Theorems · Theorem · convex and discrete geometry
Finset.centerMass_of_sum_add_sum_eq_zero
∀ {R : Type u_1} {E : Type u_3} {ι : Type u_5} [inst : Field R] [inst_1 : AddCommGroup E] [inst_2 : Module R E]
{w : ι → R} {z : ι → E} {s t : Finset ι},
∑ i ∈ s, w i + ∑ i ∈ t, w i = 0 → ∑ i ∈ s, w i • z i + ∑ i ∈ t, w i • z i = 0 → s.centerMass w z = t.centerMass w z- Defined in
- Mathlib.Analysis.Convex.Combination
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 46 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- Finsetstatement and proof · cited by 13,712
- AddCommGroupstatement and proof · cited by 12,871
- Fieldstatement and proof · cited by 7,404
- Finset.sumstatement and proof · cited by 5,195
- neg_smulproof · cited by 306
- smul_negproof · cited by 181
- Finset.centerMassstatement · cited by 49
- inv_negproof · cited by 42
- eq_neg_of_add_eq_zero_leftproof · cited by 21
- eq_neg_of_add_eq_zero_rightproof · cited by 8
Cited by1
Results whose statement or proof uses this declaration.
- Convex.radon_partitionproof · cited by 1