l3lab/ntp-mathlib huggingface dataset:
theorem continuousLinearMap_symm_apply' {b : B} (hb : b ∈ e₁.baseSet ∩ e₂.baseSet)
    (L : F₁ →SL[σ] F₂) :
    (continuousLinearMap σ e₁ e₂).symm b L =
      (e₂.symmL 𝕜₂ b).comp (L.comp <| e₁.continuousLinearMapAt 𝕜₁ b) := by
  rw [symm_apply]; rfl; 

Proof state:
case hb
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
ita : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
inst✝³ : Trivialization.IsLinear 𝕜₁ e₁
inst✝² : Trivialization.IsLinear 𝕜₁ e₁'
inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂
inst✝ : Trivialization.IsLinear 𝕜₂ e₂'
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ →SL[σ] F₂
⊢ b ∈ (continuousLinearMap σ e₁ e₂).baseSet

Next tactic:
exact hb
theorem continuousLinearMapCoordChange_apply (b : B)
    (hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) (L : F₁ →SL[σ] F₂) :
    continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b L =
      (continuousLinearMap σ e₁' e₂' ⟨b, (continuousLinearMap σ e₁ e₂).symm b L⟩).2 := by
  

Proof state:
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
ita : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
inst✝³ : Trivialization.IsLinear 𝕜₁ e₁
inst✝² : Trivialization.IsLinear 𝕜₁ e₁'
inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂
inst✝ : Trivialization.IsLinear 𝕜₂ e₂'
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)
L : F₁ →SL[σ] F₂
⊢ (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b) L =
    (↑(continuousLinearMap σ e₁' e₂') { proj := b, snd := Pretrivialization.symm (continuousLinearMap σ e₁ e₂) b L }).2

Next tactic:
ext v
theorem continuousLinearMapCoordChange_apply (b : B)
    (hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) (L : F₁ →SL[σ] F₂) :
    continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b L =
      (continuousLinearMap σ e₁' e₂' ⟨b, (continuousLinearMap σ e₁ e₂).symm b L⟩).2 := by
  ext v
  

Proof state:
case h
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
ita : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
inst✝³ : Trivialization.IsLinear 𝕜₁ e₁
inst✝² : Trivialization.IsLinear 𝕜₁ e₁'
inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂
inst✝ : Trivialization.IsLinear 𝕜₂ e₂'
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)
L : F₁ →SL[σ] F₂
v : F₁
⊢ ((continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b) L) v =
    (↑(continuousLinearMap σ e₁' e₂') { proj := b, snd := Pretrivialization.symm (continuousLinearMap σ e₁ e₂) b L }).2
      v

Next tactic:
simp_rw [continuousLinearMapCoordChange, ContinuousLinearEquiv.coe_coe,
    ContinuousLinearEquiv.arrowCongrSL_apply, continuousLinearMap_apply,
    continuousLinearMap_symm_apply' σ e₁ e₂ hb.1, comp_apply, ContinuousLinearEquiv.coe_coe,
    ContinuousLinearEquiv.symm_symm, Trivialization.continuousLinearMapAt_apply,
    Trivialization.symmL_apply]
theorem continuousLinearMapCoordChange_apply (b : B)
    (hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) (L : F₁ →SL[σ] F₂) :
    continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b L =
      (continuousLinearMap σ e₁' e₂' ⟨b, (continuousLinearMap σ e₁ e₂).symm b L⟩).2 := by
  ext v
  simp_rw [continuousLinearMapCoordChange, ContinuousLinearEquiv.coe_coe,
    ContinuousLinearEquiv.arrowCongrSL_apply, continuousLinearMap_apply,
    continuousLinearMap_symm_apply' σ e₁ e₂ hb.1, comp_apply, ContinuousLinearEquiv.coe_coe,
    ContinuousLinearEquiv.symm_symm, Trivialization.continuousLinearMapAt_apply,
    Trivialization.symmL_apply]
  

Proof state:
case h
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
ita : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
inst✝³ : Trivialization.IsLinear 𝕜₁ e₁
inst✝² : Trivialization.IsLinear 𝕜₁ e₁'
inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂
inst✝ : Trivialization.IsLinear 𝕜₂ e₂'
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)
L : F₁ →SL[σ] F₂
v : F₁
⊢ (Trivialization.coordChangeL 𝕜₂ e₂ e₂' b) (L ((Trivialization.coordChangeL 𝕜₁ e₁' e₁ b) v)) =
    (Trivialization.linearMapAt 𝕜₂ e₂'
        { proj := b, snd := Pretrivialization.symm (continuousLinearMap σ e₁ e₂) b L }.proj)
      (Trivialization.symm e₂ b (L ((Trivialization.linearMapAt 𝕜₁ e₁ b) (Trivialization.symm e₁' b v))))

