Theorems · Theorem · commutative algebra
Algebra.adjoin_algebraMap_image_union_eq_adjoin_adjoin
∀ (R : Type uR) {S : Type uS} {A : Type uA} [inst : CommSemiring R] [inst_1 : CommSemiring S] [inst_2 : Semiring A]
[inst_3 : Algebra R S] [inst_4 : Algebra R A] [inst_5 : Algebra S A] [inst_6 : IsScalarTower R S A] (s : Set S)
(t : Set A),
Algebra.adjoin R (⇑(algebraMap S A) '' s ∪ t) =
Subalgebra.restrictScalars R (Algebra.adjoin (↥(Algebra.adjoin R s)) t)- Defined in
- Mathlib.RingTheory.Adjoin.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites27
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- Set.imagestatement and proof · cited by 5,609
- Algebra.algebraMapstatement and proof · cited by 4,706
- Set.rangeproof · cited by 4,705
- IsScalarTowerstatement and proof · cited by 3,896
- le_antisymmproof · cited by 2,068
- Subalgebrastatement · cited by 1,353
Cited by3
Results whose statement or proof uses this declaration.
- Algebra.adjoin_union_eq_adjoin_adjoinproof · cited by 8
- Algebra.fg_trans'proof · cited by 2
- IsTranscendenceBasis.sumElim_compproof · cited by 1