algebra_isAlgebraic_of_algebra_isAlgebraic_bot_left

English

With Nontrivial R, the bottom subalgebra is algebraic over R on the right as well.

Русский

При немомерном R нижний элемент алгебраичен над R справа.

LaTeX

$$$[Nontrivial R] \Rightarrow Algebra.IsAlgebraic(R, \; (⊥ : Subalgebra R S))$$$

Lean4

theorem algebra_isAlgebraic_of_algebra_isAlgebraic_bot_left [Algebra.IsAlgebraic (⊥ : Subalgebra R S) S] :
    Algebra.IsAlgebraic R S :=
  Algebra.IsAlgebraic.of_ringHom_of_comp_eq (algebraMap R (⊥ : Subalgebra R S)) (RingHom.id S)
    (by rintro ⟨_, r, rfl⟩; exact ⟨r, rfl⟩) Function.injective_id (by ext; rfl)

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