Theorems · Theorem · geometry

collinear_iff_of_mem

∀ {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] {s : Set P} {p₀ : P}, p₀ ∈ s → (Collinear k s ↔ ∃ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀)

Given a point p₀ in a set of points, that set is collinear if and only if the points can all be expressed as multiples of the same vector, added to p₀.

Defined in

Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Cited by

4 results in Mathlib

Foundations

Depth 121 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingAddCommGroupModuleAddTorsor