Theorems · Theorem · algebraic topology

FiberPrebundle.mk.inj

∀ {B : Type u_2} {F : Type u_3} {E : B → Type u_5} {inst : TopologicalSpace B} {inst_1 : TopologicalSpace F}
  {inst_2 : (x : B) → TopologicalSpace (E x)}
  {pretrivializationAtlas : Set (Bundle.Pretrivialization F Bundle.TotalSpace.proj)}
  {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}
  {continuous_trivChange :
    ∀ e ∈ pretrivializationAtlas,
      ∀ e' ∈ pretrivializationAtlas, ContinuousOn (↑e ∘ ↑e'.symm) (e'.target ∩ ↑e'.symm ⁻¹' e.source)}
  {totalSpaceMk_isInducing : ∀ (b : B), Topology.IsInducing (↑(pretrivializationAt b) ∘ Bundle.TotalSpace.mk b)}
  {pretrivializationAtlas_1 : Set (Bundle.Pretrivialization F Bundle.TotalSpace.proj)}
  {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}
  {continuous_trivChange_1 :
    ∀ e ∈ pretrivializationAtlas_1,
      ∀ e' ∈ pretrivializationAtlas_1, ContinuousOn (↑e ∘ ↑e'.symm) (e'.target ∩ ↑e'.symm ⁻¹' e.source)}
  {totalSpaceMk_isInducing_1 : ∀ (b : B), Topology.IsInducing (↑(pretrivializationAt_1 b) ∘ Bundle.TotalSpace.mk b)},
  { pretrivializationAtlas := pretrivializationAtlas, pretrivializationAt := pretrivializationAt,
        mem_base_pretrivializationAt := mem_base_pretrivializationAt,
        pretrivialization_mem_atlas := pretrivialization_mem_atlas, continuous_trivChange := continuous_trivChange,
        totalSpaceMk_isInducing := totalSpaceMk_isInducing } =
      { pretrivializationAtlas := pretrivializationAtlas_1, pretrivializationAt := pretrivializationAt_1,
        mem_base_pretrivializationAt := mem_base_pretrivializationAt_1,
        pretrivialization_mem_atlas := pretrivialization_mem_atlas_1, continuous_trivChange := continuous_trivChange_1,
        totalSpaceMk_isInducing := totalSpaceMk_isInducing_1 } →
    pretrivializationAtlas = pretrivializationAtlas_1 ∧ pretrivializationAt = pretrivializationAt_1

Defined in

Mathlib.Topology.FiberBundle.Basic

Cited by

1 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound