Mathlib Map

Theorems · Theorem · order theory

List.all_zero_of_le_zero_le_of_sum_eq_zero

∀ {M : Type u_3} [inst : AddCommMonoid M] [inst_1 : PartialOrder M] [IsOrderedAddMonoid M] {l : List M},
  (∀ x ∈ l, 0 ≤ x) → l.sum = 0 → ∀ {x : M}, x ∈ l → x = 0
Defined in
Mathlib.Algebra.Order.BigOperators.Group.List
Cited by
2 results in Mathlib
Foundations
Depth 13 from the axioms · uses propext, Quot.sound
Assumes
AddCommMonoidPartialOrderIsOrderedAddMonoid
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

List.sum_eq_zero_iff · cited by 2List.sum_eq_zero_iffMultiset.all_zero_of_le_zero_le_of_sum_eq_zero · cited by 0Multiset.all_zero_of_le_z…AddCommMonoid · cited by 12281AddCommMonoidPartialOrder · cited by 6410PartialOrderle_antisymm · cited by 2068le_antisymmIsOrderedAddMonoid · cited by 1659IsOrderedAddMonoidList.single_le_sum · cited by 3List.single_le_sumList.all_zero_of_le_zero_le_o…CITED BYCITES

Cites5

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by2

Results whose statement or proof uses this declaration.