Theorems · Theorem · geometry

affineCombination_mem_affineSpan_of_nonempty

∀ {ι : Type u_1} {k : Type u_2} {V : Type u_3} {P : Type u_4} [inst : Ring k] [inst_1 : AddCommGroup V]
  [inst_2 : Module k V] [inst_3 : AddTorsor V P] [Nonempty ι] {s : Finset ι} {w : ι → k},
  ∑ i ∈ s, w i = 1 → ∀ (p : ι → P), (Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)

An affineCombination with sum of weights 1 is in the affineSpan of an indexed family, if the family is nonempty.

Defined in

Mathlib.LinearAlgebra.AffineSpace.Combination

Cited by

2 results in Mathlib

Foundations

Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddTorsorNonempty