instIsFractionRingSubtypeMemSubalgebraAdjoinAdjoin

English

Subtypes in the adjoin AdjoinAdjoin context inherit algebraicity from the ambient setting.

Русский

Подтипы в контексте adjoinAdjoin наследуют алгебраическую зависимость от окружающей среды.

LaTeX

$$Algebra.IsAlgebraic (Subtype fun x => SetLike.instMembership.mem (Algebra.adjoin F S) x) (Subtype fun x => SetLike.instMembership.mem (IntermediateField.adjoin F S) x)$$

Lean4

scoped instance : IsFractionRing (Algebra.adjoin F S) (adjoin F S) :=
  .of_field _ _ fun ⟨_, h⟩ ↦
    have ⟨x, hx, y, hy, eq⟩ := mem_adjoin_iff_div.mp h
    ⟨⟨x, hx⟩, ⟨y, hy⟩, Subtype.ext eq⟩

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