Next tactic:
rw [e₂.coordChangeL_apply e₂', e₁'.coordChangeL_apply e₁, e₁.coe_linearMapAt_of_mem hb.1.1,
    e₂'.coe_linearMapAt_of_mem hb.2.2]
/-- The continuous `σ`-semilinear maps between two topological vector bundles form a
`VectorPrebundle` (this is an auxiliary construction for the
`VectorBundle` instance, in which the pretrivializations are collated but no topology
on the total space is yet provided). -/
def Bundle.ContinuousLinearMap.vectorPrebundle :
    VectorPrebundle 𝕜₂ (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂) where
  pretrivializationAtlas :=
    {e | ∃ (e₁ : Trivialization F₁ (π F₁ E₁)) (e₂ : Trivialization F₂ (π F₂ E₂))
      (_ : MemTrivializationAtlas e₁) (_ : MemTrivializationAtlas e₂),
        e = Pretrivialization.continuousLinearMap σ e₁ e₂}
  pretrivialization_linear' := by
    

Proof state:
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₂ E₂
inst✝² : VectorBundle 𝕜₂ F₂ E₂
inst✝¹ : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝ : ∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)
⊢ ∀
    e ∈
      {e |
        ∃ e₁ e₂, ∃ (x : MemTrivializationAtlas e₁) (x_1 : MemTrivializationAtlas e₂), e = continuousLinearMap σ e₁ e₂},
    Pretrivialization.IsLinear 𝕜₂ e

Next tactic:
rintro _ ⟨e₁, he₁, e₂, he₂, rfl⟩
/-- The continuous `σ`-semilinear maps between two topological vector bundles form a
`VectorPrebundle` (this is an auxiliary construction for the
`VectorBundle` instance, in which the pretrivializations are collated but no topology
on the total space is yet provided). -/
def Bundle.ContinuousLinearMap.vectorPrebundle :
    VectorPrebundle 𝕜₂ (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂) where
  pretrivializationAtlas :=
    {e | ∃ (e₁ : Trivialization F₁ (π F₁ E₁)) (e₂ : Trivialization F₂ (π F₂ E₂))
      (_ : MemTrivializationAtlas e₁) (_ : MemTrivializationAtlas e₂),
        e = Pretrivialization.continuousLinearMap σ e₁ e₂}
  pretrivialization_linear' := by
    rintro _ ⟨e₁, he₁, e₂, he₂, rfl⟩
    

Proof state:
case intro.intro.intro.intro
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁✝ e₁' : Trivialization F₁ TotalSpace.proj
e₂✝ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₂ E₂
inst✝² : VectorBundle 𝕜₂ F₂ E₂
inst✝¹ : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝ : ∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)
e₁ : Trivialization F₁ TotalSpace.proj
he₁ : Trivialization F₂ TotalSpace.proj
e₂ : MemTrivializationAtlas e₁
he₂ : MemTrivializationAtlas he₁
⊢ Pretrivialization.IsLinear 𝕜₂ (continuousLinearMap σ e₁ he₁)

Next tactic:
infer_instance
/-- The continuous `σ`-semilinear maps between two topological vector bundles form a
`VectorPrebundle` (this is an auxiliary construction for the
`VectorBundle` instance, in which the pretrivializations are collated but no topology
on the total space is yet provided). -/
def Bundle.ContinuousLinearMap.vectorPrebundle :
    VectorPrebundle 𝕜₂ (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂) where
  pretrivializationAtlas :=
    {e | ∃ (e₁ : Trivialization F₁ (π F₁ E₁)) (e₂ : Trivialization F₂ (π F₂ E₂))
      (_ : MemTrivializationAtlas e₁) (_ : MemTrivializationAtlas e₂),
        e = Pretrivialization.continuousLinearMap σ e₁ e₂}
  pretrivialization_linear' := by
    rintro _ ⟨e₁, he₁, e₂, he₂, rfl⟩
    infer_instance
  pretrivializationAt x :=
    Pretrivialization.continuousLinearMap σ (trivializationAt F₁ E₁ x) (trivializationAt F₂ E₂ x)
  mem_base_pretrivializationAt x :=
    ⟨mem_baseSet_trivializationAt F₁ E₁ x, mem_baseSet_trivializationAt F₂ E₂ x⟩
  pretrivialization_mem_atlas x :=
    ⟨trivializationAt F₁ E₁ x, trivializationAt F₂ E₂ x, inferInstance, inferInstance, rfl⟩
  exists_coordChange := by
    

