English
For odd n, the map a → a^n is strictly increasing on the ordered ring.
Русский
Для нечетного n отображение a → a^n strictly возрастает на упорядоченной структуре.
LaTeX
$$$a < b \Rightarrow a^n < b^n$ (n odd)$$
Lean4
theorem strictMono_pow (hn : Odd n) : StrictMono fun a : R => a ^ n :=
by
have hn₀ : n ≠ 0 := by rintro rfl; simp [Odd, eq_comm (a := 0)] at hn
intro a b hab
obtain ha | ha := le_total 0 a
· exact pow_lt_pow_left₀ hab ha hn₀
obtain hb | hb := lt_or_ge 0 b
· exact (hn.pow_nonpos ha).trans_lt (pow_pos hb _)
obtain ⟨c, hac⟩ := exists_add_of_le ha
obtain ⟨d, hbd⟩ := exists_add_of_le hb
have hd := nonneg_of_le_add_right (hb.trans_eq hbd)
refine lt_of_add_lt_add_right (a := c ^ n + d ^ n) ?_
dsimp
calc
a ^ n + (c ^ n + d ^ n) = d ^ n := by rw [← add_assoc, hn.pow_add_pow_eq_zero hac.symm, zero_add]
_ < c ^ n := (pow_lt_pow_left₀ ?_ hd hn₀)
_ = b ^ n + (c ^ n + d ^ n) := by rw [add_left_comm, hn.pow_add_pow_eq_zero hbd.symm, add_zero]
refine lt_of_add_lt_add_right (a := a + b) ?_
rwa [add_rotate', ← hbd, add_zero, add_left_comm, ← add_assoc, ← hac, zero_add]
