awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable (hypothesis)
Goal:
𝕜 : Type u_1
F₁ : Type u_2
F₂ : Type u_3
B₁ : Type u_4
B₂ : Type u_5
M : Type u_6
E₁ : B₁ → Type u_7
E₂ : B₂ → Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : (x : B₁) → AddCommGroup (E₁ x)
inst_2 : (x : B₁) → Module.{u_1, u_7} 𝕜 (E₁ x)
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
inst_5 : TopologicalSpace (TotalSpace F₁ E₁)
inst_6 : (x : B₁) → TopologicalSpace (E₁ x)
inst_7 : (x : B₂) → AddCommGroup (E₂ x)
inst_8 : (x : B₂) → Module.{u_1, u_8} 𝕜 (E₂ x)
inst_9 : NormedAddCommGroup F₂
inst_10 : NormedSpace 𝕜 F₂
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : (x : B₂) → TopologicalSpace (E₂ x)
EB₁ : Type u_9
inst_13 : NormedAddCommGroup EB₁
inst_14 : NormedSpace 𝕜 EB₁
HB₁ : Type u_10
inst_15 : TopologicalSpace HB₁
IB₁ : ModelWithCorners 𝕜 EB₁ HB₁
inst_16 : TopologicalSpace B₁
inst_17 : ChartedSpace HB₁ B₁
EB₂ : Type u_11
inst_18 : NormedAddCommGroup EB₂
inst_19 : NormedSpace 𝕜 EB₂
HB₂ : Type u_12
inst_20 : TopologicalSpace HB₂
IB₂ : ModelWithCorners 𝕜 EB₂ HB₂
inst_21 : TopologicalSpace B₂
inst_22 : ChartedSpace HB₂ B₂
EM : Type u_13
inst_23 : NormedAddCommGroup EM
inst_24 : NormedSpace 𝕜 EM
HM : Type u_14
inst_25 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_26 : TopologicalSpace M
inst_27 : ChartedSpace HM M
inst_28 : FiberBundle F₁ E₁
inst_29 : VectorBundle 𝕜 F₁ E₁
inst_30 : FiberBundle F₂ E₂
inst_31 : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b₁ m)) (E₂ (b₂ m))
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ : (MDiffAt[s] fun (m : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) m₀
hv :
  (MDiffAt[s] fun (x : M) ↦ TotalSpace.proj.{u_4, u_2, u_7} (F := F₁) (E := E₁) ⟨b₁ x, v x⟩) m₀ ∧
    (MDiffAt[s] fun (x : M) ↦
        ((↑(trivializationAt F₁ E₁ (TotalSpace.proj.{u_4, u_2, u_7} (F := F₁) (E := E₁) ⟨b₁ m₀, v m₀⟩)) :
              TotalSpace F₁ E₁ → B₁ × F₁)
            ⟨b₁ x, v x⟩).2)
      m₀
hb₂ : MDiffAt[s] b₂ m₀
⊢ EventuallyEq.{u_6, u_3} (β := F₂) (𝓝[insert m₀ s] m₀)
    (fun (x : M) ↦
      ((↑(trivializationAt F₂ E₂
                (TotalSpace.proj.{u_5, u_3, u_8} (F := F₂) (E := E₂)
                  ⟨b₂ m₀, (ϕ m₀ : (a : E₁ (b₁ m₀)) → E₂ (b₂ m₀)) (v m₀)⟩)) :
            TotalSpace F₂ E₂ → B₂ × F₂)
          ⟨b₂ x, (ϕ x : (a : E₁ (b₁ x)) → E₂ (b₂ x)) (v x)⟩).2)
    fun (x : M) ↦
    (inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ x) (b₂ m₀) (b₂ x) (ϕ x) : (a : F₁) → F₂)
      ((↑(trivializationAt F₁ E₁ (TotalSpace.proj.{u_4, u_2, u_7} (F := F₁) (E := E₁) ⟨b₁ m₀, v m₀⟩)) :
            TotalSpace F₁ E₁ → B₁ × F₁)
          ⟨b₁ x, v x⟩).2
have A : ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₁ m ∈ Trivialization.baseSet (trivializationAt F₁ E₁ (b₁ m₀))
Tactic: have A : ∀ᶠ m in 𝓝[insert m₀ s] m₀, b₁ m ∈ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet := by
    apply hv.1.insert.continuousWithinAt
    apply (trivializationAt F₁ E₁ (b₁ m₀)).open_baseSet.mem_nhds
    exact FiberBundle.mem_baseSet_trivializationAt' (b₁ m₀)
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable (goal)
Goal:
𝕜 : Type u_1
F₁ : Type u_2
F₂ : Type u_3
B₁ : Type u_4
B₂ : Type u_5
M : Type u_6
E₁ : B₁ → Type u_7
E₂ : B₂ → Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : (x : B₁) → AddCommGroup (E₁ x)
inst_2 : (x : B₁) → Module.{u_1, u_7} 𝕜 (E₁ x)
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
inst_5 : TopologicalSpace (TotalSpace F₁ E₁)
inst_6 : (x : B₁) → TopologicalSpace (E₁ x)
inst_7 : (x : B₂) → AddCommGroup (E₂ x)
inst_8 : (x : B₂) → Module.{u_1, u_8} 𝕜 (E₂ x)
inst_9 : NormedAddCommGroup F₂
inst_10 : NormedSpace 𝕜 F₂
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : (x : B₂) → TopologicalSpace (E₂ x)
EB₁ : Type u_9
inst_13 : NormedAddCommGroup EB₁
inst_14 : NormedSpace 𝕜 EB₁
HB₁ : Type u_10
inst_15 : TopologicalSpace HB₁
IB₁ : ModelWithCorners 𝕜 EB₁ HB₁
inst_16 : TopologicalSpace B₁
inst_17 : ChartedSpace HB₁ B₁
EB₂ : Type u_11
inst_18 : NormedAddCommGroup EB₂
inst_19 : NormedSpace 𝕜 EB₂
HB₂ : Type u_12
inst_20 : TopologicalSpace HB₂
IB₂ : ModelWithCorners 𝕜 EB₂ HB₂
inst_21 : TopologicalSpace B₂
inst_22 : ChartedSpace HB₂ B₂
EM : Type u_13
inst_23 : NormedAddCommGroup EM
inst_24 : NormedSpace 𝕜 EM
HM : Type u_14
inst_25 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_26 : TopologicalSpace M
inst_27 : ChartedSpace HM M
inst_28 : FiberBundle F₁ E₁
inst_29 : VectorBundle 𝕜 F₁ E₁
inst_30 : FiberBundle F₂ E₂
inst_31 : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b₁ m)) (E₂ (b₂ m))
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ : (MDiffAt[s] fun (m : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) m₀
hv :
  (MDiffAt[s] fun (x : M) ↦ TotalSpace.proj.{u_4, u_2, u_7} (F := F₁) (E := E₁) ⟨b₁ x, v x⟩) m₀ ∧
    (MDiffAt[s] fun (x : M) ↦
        ((↑(trivializationAt F₁ E₁ (TotalSpace.proj.{u_4, u_2, u_7} (F := F₁) (E := E₁) ⟨b₁ m₀, v m₀⟩)) :
              TotalSpace F₁ E₁ → B₁ × F₁)
            ⟨b₁ x, v x⟩).2)
      m₀
hb₂ : MDiffAt[s] b₂ m₀
A : ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₁ m ∈ Trivialization.baseSet (trivializationAt F₁ E₁ (b₁ m₀))
⊢ ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₂ m ∈ Trivialization.baseSet (trivializationAt F₂ E₂ (b₂ m₀))
have a : Trivialization.baseSet (trivializationAt F₂ E₂ (b₂ m₀)) ∈ 𝓝 (b₂ m₀)
Tactic: apply hb₂.insert.continuousWithinAt
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable (goal)
Goal:
𝕜 : Type u_1
F₁ : Type u_2
F₂ : Type u_3
B₁ : Type u_4
B₂ : Type u_5
M : Type u_6
E₁ : B₁ → Type u_7
E₂ : B₂ → Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : (x : B₁) → AddCommGroup (E₁ x)
inst_2 : (x : B₁) → Module.{u_1, u_7} 𝕜 (E₁ x)
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
inst_5 : TopologicalSpace (TotalSpace F₁ E₁)
inst_6 : (x : B₁) → TopologicalSpace (E₁ x)
inst_7 : (x : B₂) → AddCommGroup (E₂ x)
inst_8 : (x : B₂) → Module.{u_1, u_8} 𝕜 (E₂ x)
