Theorems · Theorem · ring theory
Subalgebra.mul_mem
∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (S : Subalgebra R A)
{x y : A}, x ∈ S → y ∈ S → x * y ∈ S- Defined in
- Mathlib.Algebra.Algebra.Subalgebra.Basic
- Cited by
- 26 results in Mathlib
- Foundations
- Depth 22 from the axioms · uses no axioms
- Assumes
- CommSemiringSemiringAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- Subalgebrastatement and proof · cited by 1,353
- MulMemClass.mul_memproof · cited by 173
Cited by26
Results whose statement or proof uses this declaration.
- MvPolynomial.adjoin_range_Xproof · cited by 4
- Algebra.adjoin_adjoin_coe_preimageproof · cited by 3
- IsPrimitiveRoot.adjoin_isCyclotomicExtensionproof · cited by 3
- Subalgebra.SeparatesPoints.rclike_to_realproof · cited by 2
- IsLocalization.exists_smul_mem_of_mem_adjoinproof · cited by 2
- Algebra.isCyclotomicExtension_adjoin_of_exists_isPrimitiveRootproof · cited by 2
- Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoinproof · cited by 2
- Polynomial.IsWeaklyEisensteinAt.exists_mem_adjoin_mul_eq_pow_natDegreeproof · cited by 1
- IsDedekindDomain.range_sup_range_eq_top_of_isCoprime_differentIdealproof · cited by 1
- integral_mulExpNegMulSq_comp_eqproof · cited by 1
- Algebra.EssFiniteType.auxproof · cited by 1