Definition df-lvec 20410

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

Step	Hyp	Ref	Expression
1		clvec 20409	. 2 class LVec
2		vf	. . . . . 6 setvar 𝑓
3	2	cv 1538	. . . . 5 class 𝑓
4		csca 17010	. . . . 5 class Scalar
5	3, 4	cfv 6458	. . . 4 class (Scalar‘𝑓)
6		cdr 20036	. . . 4 class DivRing
7	5, 6	wcel 2104	. . 3 wff (Scalar‘𝑓) ∈ DivRing
8		clmod 20168	. . 3 class LMod
9	7, 2, 8	crab 3284	. 2 class {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
10	1, 9	wceq 1539	1 wff LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}

This definition is referenced by:  islvec  20411  bj-vecssmod  35496