Theorems · Theorem · algebraic topology

VectorPrebundle.mk.inj

∀ {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} {inst : NontriviallyNormedField R}
  {inst_1 : (x : B) → AddCommMonoid (E x)} {inst_2 : (x : B) → Module R (E x)} {inst_3 : NormedAddCommGroup F}
  {inst_4 : NormedSpace R F} {inst_5 : TopologicalSpace B} {inst_6 : (x : B) → TopologicalSpace (E x)}
  {pretrivializationAtlas : Set (Bundle.Pretrivialization F Bundle.TotalSpace.proj)}
  {pretrivialization_linear' : ∀ e ∈ pretrivializationAtlas, Bundle.Pretrivialization.IsLinear R e}
  {pretrivializationAt : B → Bundle.Pretrivialization F Bundle.TotalSpace.proj}
  {mem_base_pretrivializationAt : ∀ (x : B), x ∈ (pretrivializationAt x).baseSet}
  {pretrivialization_mem_atlas : ∀ (x : B), pretrivializationAt x ∈ pretrivializationAtlas}
  {exists_coordChange :
    ∀ e ∈ pretrivializationAtlas,
      ∀ e' ∈ pretrivializationAtlas,
        ∃ f,
          ContinuousOn f (e.baseSet ∩ e'.baseSet) ∧
            ∀ b ∈ e.baseSet ∩ e'.baseSet, ∀ (v : F), (f b) v = (↑e' ⟨b, e.symm b v⟩).2}
  {totalSpaceMk_isInducing : ∀ (b : B), Topology.IsInducing (↑(pretrivializationAt b) ∘ Bundle.TotalSpace.mk b)}
  {pretrivializationAtlas_1 : Set (Bundle.Pretrivialization F Bundle.TotalSpace.proj)}
  {pretrivialization_linear'_1 : ∀ e ∈ pretrivializationAtlas_1, Bundle.Pretrivialization.IsLinear R e}
  {pretrivializationAt_1 : B → Bundle.Pretrivialization F Bundle.TotalSpace.proj}
  {mem_base_pretrivializationAt_1 : ∀ (x : B), x ∈ (pretrivializationAt_1 x).baseSet}
  {pretrivialization_mem_atlas_1 : ∀ (x : B), pretrivializationAt_1 x ∈ pretrivializationAtlas_1}
  {exists_coordChange_1 :
    ∀ e ∈ pretrivializationAtlas_1,
      ∀ e' ∈ pretrivializationAtlas_1,
        ∃ f,
          ContinuousOn f (e.baseSet ∩ e'.baseSet) ∧
            ∀ b ∈ e.baseSet ∩ e'.baseSet, ∀ (v : F), (f b) v = (↑e' ⟨b, e.symm b v⟩).2}
  {totalSpaceMk_isInducing_1 : ∀ (b : B), Topology.IsInducing (↑(pretrivializationAt_1 b) ∘ Bundle.TotalSpace.mk b)},
  { pretrivializationAtlas := pretrivializationAtlas, pretrivialization_linear' := pretrivialization_linear',
        pretrivializationAt := pretrivializationAt, mem_base_pretrivializationAt := mem_base_pretrivializationAt,
        pretrivialization_mem_atlas := pretrivialization_mem_atlas, exists_coordChange := exists_coordChange,
        totalSpaceMk_isInducing := totalSpaceMk_isInducing } =
      { pretrivializationAtlas := pretrivializationAtlas_1, pretrivialization_linear' := pretrivialization_linear'_1,
        pretrivializationAt := pretrivializationAt_1, mem_base_pretrivializationAt := mem_base_pretrivializationAt_1,
        pretrivialization_mem_atlas := pretrivialization_mem_atlas_1, exists_coordChange := exists_coordChange_1,
        totalSpaceMk_isInducing := totalSpaceMk_isInducing_1 } →
    pretrivializationAtlas = pretrivializationAtlas_1 ∧ pretrivializationAt = pretrivializationAt_1

Defined in

Mathlib.Topology.VectorBundle.Basic

Cited by

1 results in Mathlib

Foundations

Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound