Theorems · Theorem · linear algebra

MultilinearMap.piFamily_single

∀ {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 : Semiring R] [inst_1 : (i : ι) → (k : κ i) → AddCommMonoid (M i k)]
  [inst_2 : (p : (i : ι) → κ i) → AddCommMonoid (N p)] [inst_3 : (i : ι) → (k : κ i) → Module R (M i k)]
  [inst_4 : (p : (i : ι) → κ i) → Module R (N p)] [inst_5 : Fintype ι] [inst_6 : (i : ι) → DecidableEq (κ i)]
  (f : (p : (i : ι) → κ i) → MultilinearMap R (fun i => M i (p i)) (N p)) (p : (i : ι) → κ i) (m : (i : ι) → M i (p i)),
  ((MultilinearMap.piFamily f) fun i => Pi.single (p i) (m i)) = Pi.single p ((f p) m)

When applied to a family of finitely-supported functions each supported on a single element, piFamily is itself supported on a single element, with value equal to the map f applied at that point.

Defined in

Mathlib.LinearAlgebra.Multilinear.Pi

Cited by

1 results in Mathlib

Foundations

Depth 59 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidAddCommMonoidModuleModuleFintypeDecidableEq