Theorems · Theorem · group theory
sum_mem
∀ {B : Type u_3} {S : B} {M : Type u_4} [inst : AddCommMonoid M] [inst_1 : SetLike B M] [AddSubmonoidClass B M]
{ι : Type u_5} {t : Finset ι} {f : ι → M}, (∀ c ∈ t, f c ∈ S) → ∑ c ∈ t, f c ∈ SSum of elements in an AddSubmonoid of an AddCommMonoid indexed by a Finset
is in the AddSubmonoid.
- Cited by
- 64 results in Mathlib
- Foundations
- Depth 61 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.
- Finsetstatement and proof · cited by 13,712
- AddCommMonoidstatement and proof · cited by 12,281
- Finset.sumstatement · cited by 5,195
- SetLikestatement and proof · cited by 1,084
- Multiset.mapproof · cited by 876
- Finset.valproof · cited by 438
- AddSubmonoidClassstatement and proof · cited by 346
- Multiset.mem_mapproof · cited by 72
- multiset_sum_memproof · cited by 10
Cited by64
Results whose statement or proof uses this declaration.
- Submodule.sum_memproof · cited by 42
- Finsupp.range_linearCombinationproof · cited by 24
- Subalgebra.sum_memproof · cited by 13
- Submodule.sum_smul_memproof · cited by 8
- Finsupp.span_image_eq_map_linearCombinationproof · cited by 7
- LinearMap.BilinForm.dualSubmodule_span_of_basisproof · cited by 7
- Module.Basis.mem_span_iff_repr_memproof · cited by 6
- DirectSum.AddSubmonoidClass.IsHomogeneous.mem_iffproof · cited by 4
- Submodule.mem_span_iff_exists_finset_subsetproof · cited by 3
- IsSumSq.sumproof · cited by 3
- Finsupp.supported_eq_span_singleproof · cited by 3
- Algebra.Generators.map_toComp_kerproof · cited by 3