Theorems · Definition · algebraic topology

VectorPrebundle.recOn

{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)] →
                      {motive : VectorPrebundle R F E → Sort u} →
                        (t : VectorPrebundle R F E) →
                          ((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)) →
                                          motive
                                            { 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 }) →
                            motive t

Defined in

Mathlib.Topology.VectorBundle.Basic

Cited by

0 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldAddCommMonoidModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpace