isEmpty_of_isAlgebraic — Mathlib

English

If A is algebraic over R, then any algebraically independent family must have empty index set.

Русский

Если A алгебраично над R, то любая алгебраически независимая семейство должно иметь пустой индекс.

LaTeX

$$$[\text{Algebra.IsAlgebraic } R A] \Rightarrow IsEmpty ι \text{ for every } x: ι \to A \ (AlgebraicIndependent R x)$$$

Lean4

/-- If `A/R` is algebraic, then all algebraically independent families are empty. -/
theorem isEmpty_of_isAlgebraic [Algebra.IsAlgebraic R A] : IsEmpty ι :=
  by
  rcases isEmpty_or_nonempty ι with h | ⟨⟨i⟩⟩
  · exact h
  exact False.elim (hx.transcendental i (Algebra.IsAlgebraic.isAlgebraic _))

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