Theorems · Theorem · order theory
Finset.sum_eq_zero_iff_of_nonneg
∀ {ι : Type u_1} {N : Type u_5} [inst : AddCommMonoid N] [inst_1 : PartialOrder N] {f : ι → N} {s : Finset ι}
[AddLeftMono N], (∀ i ∈ s, 0 ≤ f i) → (∑ i ∈ s, f i = 0 ↔ ∀ i ∈ s, f i = 0)- Cited by
- 19 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.
Cites13
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
- PartialOrderstatement and proof · cited by 6,410
- Finset.sumstatement and proof · cited by 5,195
- AddLeftMonostatement and proof · cited by 687
- Finset.sum_insertproof · cited by 196
- Finset.induction_onproof · cited by 167
- Finset.mem_insert_selfproof · cited by 128
- Finset.mem_insert_of_memproof · cited by 109
- Finset.sum_nonnegproof · cited by 91
- Finset.notMem_emptyproof · cited by 40
- Finset.forall_mem_insertproof · cited by 19
Cited by19
Results whose statement or proof uses this declaration.
- Convex.centerMass_memproof · cited by 4
- Real.geom_mean_eq_arith_mean_weighted_iff_of_pos'proof · cited by 4
- Finset.sum_sq_le_sum_mul_sum_of_sq_le_mulproof · cited by 3
- Fintype.sum_eq_zero_iff_of_nonnegproof · cited by 3
- Finset.sum_pos_iff_of_nonnegproof · cited by 3
- StrictConvex.centerMass_mem_interiorproof · cited by 2
- Finset.sum_eq_zero_iff_of_nonposproof · cited by 2
- NumberField.mixedEmbedding.convexBodySumFun_eq_zero_iffproof · cited by 2
- dotProduct_self_eq_zeroproof · cited by 1
- Affine.Simplex.closedInterior_eq_interior_unionproof · cited by 1
- Affine.Simplex.closedInterior_inter_shift_zeroproof · cited by 1
- ENNReal.lintegral_prod_norm_pow_leproof · cited by 1