Theorems · Theorem · linear algebra

MultilinearMap.freeDFinsuppEquiv_single

∀ {ι : Type uι} {κ : ι → Type uκ} {R : Type uR} {ι' : Type u_1} [inst : DecidableEq ι] [inst_1 : Fintype ι]
  [inst_2 : CommSemiring R] [inst_3 : (i : ι) → Fintype (κ i)] [inst_4 : (i : ι) → DecidableEq (κ i)]
  [inst_5 : DecidableEq ι'] (p : ((i : ι) → κ i) × ι') (r : R) (x : (i : ι) → Π₀ (x : κ i), R),
  (MultilinearMap.freeDFinsuppEquiv fun₀ | p => r) x = r • fun₀ | p.2 => ∏ i, (x i) (p.1 i)

When freeDFinsuppEquiv is applied to a map with a single value of one the resulting multilinear map sends inputs to a single value in the codomain, taking a product over images from each component of the domain.

Defined in

Mathlib.LinearAlgebra.Multilinear.DFinsupp

Cited by

2 results in Mathlib

Foundations

Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableEqFintypeCommSemiringFintypeDecidableEqDecidableEq