inner_nonneg_of_dist_le_radius — Mathlib

English

Given a point p1 on a sphere s and p2 inside s, the inner product ⟪p1−p2, p1−center⟫ is strictly positive or p1=p2.

Русский

Пусть точка p1 лежит на сфере s, а q внутри сферы; тогда скалярное произведение ⟪p1−p2, p1−центр⟫ strictly положительно или p1 = p2.

LaTeX

$$0 < ⟨p1−p2, p1−center⟩ ∨ p1 = p2$$

Lean4

/-- Given a point on a sphere and a point not outside it, the inner product between the
difference of those points and the radius vector is nonnegative. -/
theorem inner_nonneg_of_dist_le_radius {s : Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : dist p₂ s.center ≤ s.radius) :
    0 ≤ ⟪p₁ -ᵥ p₂, p₁ -ᵥ s.center⟫ :=
  by
  rcases inner_pos_or_eq_of_dist_le_radius hp₁ hp₂ with (h | rfl)
  · exact h.le
  · simp

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