Proof state:
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₂ E₂
inst✝² : VectorBundle 𝕜₂ F₂ E₂
inst✝¹ : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝ : ∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)
⊢ ∀
    e ∈
      {e |
        ∃ e₁ e₂, ∃ (x : MemTrivializationAtlas e₁) (x_1 : MemTrivializationAtlas e₂), e = continuousLinearMap σ e₁ e₂},
    ∀
      e' ∈
        {e |
          ∃ e₁ e₂,
            ∃ (x : MemTrivializationAtlas e₁) (x_1 : MemTrivializationAtlas e₂), e = continuousLinearMap σ e₁ e₂},
      ∃ f,
        ContinuousOn f (e.baseSet ∩ e'.baseSet) ∧
          ∀ b ∈ e.baseSet ∩ e'.baseSet,
            ∀ (v : F₁ →SL[σ] F₂), (f b) v = (↑e' { proj := b, snd := Pretrivialization.symm e b v }).2

Next tactic:
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩ _ ⟨e₁', e₂', he₁', he₂', rfl⟩
/-- The continuous `σ`-semilinear maps between two topological vector bundles form a
`VectorPrebundle` (this is an auxiliary construction for the
`VectorBundle` instance, in which the pretrivializations are collated but no topology
on the total space is yet provided). -/
def Bundle.ContinuousLinearMap.vectorPrebundle :
    VectorPrebundle 𝕜₂ (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂) where
  pretrivializationAtlas :=
    {e | ∃ (e₁ : Trivialization F₁ (π F₁ E₁)) (e₂ : Trivialization F₂ (π F₂ E₂))
      (_ : MemTrivializationAtlas e₁) (_ : MemTrivializationAtlas e₂),
        e = Pretrivialization.continuousLinearMap σ e₁ e₂}
  pretrivialization_linear' := by
    rintro _ ⟨e₁, he₁, e₂, he₂, rfl⟩
    infer_instance
  pretrivializationAt x :=
    Pretrivialization.continuousLinearMap σ (trivializationAt F₁ E₁ x) (trivializationAt F₂ E₂ x)
  mem_base_pretrivializationAt x :=
    ⟨mem_baseSet_trivializationAt F₁ E₁ x, mem_baseSet_trivializationAt F₂ E₂ x⟩
  pretrivialization_mem_atlas x :=
    ⟨trivializationAt F₁ E₁ x, trivializationAt F₂ E₂ x, inferInstance, inferInstance, rfl⟩
  exists_coordChange := by
    rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩ _ ⟨e₁', e₂', he₁', he₂', rfl⟩
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁✝ e₁'✝ : Trivialization F₁ TotalSpace.proj
e₂✝ e₂'✝ : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₂ E₂
inst✝² : VectorBundle 𝕜₂ F₂ E₂
inst✝¹ : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝ : ∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
he₁ : MemTrivializationAtlas e₁
he₂ : MemTrivializationAtlas e₂
e₁' : Trivialization F₁ TotalSpace.proj
e₂' : Trivialization F₂ TotalSpace.proj
he₁' : MemTrivializationAtlas e₁'
he₂' : MemTrivializationAtlas e₂'
⊢ ∃ f,
    ContinuousOn f ((continuousLinearMap σ e₁ e₂).baseSet ∩ (continuousLinearMap σ e₁' e₂').baseSet) ∧
      ∀ b ∈ (continuousLinearMap σ e₁ e₂).baseSet ∩ (continuousLinearMap σ e₁' e₂').baseSet,
        ∀ (v : F₁ →SL[σ] F₂),
          (f b) v =
            (↑(continuousLinearMap σ e₁' e₂')
                { proj := b, snd := Pretrivialization.symm (continuousLinearMap σ e₁ e₂) b v }).2

