Theorems · Theorem · geometry

AffineIndependent.affineCombination_mem_shift_iff

∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : Ring k] [inst_1 : AddCommGroup V] [inst_2 : AddTorsor V P]
  [inst_3 : Module k V] {ι : Type u_4} [inst_4 : Fintype ι] [Nontrivial ι] {p : ι → P},
  AffineIndependent k p →
    ∀ (i : ι) {w : ι → k},
      ∑ i, w i = 1 →
        ∀ (c : k), (Finset.affineCombination k Finset.univ p) w ∈ (affineSpan k (p '' {i}ᶜ)).shift (p i) c ↔ w i = 1 - c

The iff version of affineCombination_mem_shift for affine independent points.

Defined in

Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift

Cited by

3 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupAddTorsorModuleFintypeNontrivial