Theorems · Theorem · commutative algebra
Algebra.adjoin_union_eq_adjoin_adjoin
∀ (R : Type uR) {A : Type uA} [inst : CommSemiring R] [inst_1 : CommSemiring A] [inst_2 : Algebra R A] (s t : Set A),
Algebra.adjoin R (s ∪ t) = Subalgebra.restrictScalars R (Algebra.adjoin (↥(Algebra.adjoin R s)) t)- Defined in
- Mathlib.RingTheory.Adjoin.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Subalgebrastatement · cited by 1,353
- Algebra.adjoinstatement and proof · cited by 535
- Set.image_congrproof · cited by 533
- Set.image_id'proof · cited by 86
- Subalgebra.restrictScalarsstatement and proof · cited by 36
- Algebra.adjoin_algebraMap_image_union_eq_adjoin_adjoinproof · cited by 3
Cited by8
Results whose statement or proof uses this declaration.
- Algebra.restrictScalars_adjoinproof · cited by 3
- Algebra.FinitePresentation.of_restrict_scalars_finitePresentationproof · cited by 2
- Algebra.adjoin_eq_adjoin_unionproof · cited by 1
- IsCyclotomicExtension.union_rightproof · cited by 1
- Algebra.adjoin_res_eq_adjoin_resproof · cited by 0
- Polynomial.IsSplittingField.mulproof · cited by 0
- Polynomial.lift_of_splitsproof · cited by 0
- Algebra.fg_transproof · cited by 0