inst_9 : NormedAddCommGroup F₂
inst_10 : NormedSpace 𝕜 F₂
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : (x : B₂) → TopologicalSpace (E₂ x)
EB₁ : Type u_9
inst_13 : NormedAddCommGroup EB₁
inst_14 : NormedSpace 𝕜 EB₁
HB₁ : Type u_10
inst_15 : TopologicalSpace HB₁
IB₁ : ModelWithCorners 𝕜 EB₁ HB₁
inst_16 : TopologicalSpace B₁
inst_17 : ChartedSpace HB₁ B₁
EB₂ : Type u_11
inst_18 : NormedAddCommGroup EB₂
inst_19 : NormedSpace 𝕜 EB₂
HB₂ : Type u_12
inst_20 : TopologicalSpace HB₂
IB₂ : ModelWithCorners 𝕜 EB₂ HB₂
inst_21 : TopologicalSpace B₂
inst_22 : ChartedSpace HB₂ B₂
EM : Type u_13
inst_23 : NormedAddCommGroup EM
inst_24 : NormedSpace 𝕜 EM
HM : Type u_14
inst_25 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_26 : TopologicalSpace M
inst_27 : ChartedSpace HM M
inst_28 : FiberBundle F₁ E₁
inst_29 : VectorBundle 𝕜 F₁ E₁
inst_30 : FiberBundle F₂ E₂
inst_31 : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b₁ m)) (E₂ (b₂ m))
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ : (MDiffAt[s] fun (m : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) m₀
hv :
  (MDiffAt[s] fun (x : M) ↦ TotalSpace.proj.{u_4, u_2, u_7} (F := F₁) (E := E₁) ⟨b₁ x, v x⟩) m₀ ∧
    (MDiffAt[s] fun (x : M) ↦
        ((↑(trivializationAt F₁ E₁ (TotalSpace.proj.{u_4, u_2, u_7} (F := F₁) (E := E₁) ⟨b₁ m₀, v m₀⟩)) :
              TotalSpace F₁ E₁ → B₁ × F₁)
            ⟨b₁ x, v x⟩).2)
      m₀
hb₂ : MDiffAt[s] b₂ m₀
A : ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₁ m ∈ Trivialization.baseSet (trivializationAt F₁ E₁ (b₁ m₀))
⊢ Trivialization.baseSet (trivializationAt F₂ E₂ (b₂ m₀)) ∈ 𝓝 (b₂ m₀)
have a : b₂ m₀ ∈ Trivialization.baseSet (trivializationAt F₂ E₂ (b₂ m₀))
Tactic: apply (trivializationAt F₂ E₂ (b₂ m₀)).open_baseSet.mem_nhds
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable (hypothesis)
Goal:
𝕜 : Type u_1
F₁ : Type u_2
F₂ : Type u_3
B₁ : Type u_4
B₂ : Type u_5
M : Type u_6
E₁ : B₁ → Type u_7
E₂ : B₂ → Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : (x : B₁) → AddCommGroup (E₁ x)
inst_2 : (x : B₁) → Module.{u_1, u_7} 𝕜 (E₁ x)
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
inst_5 : TopologicalSpace (TotalSpace F₁ E₁)
inst_6 : (x : B₁) → TopologicalSpace (E₁ x)
inst_7 : (x : B₂) → AddCommGroup (E₂ x)
inst_8 : (x : B₂) → Module.{u_1, u_8} 𝕜 (E₂ x)
inst_9 : NormedAddCommGroup F₂
inst_10 : NormedSpace 𝕜 F₂
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : (x : B₂) → TopologicalSpace (E₂ x)
EB₁ : Type u_9
inst_13 : NormedAddCommGroup EB₁
inst_14 : NormedSpace 𝕜 EB₁
HB₁ : Type u_10
inst_15 : TopologicalSpace HB₁
IB₁ : ModelWithCorners 𝕜 EB₁ HB₁
inst_16 : TopologicalSpace B₁
inst_17 : ChartedSpace HB₁ B₁
EB₂ : Type u_11
inst_18 : NormedAddCommGroup EB₂
inst_19 : NormedSpace 𝕜 EB₂
HB₂ : Type u_12
inst_20 : TopologicalSpace HB₂
IB₂ : ModelWithCorners 𝕜 EB₂ HB₂
inst_21 : TopologicalSpace B₂
inst_22 : ChartedSpace HB₂ B₂
EM : Type u_13
inst_23 : NormedAddCommGroup EM
