extendScalars_of_isIntegral — Mathlib

English

If for all x in S outside t, x is algebraic over adjoin R t, then for any a ∈ A, if a is algebraic over adjoin R s, then a is algebraic over adjoin R t.

Русский

Если для всех x ∈ S \\ t x алгебраичен над adjoin R t, тогда для любого a ∈ A, если a алгебраичен над adjoin R s, то a алгебраичен над adjoin R t.

LaTeX

$$$\\forall x \\in S, x \\notin t \\Rightarrow IsAlgebraic(\\Subtype\\ldots)(x) \\Rightarrow (IsAlgebraic(\\ldots) a) \\Rightarrow IsAlgebraic(\\ldots) a$$$

Lean4

theorem extendScalars_of_isIntegral [Algebra.IsIntegral R S] : Transcendental S a :=
  by
  contrapose ha
  rw [Transcendental, not_not] at ha ⊢
  exact ha.restrictScalars_of_isIntegral _

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