables: wff setvar class
This definition is referenced by:  islvec  18868  bj-vecssmod  32120
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