Theorems · Theorem · geometry

Finset.centroid_eq_centroid_image_of_inj_on

∀ (k : Type u_1) {V : Type u_2} {P : Type u_3} [inst : DivisionRing k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
  [inst_3 : AddTorsor V P] {ι : Type u_4} (s : Finset ι) {p : ι → P},
  (∀ i ∈ s, ∀ j ∈ s, p i = p j → i = j) →
    ∀ {ps : Set P} [inst_4 : Fintype ↑ps],
      ps = p '' ↑s → Finset.centroid k s p = Finset.centroid k Finset.univ fun x => ↑x

An indexed family of points that is injective on the given Finset has the same centroid as the image of that Finset. This is stated in terms of a set equal to the image to provide control of definitional equality for the index type used for the centroid of the image.

Defined in

Mathlib.LinearAlgebra.AffineSpace.Centroid

Cited by

1 results in Mathlib

Foundations

Depth 56 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingAddCommGroupModuleAddTorsorFintype