one_lt_mul_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 порядке моноида с правой монотонностью, 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.one_lt_mul_of_le_of_lt`. -/
@[to_additive Right.add_pos_of_nonneg_of_pos /-- Assumes right covariance.
    The lemma assuming left covariance is `Left.add_pos_of_nonneg_of_pos`. -/
    ]
theorem one_lt_mul_of_le_of_lt [MulRightMono α] {a b : α} (ha : 1 ≤ a) (hb : 1 < b) : 1 < a * b :=
  lt_mul_of_one_le_of_lt ha hb

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