mem_of_source_eq_empty — Mathlib

English

ContDiffOn 0 f s equals ContinuousOn f s.

Русский

ContDiffOn 0 f s равняется ContinuousOn f s.

LaTeX

$$$ContDiffOn\ 𝕜\ 0\ f\ s = ContinuousOn\ f\ s$$$

Lean4

/-- Any change of coordinates with empty source belongs to `contDiffGroupoid`. -/
theorem mem_of_source_eq_empty (f : OpenPartialHomeomorph H H) (hf : f.source = ∅) : f ∈ contDiffGroupoid n I :=
  by
  constructor
  · intro x ⟨hx, _⟩
    rw [mem_preimage] at hx
    simp_all only [mem_empty_iff_false]
  · intro x ⟨hx, _⟩
    have : f.target = ∅ := by simp [← f.image_source_eq_target, hf]
    simp_all

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