closedBall_subset_trichotomy — Mathlib

English

Two closed balls with the same radius are either equal or disjoint.

Русский

Два замкнутых шара одинакового радиуса либо равны, либо дисjoint.

LaTeX

$$closedBall(x,r) = closedBall(y,r) ∨ Disjoint(closedBall(x,r), closedBall(y,r))$$

Lean4

theorem closedBall_subset_trichotomy :
    closedBall x r ⊆ closedBall y s ∨ closedBall y s ⊆ closedBall x r ∨ Disjoint (closedBall x r) (closedBall y s) :=
  by
  wlog hrs : r ≤ s generalizing x y r s
  · rw [disjoint_comm, ← or_assoc, or_comm (b := _ ⊆ _), or_assoc]
    exact this y x s r (lt_of_not_ge hrs).le
  · refine Set.disjoint_or_nonempty_inter (closedBall x r) (closedBall y s) |>.symm.imp (fun h ↦ ?_) (Or.inr ·)
    obtain ⟨hxz, hyz⟩ := (Set.mem_inter_iff _ _ _).mp h.some_mem
    have hx := closedBall_subset_closedBall hrs (x := x)
    rwa [closedBall_eq_of_mem hyz |>.trans (closedBall_eq_of_mem <| hx hxz).symm]

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