awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
B : Type u_3
inst_1 : TopologicalSpace B
F₁ : Type u_4
inst_2 : NormedAddCommGroup F₁
inst_3 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_4 : (x : B) → AddCommGroup (E₁ x)
inst_5 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_6 : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst_7 : NormedAddCommGroup F₂
inst_8 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_9 : (x : B) → AddCommGroup (E₂ x)
inst_10 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : (x : B) → TopologicalSpace (E₁ x)
inst_13 : FiberBundle F₁ E₁
inst_14 : (x : B) → TopologicalSpace (E₂ x)
inst_15 : FiberBundle F₂ E₂
e₁ : Trivialization (B := B) F₁ (Z := TotalSpace F₁ E₁) (TotalSpace.proj (E := E₁))
e₂ : Trivialization (B := B) F₂ (Z := TotalSpace F₂ E₂) (TotalSpace.proj (E := E₂))
inst_16 : Trivialization.IsLinear 𝕜 (E := E₁) e₁
inst_17 : Trivialization.IsLinear 𝕜 (E := E₂) e₂
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ [⋀^ι]→L[𝕜] F₂
⊢ b ∈
    baseSet (F := F₁ [⋀^ι]→L[𝕜] F₂) (Z :=
      TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
      (continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂)

Tactic:
exact hb
-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
B : Type u_3
inst_1 : TopologicalSpace B
F₁ : Type u_4
inst_2 : NormedAddCommGroup F₁
inst_3 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_4 : (x : B) → AddCommGroup (E₁ x)
inst_5 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_6 : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst_7 : NormedAddCommGroup F₂
inst_8 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_9 : (x : B) → AddCommGroup (E₂ x)
inst_10 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : (x : B) → TopologicalSpace (E₁ x)
inst_13 : FiberBundle F₁ E₁
inst_14 : (x : B) → TopologicalSpace (E₂ x)
inst_15 : FiberBundle F₂ E₂
e₁ e₁' : Trivialization (B := B) F₁ (Z := TotalSpace F₁ E₁) (TotalSpace.proj (E := E₁))
e₂ e₂' : Trivialization (B := B) F₂ (Z := TotalSpace F₂ E₂) (TotalSpace.proj (E := E₂))
inst_16 : Trivialization.IsLinear 𝕜 (E := E₁) e₁
inst_17 : Trivialization.IsLinear 𝕜 (E := E₁) e₁'
inst_18 : Trivialization.IsLinear 𝕜 (E := E₂) e₂
inst_19 : Trivialization.IsLinear 𝕜 (E := E₂) e₂'
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)
L : F₁ [⋀^ι]→L[𝕜] F₂
v : ι → F₁
⊢ (b ∈ e₁.baseSet ∧ b ∈ e₂.baseSet) ∧ b ∈ e₁'.baseSet ∧ b ∈ e₂'.baseSet

