Theorems · Theorem · geometry

AffineSubspace.direction_affineSpan_insert

∀ {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] {s : AffineSubspace k P} {p₁ p₂ : P},
  p₁ ∈ s → (affineSpan k (insert p₂ ↑s)).direction = k ∙ (p₂ -ᵥ p₁) ⊔ s.direction

The direction of the span of the result of adding a point to a nonempty affine subspace is the sup of the direction of that subspace and of any one difference between that point and a point in the subspace.

Defined in

Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic

Cited by

2 results in Mathlib

Foundations

Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddTorsor