piFamily_compLinearMap_lsingle — Mathlib

English

The piFamily of the zero family is the zero map.

Русский

piFamily нулевой семейства является нулевым отображением.

LaTeX

$$$\piFamily 0 = 0$$$

Lean4

@[simp]
theorem piFamily_compLinearMap_lsingle [Fintype ι] [∀ i, DecidableEq (κ i)]
    (f : Π (p : Π i, κ i), MultilinearMap R (fun i ↦ M i (p i)) (N p)) (p : ∀ i, κ i) :
    (piFamily f).compLinearMap (fun i => LinearMap.single _ _ (p i)) =
      (LinearMap.single _ _ p).compMultilinearMap (f p) :=
  MultilinearMap.ext <| piFamily_single f p

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