smul_lt_smul_of_le_of_lt — Mathlib

English

If a1 ≤ a2 and b1 < b2 and 0 < a1 and 0 ≤ b2, then a1 • b1 < a2 • b2.

Русский

Если a1 ≤ a2 и b1 < b2 и 0 < a1 и 0 ≤ b2, то a1 • b1 < a2 • b2.

LaTeX

$$$a_1 \le a_2 \;\land\; b_1 < b_2 \;\land\; 0 < a_1 \;\land\; 0 \le b_2\Rightarrow a_1 \cdot b_1 < a_2 \cdot b_2$$$

Lean4

theorem smul_lt_smul_of_le_of_lt [PosSMulStrictMono α β] [SMulPosMono α β] (ha : a₁ ≤ a₂) (hb : b₁ < b₂) (h₁ : 0 < a₁)
    (h₂ : 0 ≤ b₂) : a₁ • b₁ < a₂ • b₂ :=
  (smul_lt_smul_of_pos_left hb h₁).trans_le (smul_le_smul_of_nonneg_right ha h₂)

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