inv_le_iff_le_mul — Mathlib

English

If for every NNReal r we have r ≤ x implies r ≤ y, then x ≤ y.

Русский

Если для каждого NNReal r верно r ≤ x ⇒ r ≤ y, тогда x ≤ y.

LaTeX

$$$\\forall r \\in \\mathbb{NNReal},\\; (r \\\\le x) \\,\\Rightarrow \\, (r \\\\le y) \\Rightarrow x \\\\le y$$$

Lean4

theorem inv_le_iff_le_mul (h₁ : b = ∞ → a ≠ 0) (h₂ : a = ∞ → b ≠ 0) : a⁻¹ ≤ b ↔ 1 ≤ a * b :=
  by
  rw [← one_div, ENNReal.div_le_iff_le_mul, mul_comm]
  exacts [or_not_of_imp h₁, not_or_of_imp h₂]

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