of_addGroup_of_nhds_zero — Mathlib

English

If every R_i is a topological semiring, then the family map i ↦ R_i has ContinuousAdd.

Русский

Если каждый R_i — топологическое полукольцо, то семейство R_i имеет ContinuousAdd.

LaTeX

$$ContinuousAdd ((i : ι) → R i)$$

Lean4

theorem of_addGroup_of_nhds_zero [IsTopologicalAddGroup R]
    (hmul : Tendsto (uncurry ((· * ·) : R → R → R)) (𝓝 0 ×ˢ 𝓝 0) <| 𝓝 0)
    (hmul_left : ∀ x₀ : R, Tendsto (fun x : R => x₀ * x) (𝓝 0) <| 𝓝 0)
    (hmul_right : ∀ x₀ : R, Tendsto (fun x : R => x * x₀) (𝓝 0) <| 𝓝 0) : IsTopologicalRing R where
  continuous_mul := by
    refine continuous_of_continuousAt_zero₂ (AddMonoidHom.mul (R := R)) ?_ ?_ ?_ <;>
      simpa only [ContinuousAt, mul_zero, zero_mul, nhds_prod_eq, AddMonoidHom.mul_apply]

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