Theorems · Theorem · geometry

vectorSpan_image_eq_span_vsub_set_right_ne

∀ (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] {ι : Type u_4} (p : ι → P) {s : Set ι} {i : ι},
  i ∈ s → vectorSpan k (p '' s) = Submodule.span k ((fun x => x -ᵥ p i) '' p '' (s \ {i}))

The vectorSpan of the image of a function is the span of the pairwise subtractions with a given point on the right, excluding the subtraction of that point from itself.

Defined in

Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic

Cited by

1 results in Mathlib

Foundations

Depth 59 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddTorsor