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Definition df-lvec 19851
 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.)
Assertion
Ref Expression
df-lvec LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}

Detailed syntax breakdown of Definition df-lvec
StepHypRef Expression
1 clvec 19850 . 2 class LVec
2 vf . . . . . 6 setvar 𝑓
32cv 1537 . . . . 5 class 𝑓
4 csca 16547 . . . . 5 class Scalar
53, 4cfv 6328 . . . 4 class (Scalar‘𝑓)
6 cdr 19478 . . . 4 class DivRing
75, 6wcel 2115 . . 3 wff (Scalar‘𝑓) ∈ DivRing
8 clmod 19610 . . 3 class LMod
97, 2, 8crab 3130 . 2 class {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
101, 9wceq 1538 1 wff LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
 Colors of variables: wff setvar class This definition is referenced by:  islvec  19852  bj-vecssmod  34581
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