smul_lt_smul_of_lt_of_le' — Mathlib

English

If a1 < a2 and b ≥ 0 with hb ≥ 0, then a1 • b < a2 • b.

Русский

Если a1 < a2 и b ≥ 0, тогда a1 • b < a2 • b.

LaTeX

$$$a_1 < a_2 \Rightarrow a_1 \cdot b < a_2 \cdot b \quad( b \ge 0)$$$

Lean4

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

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