Theorems · Theorem · ring theory
Subalgebra.sum_mem
∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (S : Subalgebra R A)
{ι : Type w} {t : Finset ι} {f : ι → A}, (∀ x ∈ t, f x ∈ S) → ∑ x ∈ t, f x ∈ S- Defined in
- Mathlib.Algebra.Algebra.Subalgebra.Basic
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 62 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
- Finsetstatement and proof · cited by 13,712
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Finset.sumstatement · cited by 5,195
- Subalgebrastatement and proof · cited by 1,353
- sum_memproof · cited by 64
Cited by13
Results whose statement or proof uses this declaration.
- IsIntegral.sumproof · cited by 10
- Polynomial.adjoin_Xproof · cited by 3
- exists_fg_and_mem_baseChangeproof · cited by 2
- Subalgebra.centralizer_coe_image_includeLeft_eq_center_tensorProductproof · 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
- adjoin_monomial_eq_reesAlgebraproof · cited by 1
- mem_adjoin_of_dvd_coeff_of_dvd_aevalproof · cited by 1
- polynomialFunctions_closure_eq_top'proof · cited by 1
- Algebra.discr_mul_isIntegral_mem_adjoinproof · cited by 1
- IsPrimitiveRoot.self_sub_one_pow_dvd_orderproof · cited by 1
- jacobiSum_mem_algebraAdjoin_of_pow_eq_oneproof · cited by 0