comap_exp_nhdsNE — Mathlib

English

Exponential map along nhds excluding 0 equals the same as comap re atBot; i.e., lim along nonzero neighborhood approaches 0 corresponds to Re z → -∞.

Русский

Экспонента вдоль nhds без нуля эквивалентно comap re atBot; т.е. предел при Re z → −∞.

LaTeX

$$$ \operatorname{comap} (\exp) (\mathcal{N}_{\neq 0}) = \operatorname{comap} (\operatorname{re}) (\operatorname{atBot}) $$$

Lean4

theorem comap_exp_nhdsNE : comap exp (𝓝[≠] 0) = comap re atBot :=
  by
  have : (exp ⁻¹' {0})ᶜ = Set.univ := eq_univ_of_forall exp_ne_zero
  simp [nhdsWithin, comap_exp_nhds_zero, this]

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