Theorems · Theorem · commutative algebra

IsAlgebraic.adjoin_of_forall_isAlgebraic

∀ {R : Type u_1} {S : Type u_2} {A : Type u_3} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Ring A]
  [inst_3 : Algebra R S] [inst_4 : Algebra S A] [NoZeroDivisors S] {s t : Set S},
  (∀ x ∈ s \ t, IsAlgebraic (↥(Algebra.adjoin R t)) x) →
    ∀ {a : A}, IsAlgebraic (↥(Algebra.adjoin R s)) a → IsAlgebraic (↥(Algebra.adjoin R t)) a

Defined in

Mathlib.RingTheory.Algebraic.Integral

Cited by

1 results in Mathlib

Foundations

Depth 140 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingRingAlgebraAlgebraNoZeroDivisors