Theorems · Theorem · linear algebra

MultilinearMap.map_piecewise_smul

∀ {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [inst : CommSemiring R]
  [inst_1 : (i : ι) → AddCommMonoid (M₁ i)] [inst_2 : AddCommMonoid M₂] [inst_3 : (i : ι) → Module R (M₁ i)]
  [inst_4 : Module R M₂] (f : MultilinearMap R M₁ M₂) [inst_5 : DecidableEq ι] (c : ι → R) (m : (i : ι) → M₁ i)
  (s : Finset ι), f (s.piecewise (fun i => c i • m i) m) = (∏ i ∈ s, c i) • f m

If one multiplies by c i the coordinates in a finset s, then the image under a multilinear map is multiplied by ∏ i ∈ s, c i. This is mainly an auxiliary statement to prove the result when s = univ, given in map_smul_univ, although it can be useful in its own right as it does not require the index set ι to be finite.

Defined in

Mathlib.LinearAlgebra.Multilinear.Basic

Cited by

4 results in Mathlib

Foundations

Depth 61 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidModuleModuleDecidableEq