closedInterior_eq_singleton — Mathlib

English

Over a characteristic-zero division ring, centroids of two faces are equal if and only if the faces are the same.

Русский

Нормально: центроиды двух граней равны тогда и только тогда, когда грани совпадают.

LaTeX

$$$ fs_1.centroid_k s.points = fs_2.centroid_k s.points \iff fs_1 = fs_2 $ (CharZero k)$$

Lean4

@[simp]
theorem closedInterior_eq_singleton [ZeroLEOneClass k] (s : Simplex k P 0) : s.closedInterior = {s.points 0} :=
  by
  ext p
  simp only [Simplex.closedInterior, Simplex.setInterior, Nat.reduceAdd, univ_unique, Fin.default_eq_zero, Fin.isValue,
    sum_singleton, Set.mem_Icc, Set.mem_setOf_eq, Set.mem_singleton_iff]
  constructor
  · rintro ⟨w, h0, hi, rfl⟩
    simp [affineCombination_apply, h0]
  · rintro rfl
    exact ⟨1, by simp [affineCombination_apply]⟩

Похожие утверждения