English
If f and g are strictly monotone, then the product f(x)·g(x) is strictly monotone (under suitable left/right mono assumptions).
Русский
Если f и g строго монотонны, то произведение f(x)·g(x) строго монотонно (при соответствующих предпосылках по умножению слева/спright).
LaTeX
$$$StrictMono(f) \to\; StrictMono(g) \Rightarrow StrictMono(x \mapsto f(x) \cdot g(x))$$$
Lean4
/-- The product of a monotone function and a strictly monotone function is strictly monotone. -/
@[to_additive add_strictMono /-- The sum of a monotone function and a strictly monotone function is
strictly monotone. -/
]
theorem mul_strictMono' [MulLeftStrictMono α] [MulRightMono α] {f g : β → α} (hf : Monotone f) (hg : StrictMono g) :
StrictMono fun x => f x * g x := fun _ _ h => mul_lt_mul_of_le_of_lt (hf h.le) (hg h)
