extend_image_target_mem_nhds — Mathlib

English

The image of f.target under the extended chart lies in nhdsWithin of its image.

Русский

Образ f.target под расширенным чартом лежит в nhdsWithin вокруг образа.

LaTeX

$$$ I \\,' f.target \\in \\mathcal{N}_{range(I)}((f^{\\mathrm{extend}}_I)(y)). $$$

Lean4

theorem extend_image_target_mem_nhds {x : M} (hx : x ∈ f.source) : I '' f.target ∈ 𝓝[range I] (f.extend I) x :=
  by
  rw [← f.map_extend_nhds hx, Filter.mem_map, f.extend_coe, Set.preimage_comp, I.preimage_image f.target]
  exact (f.continuousAt hx).preimage_mem_nhds (f.open_target.mem_nhds (f.map_source hx))

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