Theorems · Theorem · geometry

AffineIndependent.affineIndependent_update_of_notMem_affineSpan

∀ {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} [inst_4 : DecidableEq ι] {p : ι → P},
  AffineIndependent k p →
    ∀ {i : ι} {p₀ : P}, p₀ ∉ affineSpan k (p '' {x | x ≠ i}) → AffineIndependent k (Function.update p i p₀)

If a family is affinely independent, we update any one point with a new point does not lie in the affine span of that family, the new family is affinely independent.

Defined in

Mathlib.LinearAlgebra.AffineSpace.Independent

Cited by

1 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingAddCommGroupModuleAddTorsorDecidableEq