isAlgebraic_iff — Mathlib

English

An algebra is algebraic iff it is algebraic as the top subalgebra.

Русский

Алгебраическая надлогическая эквивалентна как вершина подалгебра.

LaTeX

$$$\\text{Algebra.IsAlgebraic } R A \\iff (\\top : Subalgebra R A).IsAlgebraic$$$

Lean4

/-- An algebra is algebraic if and only if it is algebraic as a subalgebra. -/
theorem isAlgebraic_iff : Algebra.IsAlgebraic R A ↔ (⊤ : Subalgebra R A).IsAlgebraic :=
  by
  delta Subalgebra.IsAlgebraic
  simp only [Algebra.isAlgebraic_def, Algebra.mem_top, forall_prop_of_true]

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