contMDiffOn_symm — Mathlib

English

If a VectorBundleCore has the IsContMDiff mixin, then the associated vector bundle is a ContMDiffVectorBundle.

Русский

Векторное расслоение гладко дифференцируемо по координатам во всех местах.

LaTeX

$$$$\\text{ContMDiffVectorBundle } n F Z.Fiber IB$$$$

Lean4

theorem contMDiffOn_symm (e : Trivialization F (π F E)) [MemTrivializationAtlas e] :
    ContMDiffOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) n e.toOpenPartialHomeomorph.symm e.target :=
  by
  rw [e.contMDiffOn_iff e.toOpenPartialHomeomorph.symm_mapsTo]
  refine ⟨contMDiffOn_fst.congr fun x hx ↦ e.proj_symm_apply hx, contMDiffOn_snd.congr fun x hx ↦ ?_⟩
  rw [e.apply_symm_apply hx]

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