Tactic:
simp only [mem_inter_iff] at hb

Premises used: Set.mem_inter_iff
-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
B : Type u_3
inst_1 : TopologicalSpace B
F₁ : Type u_4
inst_2 : NormedAddCommGroup F₁
inst_3 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_4 : (x : B) → AddCommGroup (E₁ x)
inst_5 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_6 : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst_7 : NormedAddCommGroup F₂
inst_8 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_9 : (x : B) → AddCommGroup (E₂ x)
inst_10 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : (x : B) → TopologicalSpace (E₁ x)
inst_13 : FiberBundle F₁ E₁
inst_14 : (x : B) → TopologicalSpace (E₂ x)
inst_15 : FiberBundle F₂ E₂
e₁ e₁' : Trivialization (B := B) F₁ (Z := TotalSpace F₁ E₁) (TotalSpace.proj (E := E₁))
e₂ e₂' : Trivialization (B := B) F₂ (Z := TotalSpace F₂ E₂) (TotalSpace.proj (E := E₂))
inst_16 : Trivialization.IsLinear 𝕜 (E := E₁) e₁
inst_17 : Trivialization.IsLinear 𝕜 (E := E₁) e₁'
inst_18 : Trivialization.IsLinear 𝕜 (E := E₂) e₂
inst_19 : Trivialization.IsLinear 𝕜 (E := E₂) e₂'
b : B
L : F₁ [⋀^ι]→L[𝕜] F₂
v : ι → F₁
hb : (b ∈ e₁.baseSet ∧ b ∈ e₂.baseSet) ∧ b ∈ e₁'.baseSet ∧ b ∈ e₂'.baseSet
⊢ Eq.{u_6 + 1} (α := F₂)
    ((((continuousAlternatingMapCoordChange 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₁' e₂ e₂' b :
                (a : F₁ [⋀^ι]→L[𝕜] F₂) → F₁ [⋀^ι]→L[𝕜] F₂)
              L :
            F₁ [⋀^ι]→L[𝕜] F₂) :
          (a : ι → F₁) → F₂)
        v :
      F₂)
    ((((↑(continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁' e₂') :
                (TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) → B × F₁ [⋀^ι]→L[𝕜] F₂)
              (TotalSpace.mk (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι) b
                (Pretrivialization.symm (E := fun (x : B) ↦
                  ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
                  (continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂ :
                    Pretrivialization (B := B) (F₁ [⋀^ι]→L[𝕜] F₂) (Z :=
                      TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦
                        ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
                      (TotalSpace.proj (E := fun (x : B) ↦
                        ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)))
                  b L))).2 :
          (a : ι → F₁) → F₂)
        v :
      F₂)

Tactic:
simp [continuousAlternatingMapCoordChange, continuousAlternatingMap_apply,
    Function.comp_def, Trivialization.coordChangeL_apply,
    continuousAlternatingMap_symm_apply' hb.left, hb]

Premises used: Bundle.Trivialization.coordChangeL_apply, Bundle.Pretrivialization.continuousAlternatingMap_symm_apply', Function.comp_def, And.left, Bundle.Pretrivialization.continuousAlternatingMap_apply, Bundle.Pretrivialization.continuousAlternatingMapCoordChange
-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
inst_1 : Fintype ι
B : Type u_3
inst_2 : TopologicalSpace B
F₁ : Type u_4
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_5 : (x : B) → AddCommGroup (E₁ x)
inst_6 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_7 : TopologicalSpace (TotalSpace F₁ E₁)
inst_8 : (x : B) → TopologicalSpace (E₁ x)
inst_9 : FiberBundle F₁ E₁
inst_10 : VectorBundle 𝕜 F₁ E₁
F₂ : Type u_6
inst_11 : NormedAddCommGroup F₂
inst_12 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_13 : (x : B) → AddCommGroup (E₂ x)
inst_14 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_15 : TopologicalSpace (TotalSpace F₂ E₂)
inst_16 : (x : B) → TopologicalSpace (E₂ x)
inst_17 : FiberBundle F₂ E₂
inst_18 : VectorBundle 𝕜 F₂ E₂
inst_19 : ∀ (x : B), IsTopologicalAddGroup.{u_7} (E₂ x)
inst_20 : ∀ (x : B), ContinuousSMul.{u_1, u_7} 𝕜 (E₂ x)
e₁ : Trivialization (B := B) F₁ (Z := TotalSpace F₁ E₁) (TotalSpace.proj (E := E₁))
he₁ : Trivialization (B := B) F₂ (Z := TotalSpace F₂ E₂) (TotalSpace.proj (E := E₂))
e₂ : MemTrivializationAtlas (E := E₁) e₁
he₂ : MemTrivializationAtlas (E := E₂) he₁
⊢ Pretrivialization.IsLinear 𝕜 (F := F₁ [⋀^ι]→L[𝕜] F₂) (E := fun (x : B) ↦
    ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
    (continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ he₁)

