Theorems · Theorem · geometry

mem_vectorSpan_pair

∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : Ring k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
  [inst_3 : AddTorsor V P] {p₁ p₂ : P} {v : V}, v ∈ vectorSpan k {p₁, p₂} ↔ ∃ r, r • (p₁ -ᵥ p₂) = v

A vector lies in the vectorSpan of two points if and only if it is a multiple of their difference.

Defined in

Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic

Cited by

2 results in Mathlib

Foundations

Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddTorsor