smul_lt_smul_of_lt_of_le — 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

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

Lean4

theorem smul_lt_smul_of_lt_of_le [PosSMulMono α β] [SMulPosStrictMono α β] (ha : a₁ < a₂) (hb : b₁ ≤ b₂) (h₁ : 0 ≤ a₁)
    (h₂ : 0 < b₂) : a₁ • b₁ < a₂ • b₂ :=
  (smul_le_smul_of_nonneg_left hb h₁).trans_lt (smul_lt_smul_of_pos_right ha h₂)

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