Theorems · Theorem · linear algebra

MultilinearMap.map_sum_finset_aux

∀ {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) {α : ι → Type u_1} (g : (i : ι) → α i → M₁ i)
  (A : (i : ι) → Finset (α i)) [inst_5 : DecidableEq ι] [inst_6 : Fintype ι] {n : ℕ},
  ∑ i, (A i).card = n → (f fun i => ∑ j ∈ A i, g i j) = ∑ r ∈ Fintype.piFinset A, f fun i => g i (r i)

If f is multilinear, then f (Σ_{j₁ ∈ A₁} g₁ j₁, ..., Σ_{jₙ ∈ Aₙ} gₙ jₙ) is the sum of f (g₁ (r 1), ..., gₙ (r n)) where r ranges over all functions with r 1 ∈ A₁, ..., r n ∈ Aₙ. This follows from multilinearity by expanding successively with respect to each coordinate. Here, we give an auxiliary statement tailored for an inductive proof. Use instead map_sum_finset.

Defined in

Mathlib.LinearAlgebra.Multilinear.Basic

Cited by

1 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidAddCommMonoidModuleModuleDecidableEqFintype