Tactic:
infer_instance
-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
inst_1 : Fintype ι
B : Type u_3
inst_2 : TopologicalSpace B
F₁ : Type u_4
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_5 : (x : B) → AddCommGroup (E₁ x)
inst_6 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_7 : TopologicalSpace (TotalSpace F₁ E₁)
inst_8 : (x : B) → TopologicalSpace (E₁ x)
inst_9 : FiberBundle F₁ E₁
inst_10 : VectorBundle 𝕜 F₁ E₁
F₂ : Type u_6
inst_11 : NormedAddCommGroup F₂
inst_12 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_13 : (x : B) → AddCommGroup (E₂ x)
inst_14 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_15 : TopologicalSpace (TotalSpace F₂ E₂)
inst_16 : (x : B) → TopologicalSpace (E₂ x)
inst_17 : FiberBundle F₂ E₂
inst_18 : VectorBundle 𝕜 F₂ E₂
inst_19 : ∀ (x : B), IsTopologicalAddGroup.{u_7} (E₂ x)
inst_20 : ∀ (x : B), ContinuousSMul.{u_1, u_7} 𝕜 (E₂ x)
b : B
simp :
  IsInducing (Y := F₁ [⋀^ι]→L[𝕜] F₂) fun (x' : ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ b) (E₂ b) ι) ↦
    ContinuousLinearMap.compContinuousAlternatingMap
      (Trivialization.continuousLinearMapAt 𝕜 (E := E₂) (trivializationAt F₂ E₂ b) b)
      (x'.compContinuousLinearMap (Trivialization.symmL 𝕜 (E := E₁) (trivializationAt F₁ E₁ b) b))
⊢ IsInducing (X := ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ b) (E₂ b) ι) (Y := B × F₁ [⋀^ι]→L[𝕜] F₂)
    (Function.comp (β :=
      TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
      (↑(continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) (trivializationAt F₁ E₁ b) (trivializationAt F₂ E₂ b)) :
        (TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) → B × F₁ [⋀^ι]→L[𝕜] F₂)
      (TotalSpace.mk (F := F₁ [⋀^ι]→L[𝕜] F₂) (E := fun (x : B) ↦
        ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι) b))

Tactic:
simp only [Function.comp_def, continuousAlternatingMap_apply, isInducing_const_prod]

Premises used: Function.comp_def, Bundle.Pretrivialization.continuousAlternatingMap_apply, Topology.isInducing_const_prod
-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
inst_1 : Fintype ι
B : Type u_3
inst_2 : TopologicalSpace B
F₁ : Type u_4
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_5 : (x : B) → AddCommGroup (E₁ x)
inst_6 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_7 : TopologicalSpace (TotalSpace F₁ E₁)
inst_8 : (x : B) → TopologicalSpace (E₁ x)
inst_9 : FiberBundle F₁ E₁
inst_10 : VectorBundle 𝕜 F₁ E₁
F₂ : Type u_6
inst_11 : NormedAddCommGroup F₂
inst_12 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_13 : (x : B) → AddCommGroup (E₂ x)
inst_14 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_15 : TopologicalSpace (TotalSpace F₂ E₂)
inst_16 : (x : B) → TopologicalSpace (E₂ x)
inst_17 : FiberBundle F₂ E₂
inst_18 : VectorBundle 𝕜 F₂ E₂
inst_19 : ∀ (x : B), IsTopologicalAddGroup.{u_7} (E₂ x)
inst_20 : ∀ (x : B), ContinuousSMul.{u_1, u_7} 𝕜 (E₂ x)
b : B
L₁ : ContinuousLinearEquiv.{u_1, u_1, u_5, u_4} (RingHom.id 𝕜) (E₁ b) F₁ :=
  Trivialization.continuousLinearEquivAt 𝕜 (E := E₁) (trivializationAt F₁ E₁ b) b ⋯
L₂ : ContinuousLinearEquiv.{u_1, u_1, u_7, u_6} (RingHom.id 𝕜) (E₂ b) F₂ :=
  Trivialization.continuousLinearEquivAt 𝕜 (E := E₂) (trivializationAt F₂ E₂ b) b ⋯
⊢ IsInducing (X := ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ b) (E₂ b) ι) (Y := F₁ [⋀^ι]→L[𝕜] F₂)
    (⇑(ContinuousLinearEquiv.toHomeomorph (L₁.continuousAlternatingMapCongr L₂)) :
      (a : E₁ b [⋀^ι]→L[𝕜] E₂ b) → F₁ [⋀^ι]→L[𝕜] F₂)

Tactic:
(L₁.continuousAlternatingMapCongr L₂).toHomeomorph.isInducing

Premises used: Homeomorph.isInducing, ContinuousLinearEquiv.continuousAlternatingMapCongr, ContinuousLinearEquiv.toHomeomorph
-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
inst_1 : Fintype ι
B : Type u_3
inst_2 : TopologicalSpace B
F₁ : Type u_4
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_5 : (x : B) → AddCommGroup (E₁ x)
inst_6 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_7 : TopologicalSpace (TotalSpace F₁ E₁)
inst_8 : (x : B) → TopologicalSpace (E₁ x)
inst_9 : FiberBundle F₁ E₁
inst_10 : VectorBundle 𝕜 F₁ E₁
F₂ : Type u_6
inst_11 : NormedAddCommGroup F₂
inst_12 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_13 : (x : B) → AddCommGroup (E₂ x)
inst_14 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_15 : TopologicalSpace (TotalSpace F₂ E₂)
inst_16 : (x : B) → TopologicalSpace (E₂ x)
inst_17 : FiberBundle F₂ E₂
inst_18 : VectorBundle 𝕜 F₂ E₂
inst_19 : ∀ (x : B), IsTopologicalAddGroup.{u_7} (E₂ x)
inst_20 : ∀ (x : B), ContinuousSMul.{u_1, u_7} 𝕜 (E₂ x)
b : B
L₁ : ContinuousLinearEquiv.{u_1, u_1, u_5, u_4} (RingHom.id 𝕜) (E₁ b) F₁ :=
  Trivialization.continuousLinearEquivAt 𝕜 (E := E₁) (trivializationAt F₁ E₁ b) b ⋯
L₂ : ContinuousLinearEquiv.{u_1, u_1, u_7, u_6} (RingHom.id 𝕜) (E₂ b) F₂ :=
  Trivialization.continuousLinearEquivAt 𝕜 (E := E₂) (trivializationAt F₂ E₂ b) b ⋯
h :
  ∀ (x : ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ b) (E₂ b) ι),
    Eq (α := F₁ [⋀^ι]→L[𝕜] F₂)
      (ContinuousLinearMap.compContinuousAlternatingMap
        (Trivialization.continuousLinearMapAt 𝕜 (E := E₂) (trivializationAt F₂ E₂ b) b)
        (x.compContinuousLinearMap (Trivialization.symmL 𝕜 (E := E₁) (trivializationAt F₁ E₁ b) b)))
      ((ContinuousLinearEquiv.toHomeomorph (L₁.continuousAlternatingMapCongr L₂) :
          (a : E₁ b [⋀^ι]→L[𝕜] E₂ b) → F₁ [⋀^ι]→L[𝕜] F₂)
        x)
⊢ IsInducing (Y := F₁ [⋀^ι]→L[𝕜] F₂) fun (x' : ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ b) (E₂ b) ι) ↦
    ContinuousLinearMap.compContinuousAlternatingMap
      (Trivialization.continuousLinearMapAt 𝕜 (E := E₂) (trivializationAt F₂ E₂ b) b)
      (x'.compContinuousLinearMap (Trivialization.symmL 𝕜 (E := E₁) (trivializationAt F₁ E₁ b) b))

