Theorems · Theorem · geometry

affineSpan_induction

∀ {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] {x : P} {s : Set P} {p : P → Prop},
  x ∈ affineSpan k s → (∀ x ∈ s, p x) → (∀ (c : k) (u v w : P), p u → p v → p w → p (c • (u -ᵥ v) +ᵥ w)) → p x

An induction principle for span membership. If p holds for all elements of s and is preserved under certain affine combinations, then p holds for all elements of the span of s.

Defined in

Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs

Cited by

1 results in Mathlib

Foundations

Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddTorsor