Theorems · Theorem · linear algebra

MultilinearMap.dfinsuppFamily_single_left_apply

∀ {ι : Type uι} {κ : ι → Type uκ} {R : Type uR} {M : (i : ι) → κ i → Type uM} {N : ((i : ι) → κ i) → Type uN}
  [inst : DecidableEq ι] [inst_1 : Fintype ι] [inst_2 : Semiring R]
  [inst_3 : (i : ι) → (k : κ i) → AddCommMonoid (M i k)] [inst_4 : (p : (i : ι) → κ i) → AddCommMonoid (N p)]
  [inst_5 : (i : ι) → (k : κ i) → Module R (M i k)] [inst_6 : (p : (i : ι) → κ i) → Module R (N p)]
  [inst_7 : (i : ι) → DecidableEq (κ i)] (p : (i : ι) → κ i) (f : MultilinearMap R (fun i => M i (p i)) (N p))
  (x : (i : ι) → Π₀ (j : κ i), M i j),
  (MultilinearMap.dfinsuppFamily (Pi.single p f)) x = fun₀ | p => f fun i => (x i) (p i)

When only one member of the family of multilinear maps is nonzero, the result consists only of the component from that member.

Defined in

Mathlib.LinearAlgebra.Multilinear.DFinsupp

Cited by

1 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableEqFintypeSemiringAddCommMonoidAddCommMonoidModuleModuleDecidableEq