map_mk_symm_nhds — Mathlib

English

For every x, the comap of the neighborhood filter along mk equals the neighborhood filter at mk.symm x.

Русский

Для каждого x комап окрестностей вдоль mk равен окрестностям mk.symm x.

LaTeX

$$$$ \operatorname{comap}(\mathrm{mk}, \mathcal{N}(x)) = \mathcal{N}(\mathrm{mk}^{-1}(x)) $$$$

Lean4

@[to_additive (attr := simp)]
theorem map_mk_symm_nhds (x : Mᵈᵐᵃ) : map (mk.symm : Mᵈᵐᵃ → M) (𝓝 x) = 𝓝 (mk.symm x) :=
  mkHomeomorph.symm.map_nhds_eq x

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