image_parallelepiped — Mathlib

English

Reindexing a family of vectors by a bijection does not change the parallelepiped.

Русский

Перестановка индексов семейства векторов не изменяет параллелепипед.

LaTeX

$$$\\mathrm{parallelepiped}(v \\circ e) = \\mathrm{parallelepiped}(v),$ где $e$ — биекция.$$

Lean4

theorem image_parallelepiped (f : E →ₗ[ℝ] F) (v : ι → E) : f '' parallelepiped v = parallelepiped (f ∘ v) :=
  by
  simp only [parallelepiped, ← image_comp]
  congr 1 with t
  simp only [Function.comp_apply, _root_.map_sum, LinearMap.map_smulₛₗ, RingHom.id_apply]

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