map_extChartAt_symm_nhdsWithin_range'

English

The nhdsWithin of x under extChartAt maps to the nhdsWithin on the intersection with range I.

Русский

nhdsWithin(extChartAt I x) при отображении extChartAt I x переходит к nhdsWithin на пересечении с range I.

LaTeX

$$$ \\text{map}(extChartAt I x)(\\mathcal{N}[s]x) = \\mathcal{N}[extChartAt I x''((extChartAt I x).source \\cap s)](extChartAt I x x)$$$

Lean4

theorem map_extChartAt_symm_nhdsWithin_range' {x y : M} (hy : y ∈ (extChartAt I x).source) :
    map (extChartAt I x).symm (𝓝[range I] extChartAt I x y) = 𝓝 y :=
  map_extend_symm_nhdsWithin_range _ <| by rwa [← extChartAt_source I]

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