English
Given a homeomorphism φ: α ≃ₜ γ and a uniform isomorphism ψ: β ≃ᵤ δ, there is a natural uniform isomorphism C(α, β) ≃ᵤ C(γ, δ) sending f to ψ ∘ f ∘ φ⁻¹.
Русский
Дано гамма-объемное преобразование φ и униформное изоморфизм ψ, существует естественное униформное соответствие между пространствами отображений: C(α, β) ≃ᵤ C(γ, δ), заданное f ↦ ψ ∘ f ∘ φ⁻¹.
LaTeX
$$$\varphi : α \simeq_t γ$, $\psi : β \simeq_u δ$ ⟹ $C(α, β) \simeq_u C(γ, δ)$, \\ f \mapsto ψ\circ f\circ φ^{-1}$$$
Lean4
/-- Any pair of a homeomorphism `X ≃ₜ Z` and an isomorphism `Y ≃ᵤ T` of uniform spaces gives rise
to an isomorphism `C(X, Y) ≃ᵤ C(Z, T)`. -/
protected def _root_.UniformEquiv.arrowCongr (φ : α ≃ₜ γ) (ψ : β ≃ᵤ δ) : C(α, β) ≃ᵤ C(γ, δ)
where
toFun f := .comp ψ.toHomeomorph <| f.comp φ.symm
invFun f := .comp ψ.symm.toHomeomorph <| f.comp φ
left_inv f := ext fun _ ↦ ψ.left_inv (f _) |>.trans <| congrArg f <| φ.left_inv _
right_inv f := ext fun _ ↦ ψ.right_inv (f _) |>.trans <| congrArg f <| φ.right_inv _
uniformContinuous_toFun := uniformContinuous_comp _ ψ.uniformContinuous |>.comp <| uniformContinuous_comp_left _
uniformContinuous_invFun := uniformContinuous_comp _ ψ.symm.uniformContinuous |>.comp <| uniformContinuous_comp_left _