Tactic:
convert! (L₁.continuousAlternatingMapCongr L₂).toHomeomorph.isInducing

Premises used: Homeomorph.isInducing, ContinuousLinearEquiv.continuousAlternatingMapCongr, ContinuousLinearEquiv.toHomeomorph
-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
inst_1 : Fintype ι
B : Type u_3
inst_2 : TopologicalSpace B
F₁ : Type u_4
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_5 : (x : B) → AddCommGroup (E₁ x)
inst_6 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_7 : TopologicalSpace (TotalSpace F₁ E₁)
inst_8 : (x : B) → TopologicalSpace (E₁ x)
inst_9 : FiberBundle F₁ E₁
inst_10 : VectorBundle 𝕜 F₁ E₁
F₂ : Type u_6
inst_11 : NormedAddCommGroup F₂
inst_12 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_13 : (x : B) → AddCommGroup (E₂ x)
inst_14 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_15 : TopologicalSpace (TotalSpace F₂ E₂)
inst_16 : (x : B) → TopologicalSpace (E₂ x)
inst_17 : FiberBundle F₂ E₂
inst_18 : VectorBundle 𝕜 F₂ E₂
inst_19 : ∀ (x : B), IsTopologicalAddGroup.{u_7} (E₂ x)
inst_20 : ∀ (x : B), ContinuousSMul.{u_1, u_7} 𝕜 (E₂ x)
b : B
L₁ : ContinuousLinearEquiv.{u_1, u_1, u_5, u_4} (RingHom.id 𝕜) (E₁ b) F₁ :=
  Trivialization.continuousLinearEquivAt 𝕜 (E := E₁) (trivializationAt F₁ E₁ b) b ⋯
L₂ : ContinuousLinearEquiv.{u_1, u_1, u_7, u_6} (RingHom.id 𝕜) (E₂ b) F₂ :=
  Trivialization.continuousLinearEquivAt 𝕜 (E := E₂) (trivializationAt F₂ E₂ b) b ⋯
x : ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ b) (E₂ b) ι
h :
  ∀ (f : ι → F₁),
    Eq.{u_6 + 1} (α := F₂)
      ((ContinuousLinearMap.compContinuousAlternatingMap
              (Trivialization.continuousLinearMapAt 𝕜 (E := E₂) (trivializationAt F₂ E₂ b) b)
              (x.compContinuousLinearMap (Trivialization.symmL 𝕜 (E := E₁) (trivializationAt F₁ E₁ b) b)) :
            (a : ι → F₁) → F₂)
          f :
        F₂)
      ((((ContinuousLinearEquiv.toHomeomorph (L₁.continuousAlternatingMapCongr L₂) :
                  (a : E₁ b [⋀^ι]→L[𝕜] E₂ b) → F₁ [⋀^ι]→L[𝕜] F₂)
                x :
              F₁ [⋀^ι]→L[𝕜] F₂) :
            (a : ι → F₁) → F₂)
          f :
        F₂)
⊢ Eq (α := F₁ [⋀^ι]→L[𝕜] F₂)
    (ContinuousLinearMap.compContinuousAlternatingMap
      (Trivialization.continuousLinearMapAt 𝕜 (E := E₂) (trivializationAt F₂ E₂ b) b)
      (x.compContinuousLinearMap (Trivialization.symmL 𝕜 (E := E₁) (trivializationAt F₁ E₁ b) b)))
    ((ContinuousLinearEquiv.toHomeomorph (L₁.continuousAlternatingMapCongr L₂) :
        (a : E₁ b [⋀^ι]→L[𝕜] E₂ b) → F₁ [⋀^ι]→L[𝕜] F₂)
      x)

