English
A pair of fundamental isomorphisms obtained from the σ-twisted homomorphism trivializations equals the pair assembled from each component trivialization via the σ-twisted linear map.
Русский
Пары основных изоморфизмΩ σ-сложно-тривиализации эквивалентны парам, собранным из компонентных тривиализаций через σ-скошенную линейную карту.
LaTeX
$$$\\text{Eq} \\Big( \\operatorname{trivializationAt} (F_1 \\to SL[\\sigma] F_2) (\\lambda x \\mapsto E_1 x \\to SL[\\sigma] E_2 x) x_0) \\big( \\operatorname{translations} \\Big)$$$
Lean4
theorem continuousWithinAt_hom_bundle {M : Type*} [TopologicalSpace M]
(f : M → TotalSpace (F₁ →SL[σ] F₂) (fun x ↦ E₁ x →SL[σ] E₂ x)) {s : Set M} {x₀ : M} :
ContinuousWithinAt f s x₀ ↔
ContinuousWithinAt (fun x ↦ (f x).1) s x₀ ∧
ContinuousWithinAt (fun x ↦ inCoordinates F₁ E₁ F₂ E₂ (f x₀).1 (f x).1 (f x₀).1 (f x).1 (f x).2) s x₀ :=
FiberBundle.continuousWithinAt_totalSpace ..
