Theorems · Theorem · algebraic topology

VectorPrebundle.mk.injEq

∀ {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

0 results in Mathlib

Foundations

Depth 170 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommMonoidModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpace