addLECancellable_iff_ne — Mathlib

English

In ENNReal, a^n = ∞ exactly when a = ∞ and n ≠ 0.

Русский

В ENNReal, a^n = ∞ точно когда a = ∞ и n ≠ 0.

LaTeX

$$$a^{n} = \infty \iff a = \infty \land n \neq 0$$$

Lean4

/-- An element `a` is `AddLECancellable` if `a + b ≤ a + c` implies `b ≤ c` for all `b` and `c`.
  This is true in `ℝ≥0∞` for all elements except `∞`. -/
@[simp]
theorem addLECancellable_iff_ne {a : ℝ≥0∞} : AddLECancellable a ↔ a ≠ ∞ :=
  by
  constructor
  · rintro h rfl
    refine zero_lt_one.not_ge (h ?_)
    simp
  · rintro h b c hbc
    apply ENNReal.le_of_add_le_add_left h hbc

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