Definition df-lvec 18404

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 e. LMod | (Scalar ` f ) e. DivRing }

Detailed syntax breakdown of Definition df-lvec

Step	Hyp	Ref	Expression
1		clvec 18403	. 2 class LVec
2		vf	. . . . . 6 setvar f
3	2	cv 1451	. . . . 5 class f
4		csca 15271	. . . . 5 class Scalar
5	3, 4	cfv 5589	. . . 4 class (Scalar ` f )
6		cdr 18053	. . . 4 class DivRing
7	5, 6	wcel 1904	. . 3 wff (Scalar ` f ) e. DivRing
8		clmod 18169	. . 3 class LMod
9	7, 2, 8	crab 2760	. 2 class { f e. LMod | (Scalar ` f ) e. DivRing }
10	1, 9	wceq 1452	1 wff LVec = { f e. LMod | (Scalar ` f ) e. DivRing }

This definition is referenced by:  islvec  18405  bj-vecssmod  31768