Tactic:
ext f
-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
inst_1 : Fintype ι
B : Type u_3
inst_2 : TopologicalSpace B
F₁ : Type u_4
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_5 : (x : B) → AddCommGroup (E₁ x)
inst_6 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_7 : TopologicalSpace (TotalSpace F₁ E₁)
inst_8 : (x : B) → TopologicalSpace (E₁ x)
inst_9 : FiberBundle F₁ E₁
inst_10 : VectorBundle 𝕜 F₁ E₁
F₂ : Type u_6
inst_11 : NormedAddCommGroup F₂
inst_12 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_13 : (x : B) → AddCommGroup (E₂ x)
inst_14 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_15 : TopologicalSpace (TotalSpace F₂ E₂)
inst_16 : (x : B) → TopologicalSpace (E₂ x)
inst_17 : FiberBundle F₂ E₂
inst_18 : VectorBundle 𝕜 F₂ E₂
inst_19 : ∀ (x : B), IsTopologicalAddGroup.{u_7} (E₂ x)
inst_20 : ∀ (x : B), ContinuousSMul.{u_1, u_7} 𝕜 (E₂ x)
b : B
L₁ : ContinuousLinearEquiv.{u_1, u_1, u_5, u_4} (RingHom.id 𝕜) (E₁ b) F₁ :=
  Trivialization.continuousLinearEquivAt 𝕜 (E := E₁) (trivializationAt F₁ E₁ b) b ⋯
L₂ : ContinuousLinearEquiv.{u_1, u_1, u_7, u_6} (RingHom.id 𝕜) (E₂ b) F₂ :=
  Trivialization.continuousLinearEquivAt 𝕜 (E := E₂) (trivializationAt F₂ E₂ b) b ⋯
x : ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ b) (E₂ b) ι
f : ι → F₁
⊢ Eq.{u_6 + 1} (α := F₂)
    ((ContinuousLinearMap.compContinuousAlternatingMap
            (Trivialization.continuousLinearMapAt 𝕜 (E := E₂) (trivializationAt F₂ E₂ b) b)
            (x.compContinuousLinearMap (Trivialization.symmL 𝕜 (E := E₁) (trivializationAt F₁ E₁ b) b)) :
          (a : ι → F₁) → F₂)
        f :
      F₂)
    ((((ContinuousLinearEquiv.toHomeomorph (L₁.continuousAlternatingMapCongr L₂) :
                (a : E₁ b [⋀^ι]→L[𝕜] E₂ b) → F₁ [⋀^ι]→L[𝕜] F₂)
              x :
            F₁ [⋀^ι]→L[𝕜] F₂) :
          (a : ι → F₁) → F₂)
        f :
      F₂)

Tactic:
simp [Trivialization.linearMapAt_def_of_mem _ (mem_baseSet_trivializationAt _ _ _), L₁, L₂]

Premises used: Bundle.Trivialization.linearMapAt_def_of_mem, FiberBundle.mem_baseSet_trivializationAt
-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
inst_1 : Fintype ι
B : Type u_3
inst_2 : TopologicalSpace B
F₁ : Type u_4
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_5 : (x : B) → AddCommGroup (E₁ x)
inst_6 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_7 : TopologicalSpace (TotalSpace F₁ E₁)
inst_8 : (x : B) → TopologicalSpace (E₁ x)
inst_9 : FiberBundle F₁ E₁
inst_10 : VectorBundle 𝕜 F₁ E₁
F₂ : Type u_6
inst_11 : NormedAddCommGroup F₂
inst_12 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_13 : (x : B) → AddCommGroup (E₂ x)
inst_14 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_15 : TopologicalSpace (TotalSpace F₂ E₂)
inst_16 : (x : B) → TopologicalSpace (E₂ x)
inst_17 : FiberBundle F₂ E₂
inst_18 : VectorBundle 𝕜 F₂ E₂
inst_19 : ∀ (x : B), IsTopologicalAddGroup.{u_7} (E₂ x)
inst_20 : ∀ (x : B), ContinuousSMul.{u_1, u_7} 𝕜 (E₂ x)
⊢ VectorBundle 𝕜 (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι

Tactic:
(vectorPrebundle 𝕜 ι F₁ E₁ F₂ E₂).toVectorBundle

Premises used: VectorPrebundle.toVectorBundle, Bundle.ContinuousAlternatingMap.vectorPrebundle
-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
inst_1 : Fintype ι
B : Type u_3
inst_2 : TopologicalSpace B
F₁ : Type u_4
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_5 : (x : B) → AddCommGroup (E₁ x)
inst_6 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_7 : TopologicalSpace (TotalSpace F₁ E₁)
inst_8 : (x : B) → TopologicalSpace (E₁ x)
inst_9 : FiberBundle F₁ E₁
inst_10 : VectorBundle 𝕜 F₁ E₁
F₂ : Type u_6
inst_11 : NormedAddCommGroup F₂
inst_12 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_13 : (x : B) → AddCommGroup (E₂ x)
inst_14 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_15 : TopologicalSpace (TotalSpace F₂ E₂)
inst_16 : (x : B) → TopologicalSpace (E₂ x)
inst_17 : FiberBundle F₂ E₂
inst_18 : VectorBundle 𝕜 F₂ E₂
inst_19 : ∀ (x : B), IsTopologicalAddGroup.{u_7} (E₂ x)
inst_20 : ∀ (x : B), ContinuousSMul.{u_1, u_7} 𝕜 (E₂ x)
e₁ : Trivialization (B := B) F₁ (Z := TotalSpace F₁ E₁) (TotalSpace.proj (E := E₁))
e₂ : Trivialization (B := B) F₂ (Z := TotalSpace F₂ E₂) (TotalSpace.proj (E := E₂))
he₁ : MemTrivializationAtlas (E := E₁) e₁
he₂ : MemTrivializationAtlas (E := E₂) e₂
⊢ MemTrivializationAtlas (E := E₁) e₁

Tactic:
infer_instance
-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
inst_1 : Fintype ι
B : Type u_3
inst_2 : TopologicalSpace B
F₁ : Type u_4
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_5 : (x : B) → AddCommGroup (E₁ x)
inst_6 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_7 : TopologicalSpace (TotalSpace F₁ E₁)
inst_8 : (x : B) → TopologicalSpace (E₁ x)
inst_9 : FiberBundle F₁ E₁
inst_10 : VectorBundle 𝕜 F₁ E₁
F₂ : Type u_6
inst_11 : NormedAddCommGroup F₂
inst_12 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_13 : (x : B) → AddCommGroup (E₂ x)
inst_14 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_15 : TopologicalSpace (TotalSpace F₂ E₂)
inst_16 : (x : B) → TopologicalSpace (E₂ x)
inst_17 : FiberBundle F₂ E₂
inst_18 : VectorBundle 𝕜 F₂ E₂
inst_19 : ∀ (x : B), IsTopologicalAddGroup.{u_7} (E₂ x)
inst_20 : ∀ (x : B), ContinuousSMul.{u_1, u_7} 𝕜 (E₂ x)
e₁ : Trivialization (B := B) F₁ (Z := TotalSpace F₁ E₁) (TotalSpace.proj (E := E₁))
e₂ : Trivialization (B := B) F₂ (Z := TotalSpace F₂ E₂) (TotalSpace.proj (E := E₂))
he₁ : MemTrivializationAtlas (E := E₁) e₁
he₂ : MemTrivializationAtlas (E := E₂) e₂
⊢ MemTrivializationAtlas (E := E₂) e₂

Tactic:
infer_instance
-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
inst_1 : Fintype ι
B : Type u_3
inst_2 : TopologicalSpace B
F₁ : Type u_4
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_5 : (x : B) → AddCommGroup (E₁ x)
inst_6 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_7 : TopologicalSpace (TotalSpace F₁ E₁)
inst_8 : (x : B) → TopologicalSpace (E₁ x)
inst_9 : FiberBundle F₁ E₁
inst_10 : VectorBundle 𝕜 F₁ E₁
F₂ : Type u_6
inst_11 : NormedAddCommGroup F₂
inst_12 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_13 : (x : B) → AddCommGroup (E₂ x)
inst_14 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_15 : TopologicalSpace (TotalSpace F₂ E₂)
inst_16 : (x : B) → TopologicalSpace (E₂ x)
inst_17 : FiberBundle F₂ E₂
inst_18 : VectorBundle 𝕜 F₂ E₂
inst_19 : ∀ (x : B), IsTopologicalAddGroup.{u_7} (E₂ x)
inst_20 : ∀ (x : B), ContinuousSMul.{u_1, u_7} 𝕜 (E₂ x)
e₁ : Trivialization (B := B) F₁ (Z := TotalSpace F₁ E₁) (TotalSpace.proj (E := E₁))
e₂ : Trivialization (B := B) F₂ (Z := TotalSpace F₂ E₂) (TotalSpace.proj (E := E₂))
he₁ : MemTrivializationAtlas (E := E₁) e₁
he₂ : MemTrivializationAtlas (E := E₂) e₂
⊢ baseSet (F := F₁ [⋀^ι]→L[𝕜] F₂) (Z :=
      TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
      (continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂) =
    e₁.baseSet ∩ e₂.baseSet

Tactic:
rfl

Premises used: rfl