Theorems · Theorem · geometry

AffineIndependent.affineIndependent_of_notMem_span

∀ {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} {p : ι → P} {i : ι},
  (AffineIndependent k fun x => p ↑x) → p i ∉ affineSpan k (p '' {x | x ≠ i}) → AffineIndependent k p

If all but one point of a family are affinely independent, and that point does not lie in the affine span of that family, the family is affinely independent.

Defined in

Mathlib.LinearAlgebra.AffineSpace.Independent

Cited by

2 results in Mathlib

Foundations

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

Assumes

DivisionRingAddCommGroupModuleAddTorsor