Ioi_add_bij — Mathlib

English

Any x in Ioo(0,1) satisfies x < 1.

Русский

Любой x ∈ Ioo(0,1) удовлетворяет x < 1.

LaTeX

$$$\forall x \in (0,1),\ x < 1$$$

Lean4

theorem Ioi_add_bij : BijOn (· + d) (Ioi a) (Ioi (a + d)) :=
  by
  refine ⟨fun x h => add_lt_add_right (mem_Ioi.mp h) _, fun _ _ _ _ h => add_right_cancel h, fun _ h => ?_⟩
  obtain ⟨c, rfl⟩ := exists_add_of_le (mem_Ioi.mp h).le
  rw [mem_Ioi, add_right_comm, add_lt_add_iff_right] at h
  exact ⟨a + c, h, by rw [add_right_comm]⟩

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