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Mirrors > Home > MPE Home > Th. List > df-lvec | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-lvec | β’ LVec = {π β LMod β£ (Scalarβπ) β DivRing} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clvec 20713 | . 2 class LVec | |
2 | vf | . . . . . 6 setvar π | |
3 | 2 | cv 1541 | . . . . 5 class π |
4 | csca 17200 | . . . . 5 class Scalar | |
5 | 3, 4 | cfv 6544 | . . . 4 class (Scalarβπ) |
6 | cdr 20357 | . . . 4 class DivRing | |
7 | 5, 6 | wcel 2107 | . . 3 wff (Scalarβπ) β DivRing |
8 | clmod 20471 | . . 3 class LMod | |
9 | 7, 2, 8 | crab 3433 | . 2 class {π β LMod β£ (Scalarβπ) β DivRing} |
10 | 1, 9 | wceq 1542 | 1 wff LVec = {π β LMod β£ (Scalarβπ) β DivRing} |
Colors of variables: wff setvar class |
This definition is referenced by: islvec 20715 bj-vecssmod 36210 |
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