Theorems · Theorem · linear algebra

MultilinearMap.map_piecewise_sub_map_piecewise

∀ {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [inst : Semiring R]
  [inst_1 : (i : ι) → AddCommGroup (M₁ i)] [inst_2 : AddCommGroup M₂] [inst_3 : (i : ι) → Module R (M₁ i)]
  [inst_4 : Module R M₂] (f : MultilinearMap R M₁ M₂) [inst_5 : LinearOrder ι] (a b v : (i : ι) → M₁ i) (s : Finset ι),
  f (s.piecewise a v) - f (s.piecewise b v) =
    ∑ i ∈ s, f fun j => if j ∈ s then if j < i then a j else if j = i then a j - b j else b j else v j

This calculates the differences between the values of a multilinear map at two arguments that differ on a finset s of ι. It requires a linear order on ι in order to express the result.

Defined in

Mathlib.LinearAlgebra.Multilinear.Basic

Cited by

0 results in Mathlib

Foundations

Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommGroupAddCommGroupModuleModuleLinearOrder