Theorems · Theorem · geometry

AffineSubspace.inter_nonempty_of_nonempty_of_sup_direction_eq_top

∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : Ring k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
  [S : AddTorsor V P] {s₁ s₂ : AffineSubspace k P},
  (↑s₁).Nonempty → (↑s₂).Nonempty → s₁.direction ⊔ s₂.direction = ⊤ → (↑s₁ ∩ ↑s₂).Nonempty

If the directions of two nonempty affine subspaces span the whole module, they have nonempty intersection.

Defined in

Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs

Cited by

1 results in Mathlib

Foundations

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

Assumes

RingAddCommGroupModuleAddTorsor