Theorems · Theorem · group theory
AddSubmonoid.sum_mem
∀ {M : Type u_4} [inst : AddCommMonoid M] (S : AddSubmonoid 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
- 9 results in Mathlib
- Foundations
- Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddCommMonoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- AddSubmonoidstatement and proof · cited by 1,178
- Multiset.mapproof · cited by 876
- Finset.valproof · cited by 438
- Multiset.mem_mapproof · cited by 72
- AddSubmonoid.multiset_sum_memproof · cited by 2
Cited by9
Results whose statement or proof uses this declaration.
- DividedPowers.dpow_sum'proof · cited by 2
- AddSubmonoid.mem_closure_finsetproof · cited by 2
- convex_sumproof · cited by 1
- AddSubgroup.mem_closure_range_iffproof · cited by 1
- AddSubmonoid.mem_closure_iff_exists_finset_subsetproof · cited by 1
- Ideal.Filtration.submodule_closure_singleproof · cited by 1
- AddSubmonoid.mem_closure_range_iffproof · cited by 1
- AddSubmonoid.closure_addIrreducibleproof · cited by 0
- IsLocalRing.exists_of_isUnit_sumproof · cited by 0