min_lt_max_of_mul_lt_mul — Mathlib

English

If a and b are such that a<b and c<d, then a·b < c·d implies min(a,b) < max(c,d) given left and right monotonicity.

Русский

Если a< b и c< d и произведение удовлетворяет неравенству, то min(a,b) < max(c,d) при условии монотонности слева и справа.

LaTeX

$$$$ \\forall a,b,c,d \\in \\alpha,\n a < b \\land c < d \\Rightarrow \\text{(if } a\\cdot b < c\\cdot d) \\Rightarrow \\min(a,b) < \\max(c,d) $$$$

Lean4

@[to_additive]
theorem min_lt_max_of_mul_lt_mul [MulLeftMono α] [MulRightMono α] (h : a * b < c * d) : min a b < max c d := by
  simp_rw [min_lt_iff, lt_max_iff]; contrapose! h; exact mul_le_mul' h.1.1 h.2.2

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