Theorems · Theorem · linear algebra

MultilinearMap.map_piecewise_add

∀ {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [inst : Semiring 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] (m m' : (i : ι) → M₁ i) (t : Finset ι),
  f (t.piecewise (m + m') m') = ∑ s ∈ t.powerset, f (s.piecewise m m')

If one adds to a vector m' another vector m, but only for coordinates in a finset t, then the image under a multilinear map f is the sum of f (s.piecewise m m') along all subsets s of t. This is mainly an auxiliary statement to prove the result when t = univ, given in map_add_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

3 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidAddCommMonoidModuleModuleDecidableEq