Theorems · Theorem · geometry

Collinear.mem_affineSpan_of_mem_of_ne

∀ {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},
  Collinear k s → ∀ {p₁ p₂ p₃ : P}, p₁ ∈ s → p₂ ∈ s → p₃ ∈ s → p₁ ≠ p₂ → p₃ ∈ line[k, p₁, p₂]

A point in a collinear set of points lies in the affine span of any two distinct points of that set.

Defined in

Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Cited by

7 results in Mathlib

Foundations

Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingAddCommGroupModuleAddTorsor