isLittleO_congr — Mathlib

English

IsLittleO is preserved under an OpenPartialHomeomorph: f = o[𝓝 b] g iff f∘e = o[𝓝 (e.symm b)] g∘e.

Русский

IsLittleO сохраняется при отображении через OpenPartialHomeomorph: f = o[𝓝 b] g эквивалентно f∘e = o[𝓝 (e.symm b)] g∘e.

LaTeX

$$$ f =o[\\mathcal{N} b] g \\iff (f \\circ e) =o[\\mathcal{N} (e.symm b)] (g \\circ e) $$$

Lean4

/-- Transfer `IsLittleO` over an `OpenPartialHomeomorph`. -/
theorem isLittleO_congr (e : OpenPartialHomeomorph α β) {b : β} (hb : b ∈ e.target) {f : β → E} {g : β → F} :
    f =o[𝓝 b] g ↔ (f ∘ e) =o[𝓝 (e.symm b)] (g ∘ e) :=
  by
  simp only [IsLittleO_def]
  exact forall₂_congr fun c _hc => e.isBigOWith_congr hb

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