inv_mul_right_le — Mathlib

English

If a ≤ 0, then a ≤ a·(b⁻¹·b); equality when b ≠ 0.

Русский

Если a ≤ 0, то a ≤ a·(b⁻¹·b); равенство при b ≠ 0.

LaTeX

$$$\forall a\,b\; (a \le 0) \Rightarrow a \le a \cdot (b^{-1} \cdot b) \\ \text{and } (b \neq 0 \Rightarrow a = a \cdot (b^{-1} \cdot b))$$$

Lean4

/-- Equality holds when `b ≠ 0`. See `inv_mul_cancel_right`. -/
theorem inv_mul_right_le (ha : 0 ≤ a) : a * b⁻¹ * b ≤ a := by obtain rfl | hb := eq_or_ne b 0 <;> simp [*]

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