Theorems · Theorem · geometry

AffineSubspace.card_pos_of_affineSpan_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] {ι : Type u_5} [inst_3 : Fintype ι] {p : ι → P},
  affineSpan k (Set.range p) = ⊤ → 0 < Fintype.card ι

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: Ring AddCommGroup Module AddTorsor Fintype

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Cites12

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Modulestatement and proof · cited by 20,661
AddCommGroupstatement and proof · cited by 12,871
Top.topstatement and proof · cited by 9,680
Fintypestatement and proof · cited by 7,736
Ringstatement and proof · cited by 7,463
Set.rangestatement and proof · cited by 4,705
AddTorsorstatement and proof · cited by 1,657
Fintype.cardstatement · cited by 1,386
AffineSubspacestatement · cited by 871
affineSpanstatement and proof · cited by 417
Fintype.card_pos_iffproof · cited by 31
AffineSubspace.nonempty_of_affineSpan_eq_topproof · cited by 3

Cited by1

Results whose statement or proof uses this declaration.

AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_oneproof · cited by 4