Theorems · Theorem · geometry

AffineIndependent.affineSpan_eq_of_le_of_card_eq_finrank_add_one

∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} {ι : Type u_4} [inst : DivisionRing k] [inst_1 : AddCommGroup V]
  [inst_2 : Module k V] [inst_3 : AddTorsor V P] [inst_4 : Fintype ι] {p : ι → P},
  AffineIndependent k p →
    ∀ {sp : AffineSubspace k P} [FiniteDimensional k ↥sp.direction],
      affineSpan k (Set.range p) ≤ sp →
        Fintype.card ι = Module.finrank k ↥sp.direction + 1 → affineSpan k (Set.range p) = sp

If the affineSpan of a finite affinely independent family lies in an affine subspace whose direction has dimension one less than its cardinality, it equals that subspace.

Defined in

Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Cited by

3 results in Mathlib

Foundations

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

Assumes

DivisionRingAddCommGroupModuleAddTorsorFintypeFiniteDimensional