mem_interior_iff_sbtw — Mathlib

English

A point lies in the interior of a 1-simplex if and only if it lies strictly between its two vertices.

Русский

Точка принадлежит interior 1‑слоям тогда и только тогда, когда она лежит строго между двумя вершинами.

LaTeX

$$p ∈ s.interior ⇔ Sbtw(R, s.points 0, p, s.points 1)$$

Lean4

/-- A point lies in the interior of a 1-simplex if and only if it lies strictly between its
vertices. -/
theorem mem_interior_iff_sbtw [Nontrivial R] [NoZeroSMulDivisors R V] {s : Simplex R P 1} {p : P} :
    p ∈ s.interior ↔ Sbtw R (s.points 0) p (s.points 1) :=
  by
  rw [interior_eq_image_Ioo, sbtw_iff_mem_image_Ioo_and_ne]
  simp [s.independent.injective.ne (by decide : (0 : Fin 2) ≠ 1)]

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