Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 multiplication is commutative i.e. a field. (Contributed by NM, 11-Nov-2013.) |
Ref | Expression |
---|---|
df-lvec | ⊢ LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clvec 18923 | . 2 class LVec | |
2 | vf | . . . . . 6 setvar 𝑓 | |
3 | 2 | cv 1474 | . . . . 5 class 𝑓 |
4 | csca 15771 | . . . . 5 class Scalar | |
5 | 3, 4 | cfv 5804 | . . . 4 class (Scalar‘𝑓) |
6 | cdr 18570 | . . . 4 class DivRing | |
7 | 5, 6 | wcel 1977 | . . 3 wff (Scalar‘𝑓) ∈ DivRing |
8 | clmod 18686 | . . 3 class LMod | |
9 | 7, 2, 8 | crab 2900 | . 2 class {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing} |
10 | 1, 9 | wceq 1475 | 1 wff LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing} |
Colors of variables: wff setvar class |
This definition is referenced by: islvec 18925 bj-vecssmod 32320 |
Copyright terms: Public domain | W3C validator |