Theorems · Theorem · order theory
Finset.sum_lt_sum_of_nonempty
∀ {ι : Type u_1} {M : Type u_4} [inst : AddCommMonoid M] [inst_1 : Preorder M] [IsOrderedCancelAddMonoid M]
{f g : ι → M} {s : Finset ι} [AddLeftStrictMono M], s.Nonempty → (∀ i ∈ s, f i < g i) → ∑ i ∈ s, f i < ∑ i ∈ s, g iIn an ordered additive commutative monoid, if each summand f i of one nontrivial finite sum is
strictly less than the corresponding summand g i of another nontrivial finite sum, then
s.sum f < s.sum g.
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 58 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- Preorderstatement and proof · cited by 7,952
- Finset.sumstatement · cited by 5,195
- Finset.Nonemptystatement and proof · cited by 1,001
- IsOrderedCancelAddMonoidstatement and proof · cited by 359
- AddLeftStrictMonostatement and proof · cited by 203
- Finset.val_eq_zeroproof · cited by 12
- Finset.not_nonempty_emptyproof · cited by 5
- Multiset.sum_lt_sum_of_nonemptyproof · cited by 1
Cited by10
Results whose statement or proof uses this declaration.
- Finset.sum_posproof · cited by 15
- Finset.exists_le_of_sum_leproof · cited by 3
- Nat.eq_properDivisors_of_subset_of_sum_eq_sumproof · cited by 2
- Affine.Simplex.inv_height_lt_sum_inv_heightproof · cited by 2
- Finset.sum_negproof · cited by 1
- Polynomial.exists_roots_norm_sub_lt_of_norm_coeff_sub_ltproof · cited by 1
- Finset.card_nsmul_lt_card_nsmul_of_le_of_ltproof · cited by 1
- Finset.card_nsmul_lt_card_nsmul_of_lt_of_leproof · cited by 1
- Subgroup.exists_index_le_card_of_leftCoset_coverproof · cited by 0
- AddSubgroup.exists_index_le_card_of_leftCoset_coverproof · cited by 0