cpow_inv_two_re — Mathlib

English

If n ≠ 0 and x ∈ ℝ, then 0 < ((n^x)^{-1}).

Русский

Если n ≠ 0 и x ∈ ℝ, то 0 < (n^x)^{-1}.

LaTeX

$$Let n ∈ ℕ with n ≠ 0 and x ∈ ℝ. 0 < ((n : ℂ) ^ (x : ℂ))^{-1}.$$

Lean4

theorem cpow_inv_two_re (x : ℂ) : (x ^ (2⁻¹ : ℂ)).re = √((‖x‖ + x.re) / 2) :=
  by
  rw [← ofReal_ofNat, ← ofReal_inv, cpow_ofReal_re, ← div_eq_mul_inv, ← one_div, ← Real.sqrt_eq_rpow, cos_half, ←
    sqrt_mul, ← mul_div_assoc, mul_add, mul_one, norm_mul_cos_arg]
  exacts [norm_nonneg _, (neg_pi_lt_arg _).le, arg_le_pi _]

Похожие утверждения