mul_lt_one_of_le_of_lt — Mathlib

English

If a ≤ 1 and b < 1 in a preorder monoid with right monotone multiplication, then a·b < 1.

Русский

Если a ≤ 1 и b < 1 в предельном моноиде с правой монотонностью умножения, то a·b < 1.

LaTeX

$$$\forall a,b\in \alpha\; (a \le 1) \to (b < 1) \Rightarrow a b < 1$$$

Lean4

/-- Assumes right covariance.
The lemma assuming left covariance is `Left.mul_lt_one_of_le_of_lt`. -/
@[to_additive Right.add_neg_of_nonpos_of_neg /-- Assumes right covariance.
          The lemma assuming left covariance is `Left.add_neg_of_nonpos_of_neg`. -/
    ]
theorem mul_lt_one_of_le_of_lt [MulRightMono α] {a b : α} (ha : a ≤ 1) (hb : b < 1) : a * b < 1 :=
  mul_lt_of_le_one_of_lt ha hb

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