Next tactic:
exact ⟨continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂',
      continuousOn_continuousLinearMapCoordChange,
      continuousLinearMapCoordChange_apply σ e₁ e₁' e₂ e₂'⟩
/-- The continuous `σ`-semilinear maps between two topological vector bundles form a
`VectorPrebundle` (this is an auxiliary construction for the
`VectorBundle` instance, in which the pretrivializations are collated but no topology
on the total space is yet provided). -/
def Bundle.ContinuousLinearMap.vectorPrebundle :
    VectorPrebundle 𝕜₂ (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂) where
  pretrivializationAtlas :=
    {e | ∃ (e₁ : Trivialization F₁ (π F₁ E₁)) (e₂ : Trivialization F₂ (π F₂ E₂))
      (_ : MemTrivializationAtlas e₁) (_ : MemTrivializationAtlas e₂),
        e = Pretrivialization.continuousLinearMap σ e₁ e₂}
  pretrivialization_linear' := by
    rintro _ ⟨e₁, he₁, e₂, he₂, rfl⟩
    infer_instance
  pretrivializationAt x :=
    Pretrivialization.continuousLinearMap σ (trivializationAt F₁ E₁ x) (trivializationAt F₂ E₂ x)
  mem_base_pretrivializationAt x :=
    ⟨mem_baseSet_trivializationAt F₁ E₁ x, mem_baseSet_trivializationAt F₂ E₂ x⟩
  pretrivialization_mem_atlas x :=
    ⟨trivializationAt F₁ E₁ x, trivializationAt F₂ E₂ x, inferInstance, inferInstance, rfl⟩
  exists_coordChange := by
    rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩ _ ⟨e₁', e₂', he₁', he₂', rfl⟩
    exact ⟨continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂',
      continuousOn_continuousLinearMapCoordChange,
      continuousLinearMapCoordChange_apply σ e₁ e₁' e₂ e₂'⟩
  totalSpaceMk_inducing := by
    

Proof state:
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₂ E₂
inst✝² : VectorBundle 𝕜₂ F₂ E₂
inst✝¹ : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝ : ∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)
⊢ ∀ (b : B),
    Inducing
      (↑((fun x => continuousLinearMap σ (trivializationAt F₁ E₁ x) (trivializationAt F₂ E₂ x)) b) ∘ TotalSpace.mk b)

Next tactic:
intro b
/-- The continuous `σ`-semilinear maps between two topological vector bundles form a
`VectorPrebundle` (this is an auxiliary construction for the
`VectorBundle` instance, in which the pretrivializations are collated but no topology
on the total space is yet provided). -/
def Bundle.ContinuousLinearMap.vectorPrebundle :
    VectorPrebundle 𝕜₂ (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂) where
  pretrivializationAtlas :=
    {e | ∃ (e₁ : Trivialization F₁ (π F₁ E₁)) (e₂ : Trivialization F₂ (π F₂ E₂))
      (_ : MemTrivializationAtlas e₁) (_ : MemTrivializationAtlas e₂),
        e = Pretrivialization.continuousLinearMap σ e₁ e₂}
  pretrivialization_linear' := by
    rintro _ ⟨e₁, he₁, e₂, he₂, rfl⟩
    infer_instance
  pretrivializationAt x :=
    Pretrivialization.continuousLinearMap σ (trivializationAt F₁ E₁ x) (trivializationAt F₂ E₂ x)
  mem_base_pretrivializationAt x :=
    ⟨mem_baseSet_trivializationAt F₁ E₁ x, mem_baseSet_trivializationAt F₂ E₂ x⟩
  pretrivialization_mem_atlas x :=
    ⟨trivializationAt F₁ E₁ x, trivializationAt F₂ E₂ x, inferInstance, inferInstance, rfl⟩
  exists_coordChange := by
    rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩ _ ⟨e₁', e₂', he₁', he₂', rfl⟩
    exact ⟨continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂',
      continuousOn_continuousLinearMapCoordChange,
      continuousLinearMapCoordChange_apply σ e₁ e₁' e₂ e₂'⟩
  totalSpaceMk_inducing := by
    intro b
    

Proof state:
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₂ E₂
inst✝² : VectorBundle 𝕜₂ F₂ E₂
inst✝¹ : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝ : ∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)
b : B
⊢ Inducing
    (↑((fun x => continuousLinearMap σ (trivializationAt F₁ E₁ x) (trivializationAt F₂ E₂ x)) b) ∘ TotalSpace.mk b)

Next tactic:
let L₁ : E₁ b ≃L[𝕜₁] F₁ :=
      (trivializationAt F₁ E₁ b).continuousLinearEquivAt 𝕜₁ b
        (mem_baseSet_trivializationAt _ _ _)