Theorems · Theorem · algebraic topology

Bundle.Pretrivialization.continuousLinearMapCoordChange.congr_simp

∀ {𝕜₁ : Type u_1} [inst : NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [inst_1 : NontriviallyNormedField 𝕜₂]
  (σ : 𝕜₁ →+* 𝕜₂) {B : Type u_3} {F₁ : Type u_4} [inst_2 : NormedAddCommGroup F₁] [inst_3 : NormedSpace 𝕜₁ F₁]
  {E₁ : B → Type u_5} [inst_4 : (x : B) → AddCommGroup (E₁ x)] [inst_5 : (x : B) → Module 𝕜₁ (E₁ x)]
  [inst_6 : TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [inst_7 : NormedAddCommGroup F₂]
  [inst_8 : NormedSpace 𝕜₂ F₂] {E₂ : B → Type u_7} [inst_9 : (x : B) → AddCommGroup (E₂ x)]
  [inst_10 : (x : B) → Module 𝕜₂ (E₂ x)] [inst_11 : TopologicalSpace (Bundle.TotalSpace F₂ E₂)]
  [inst_12 : TopologicalSpace B] (e₁ e₁_1 : Bundle.Trivialization F₁ Bundle.TotalSpace.proj) (e_e₁ : e₁ = e₁_1)
  (e₁' e₁'_1 : Bundle.Trivialization F₁ Bundle.TotalSpace.proj) (e_e₁' : e₁' = e₁'_1)
  (e₂ e₂_1 : Bundle.Trivialization F₂ Bundle.TotalSpace.proj) (e_e₂ : e₂ = e₂_1)
  (e₂' e₂'_1 : Bundle.Trivialization F₂ Bundle.TotalSpace.proj) (e_e₂' : e₂' = e₂'_1)
  [inst_13 : Bundle.Trivialization.IsLinear 𝕜₁ e₁] [inst_14 : Bundle.Trivialization.IsLinear 𝕜₁ e₁']
  [inst_15 : Bundle.Trivialization.IsLinear 𝕜₂ e₂] [inst_16 : Bundle.Trivialization.IsLinear 𝕜₂ e₂'] (b b_1 : B),
  b = b_1 →
    Bundle.Pretrivialization.continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b =
      Bundle.Pretrivialization.continuousLinearMapCoordChange σ e₁_1 e₁'_1 e₂_1 e₂'_1 b_1

Defined in

Mathlib.Topology.VectorBundle.Hom

Cited by

0 results in Mathlib

Foundations

Depth 165 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceTopologicalSpaceBundle.Trivialization.IsLinearBundle.Trivialization.IsLinearBundle.Trivialization.IsLinearBundle.Trivialization.IsLinear