inf_rng — Mathlib

English

Let f: α → β and u1, u2... a set of UniformSpace α; UniformContinuous[u1, iInf u] f iff UniformContinuous[u1, u] f for all u in the index set.

Русский

Пусть f: α → β и множество UniformSpace α; UniformContinuous[u1, iInf u] f эквивалентно UniformContinuous[u1, u] f для всех u.

LaTeX

$$UniformContinuous[u1, iInf u] f ↔ ∀ u ∈ some_index_set, UniformContinuous[u1, u] f$$

Lean4

theorem inf_rng {f : α → β} {u₁ : UniformSpace α} {u₂ u₃ : UniformSpace β} (h₁ : UniformContinuous[u₁, u₂] f)
    (h₂ : UniformContinuous[u₁, u₃] f) : UniformContinuous[u₁, u₂ ⊓ u₃] f :=
  tendsto_inf.mpr ⟨h₁, h₂⟩

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