English
The function log_b is strictly anti-monotone on the negative reals: if x < y < 0 then log_b x > log_b y.
Русский
Функция log_b строго убывает на отрицательных числах: если x < y < 0, то log_b x > log_b y.
LaTeX
$$$1 < b \Rightarrow \text{StrictAntiOn}(\log_b, \mathbb{R}_{<0}).$$$
Lean4
theorem strictAntiOn_logb : StrictAntiOn (logb b) (Set.Iio 0) :=
by
rintro x (hx : x < 0) y (hy : y < 0) hxy
rw [← logb_abs b y, ← logb_abs b x]
refine logb_lt_logb hb (abs_pos.2 hy.ne) ?_
rwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff]