disjoint_pi_univ_Ioc_update_left_right

English

The two pi-universal images of corresponding Ioc intervals are disjoint:

Русский

Две образа по всему индексу от соответствующих интервалов Ioc дисjoint.

LaTeX

$$$$ Disjoint\left( \pi univ\; (\lambda i. Ioc(x i, update y i0 m i)), \; \pi univ\; (\lambda i. Ioc(update x i0 m i, y i)) \right) $$$$

Lean4

theorem disjoint_pi_univ_Ioc_update_left_right {x y : ∀ i, α i} {i₀ : ι} {m : α i₀} :
    Disjoint (pi univ fun i ↦ Ioc (x i) (update y i₀ m i)) (pi univ fun i ↦ Ioc (update x i₀ m i) (y i)) :=
  by
  rw [disjoint_left]
  rintro z h₁ h₂
  refine (h₁ i₀ (mem_univ _)).2.not_gt ?_
  simpa only [Function.update_self] using (h₂ i₀ (mem_univ _)).1

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