Theorems · Theorem · geometry

affineSpan_singleton_union_vadd_eq_top_of_span_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]
  [inst_3 : AddTorsor V P] {s : Set V} (p : P),
  Submodule.span k (Set.range Subtype.val) = ⊤ → affineSpan k ({p} ∪ (fun v => v +ᵥ p) '' s) = ⊤

Suppose a set of vectors spans V. Then a point p, together with those vectors added to p, spans P.

Defined in

Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic

Cited by

1 results in Mathlib

Foundations

Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddTorsor