mul_lt_one' — 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 порядке с правой монотонностью умножения, то a·b < 1.

LaTeX

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

Lean4

/-- Assumes right covariance.
The lemma assuming left covariance is `Left.mul_lt_one'`. -/
@[to_additive /-- Assumes right covariance.
    The lemma assuming left covariance is `Left.add_neg'`. -/
    ]
theorem mul_lt_one' [MulRightMono α] {a b : α} (ha : a < 1) (hb : b < 1) : a * b < 1 :=
  mul_lt_of_lt_one_of_lt' ha hb

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