/-- For `B` a manifold and `F` a normed space, the groupoid on `B × F` consisting of local
homeomorphisms which are bi-smooth and fiberwise linear, and induce the identity on `B`.
When a (topological) vector bundle is smooth, then the composition of charts associated
to the vector bundle belong to this groupoid. -/
def smoothFiberwiseLinear : StructureGroupoid (B × F) where
  members :=
    ⋃ (φ : B → F ≃L[𝕜] F) (U : Set B) (hU : IsOpen U)
      (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => φ x : B → F →L[𝕜] F) U)
      (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (φ x).symm : B → F →L[𝕜] F) U),
        {e | e.EqOnSource (FiberwiseLinear.localHomeomorph φ hU hφ.continuousOn h2φ.continuousOn)}
  trans' := by
    simp only [mem_aux]
    rintro e e' ⟨φ, U, hU, hφ, h2φ, heφ⟩ ⟨φ', U', hU', hφ', h2φ', heφ'⟩
    refine' ⟨fun b => (φ b).trans (φ' b), _, hU.inter hU', _, _,
      Setoid.trans (PartialHomeomorph.EqOnSource.trans' heφ heφ') ⟨_, _⟩⟩
    · show
        SmoothOn IB 𝓘(𝕜, F →L[𝕜] F)
          (fun x : B => (φ' x).toContinuousLinearMap ∘L (φ x).toContinuousLinearMap) (U ∩ U')
      exact (hφ'.mono <| inter_subset_right _ _).clm_comp (hφ.mono <| inter_subset_left _ _)
    · show
        SmoothOn IB 𝓘(𝕜, F →L[𝕜] F)
          (fun x : B => (φ x).symm.toContinuousLinearMap ∘L (φ' x).symm.toContinuousLinearMap)
          (U ∩ U')
      exact (h2φ.mono <| inter_subset_left _ _).clm_comp (h2φ'.mono <| inter_subset_right _ _)
    · apply FiberwiseLinear.source_trans_localHomeomorph
    · rintro ⟨b, v⟩ -; apply FiberwiseLinear.trans_localHomeomorph_apply
  -- porting note: without introducing `e` first, the first `simp only` fails
  symm' := fun e ↦ by
    simp only [mem_iUnion]
    rintro ⟨φ, U, hU, hφ, h2φ, heφ⟩
    refine' ⟨fun b => (φ b).symm, U, hU, h2φ, _, PartialHomeomorph.EqOnSource.symm' heφ⟩
    simp_rw [ContinuousLinearEquiv.symm_symm]
    exact hφ
  id_mem' := by
    /- porting note: `simp_rw [mem_iUnion]` failed; expanding. Was:
    simp_rw [mem_iUnion]
    refine' ⟨fun b => ContinuousLinearEquiv.refl 𝕜 F, univ, isOpen_univ, _, _, ⟨_, fun b hb => _⟩⟩
    -/
    refine mem_iUnion.2 ⟨fun _ ↦ .refl 𝕜 F, mem_iUnion.2 ⟨univ, mem_iUnion.2 ⟨isOpen_univ, ?_⟩⟩⟩
    refine mem_iUnion.2 ⟨contMDiffOn_const, mem_iUnion.2 ⟨contMDiffOn_const, ?_, ?_⟩⟩
    · simp only [FiberwiseLinear.localHomeomorph, PartialHomeomorph.refl_localEquiv,
        PartialEquiv.refl_source, univ_prod_univ]
    · exact eqOn_refl id _
  locality' := by
    -- the hard work has been extracted to `locality_aux₁` and `locality_aux₂`
    simp only [mem_aux]
    intro e he
    obtain ⟨U, hU, h⟩ := SmoothFiberwiseLinear.locality_aux₁ e he
    

Proof state:
case intro.intro
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
e : PartialHomeomorph (B × F) (B × F)
he :
  ∀ x ∈ e.source,
    ∃ s,
      IsOpen s ∧
        x ∈ s ∧
          ∃ φ U,
            ∃ (hU : IsOpen U) (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x)) U) (h2φ :
              SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(ContinuousLinearEquiv.symm (φ x))) U),
              PartialHomeomorph.EqOnSource (PartialHomeomorph.restr e s)
                (FiberwiseLinear.localHomeomorph φ hU (_ : ContinuousOn (fun x => ↑(φ x)) U)
                  (_ : ContinuousOn (fun x => ↑(ContinuousLinearEquiv.symm (φ x))) U))
U : Set B
hU : e.source = U ×ˢ univ
h :
  ∀ x ∈ U,
    ∃ φ u,
      ∃ (hu : IsOpen u) (_ : u ⊆ U) (_ : x ∈ u) (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x)) u) (h2φ :
        SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(ContinuousLinearEquiv.symm (φ x))) u),
        PartialHomeomorph.EqOnSource (PartialHomeomorph.restr e (u ×ˢ univ))
          (FiberwiseLinear.localHomeomorph φ hu (_ : ContinuousOn (fun x => ↑(φ x)) u)
            (_ : ContinuousOn (fun x => ↑(ContinuousLinearEquiv.symm (φ x))) u))
⊢ ∃ φ U,
    ∃ (hU : IsOpen U) (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x)) U) (h2φ :
      SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(ContinuousLinearEquiv.symm (φ x))) U),
      PartialHomeomorph.EqOnSource e
        (FiberwiseLinear.localHomeomorph φ hU (_ : ContinuousOn (fun x => ↑(φ x)) U)
          (_ : ContinuousOn (fun x => ↑(ContinuousLinearEquiv.symm (φ x))) U))

Next tactic:
exact SmoothFiberwiseLinear.locality_aux₂ e U hU h