English
Let φ be a nonunital algebra homomorphism from A to B. If A is finite and B admits a decidable equality, then the range of φ is finite (i.e., the image set φ(A) is finite).
Русский
Пусть φ — негомоморфизм по алгебре без единицы, от A к B. Если A конечно по размеру и на B задано двоичное равенство (существует разрешимое сравнение), то образ φ(A) конечно.
LaTeX
$$$\text{If } A \text{ is finite and } \text{DecidableEq}(B), \text{ then } \#\bigl(\operatorname{im}(\varphi)\bigr) < \infty.$$$
Lean4
/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.
Note that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/
instance fintypeRange [Fintype A] [DecidableEq B] (φ : F) : Fintype (NonUnitalAlgHom.range φ) :=
Set.fintypeRange φ
