tendsto_lift₂_nhds — Mathlib

English

For x ∈ X and y ∈ Y, lift₂ f hf (mk x) (mk y) = f x y.

Русский

Для x ∈ X и y ∈ Y, lift₂ f hf (mk x) (mk y) = f x y.

LaTeX

$$lift₂ f hf (mk x) (mk y) = f x y$$

Lean4

@[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
    {l : Filter α} : Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l :=
  by
  rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
  rfl

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