contMDiff_of_contMDiffOn_union_of_isOpen

English

TheContMDiffOn on a union equals the conjunction of ContMDiffOn on each piece when the pieces are open.

Русский

ContMDiffOn на объединении равен конъюнкции ContMDiffOn на каждой части, когда части открыты.

LaTeX

$$ContMDiffOn I I' n f (s ∪ t) ↔ ContMDiffOn I I' n f s ∧ ContMDiffOn I I' n f t$$

Lean4

theorem contMDiff_of_contMDiffOn_union_of_isOpen (hf : ContMDiffOn I I' n f s) (hf' : ContMDiffOn I I' n f t)
    (hst : s ∪ t = univ) (hs : IsOpen s) (ht : IsOpen t) : ContMDiff I I' n f :=
  by
  rw [← contMDiffOn_univ, ← hst]
  exact hf.union_of_isOpen hf' hs ht

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