Theorems · Theorem · ring theory
Algebra.eq_top_iff
∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] {S : Subalgebra R A},
S = ⊤ ↔ ∀ (x : A), x ∈ S- Cited by
- 13 results in Mathlib
- Foundations
- Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringSemiringAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Top.topstatement and proof · cited by 9,680
- Subalgebrastatement and proof · cited by 1,353
- Subalgebra.extproof · cited by 28
- Algebra.mem_topproof · cited by 10
Cited by13
Results whose statement or proof uses this declaration.
- IsCyclotomicExtension.iff_adjoin_eq_topproof · cited by 13
- AlgHom.range_eq_topproof · cited by 12
- AdjoinRoot.adjoinRoot_eq_topproof · cited by 4
- Algebra.adjoin_adjoin_coe_preimageproof · cited by 3
- Algebra.FiniteType.of_restrictScalars_finiteTypeproof · cited by 3
- IsCyclotomicExtension.singleton_oneproof · cited by 2
- Algebra.FinitePresentation.of_restrict_scalars_finitePresentationproof · cited by 2
- adjoin_eq_top_of_conductor_eq_topproof · cited by 1
- IsDedekindDomain.range_sup_range_eq_top_of_isCoprime_differentIdealproof · cited by 1
- Algebra.comap_topproof · cited by 1
- Algebra.FiniteType.of_span_eq_top_sourceproof · cited by 1
- Polynomial.SplittingFieldAux.adjoin_rootSetproof · cited by 0