inst_24 : NormedSpace 𝕜 EM
HM : Type u_14
inst_25 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_26 : TopologicalSpace M
inst_27 : ChartedSpace HM M
inst_28 : FiberBundle F₁ E₁
inst_29 : VectorBundle 𝕜 F₁ E₁
inst_30 : FiberBundle F₂ E₂
inst_31 : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b₁ m)) (E₂ (b₂ m))
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ : (MDiffAt[s] fun (m : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) m₀
hv :
  (MDiffAt[s] fun (x : M) ↦ TotalSpace.proj.{u_4, u_2, u_7} (F := F₁) (E := E₁) ⟨b₁ x, v x⟩) m₀ ∧
    (MDiffAt[s] fun (x : M) ↦
        ((↑(trivializationAt F₁ E₁ (TotalSpace.proj.{u_4, u_2, u_7} (F := F₁) (E := E₁) ⟨b₁ m₀, v m₀⟩)) :
              TotalSpace F₁ E₁ → B₁ × F₁)
            ⟨b₁ x, v x⟩).2)
      m₀
hb₂ : MDiffAt[s] b₂ m₀
A : ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₁ m ∈ Trivialization.baseSet (trivializationAt F₁ E₁ (b₁ m₀))
⊢ EventuallyEq.{u_6, u_3} (β := F₂) (𝓝[insert m₀ s] m₀)
    (fun (x : M) ↦
      ((↑(trivializationAt F₂ E₂
                (TotalSpace.proj.{u_5, u_3, u_8} (F := F₂) (E := E₂)
                  ⟨b₂ m₀, (ϕ m₀ : (a : E₁ (b₁ m₀)) → E₂ (b₂ m₀)) (v m₀)⟩)) :
            TotalSpace F₂ E₂ → B₂ × F₂)
          ⟨b₂ x, (ϕ x : (a : E₁ (b₁ x)) → E₂ (b₂ x)) (v x)⟩).2)
    fun (x : M) ↦
    (inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ x) (b₂ m₀) (b₂ x) (ϕ x) : (a : F₁) → F₂)
      ((↑(trivializationAt F₁ E₁ (TotalSpace.proj.{u_4, u_2, u_7} (F := F₁) (E := E₁) ⟨b₁ m₀, v m₀⟩)) :
            TotalSpace F₁ E₁ → B₁ × F₁)
          ⟨b₁ x, v x⟩).2
have A' : ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₂ m ∈ Trivialization.baseSet (trivializationAt F₂ E₂ (b₂ m₀))
Tactic: have A' : ∀ᶠ m in 𝓝[insert m₀ s] m₀, b₂ m ∈ (trivializationAt F₂ E₂ (b₂ m₀)).baseSet := by
    apply hb₂.insert.continuousWithinAt
    apply (trivializationAt F₂ E₂ (b₂ m₀)).open_baseSet.mem_nhds
    exact FiberBundle.mem_baseSet_trivializationAt' (b₂ m₀)
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable (hypothesis)
Goal:
𝕜 : Type u_1
F₁ : Type u_2
F₂ : Type u_3
B₁ : Type u_4
B₂ : Type u_5
M : Type u_6
E₁ : B₁ → Type u_7
E₂ : B₂ → Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : (x : B₁) → AddCommGroup (E₁ x)
inst_2 : (x : B₁) → Module.{u_1, u_7} 𝕜 (E₁ x)
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
inst_5 : TopologicalSpace (TotalSpace F₁ E₁)
inst_6 : (x : B₁) → TopologicalSpace (E₁ x)
inst_7 : (x : B₂) → AddCommGroup (E₂ x)
inst_8 : (x : B₂) → Module.{u_1, u_8} 𝕜 (E₂ x)
inst_9 : NormedAddCommGroup F₂
inst_10 : NormedSpace 𝕜 F₂
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : (x : B₂) → TopologicalSpace (E₂ x)
EB₁ : Type u_9
inst_13 : NormedAddCommGroup EB₁
inst_14 : NormedSpace 𝕜 EB₁
HB₁ : Type u_10
inst_15 : TopologicalSpace HB₁
IB₁ : ModelWithCorners 𝕜 EB₁ HB₁
inst_16 : TopologicalSpace B₁
inst_17 : ChartedSpace HB₁ B₁
EB₂ : Type u_11
inst_18 : NormedAddCommGroup EB₂
inst_19 : NormedSpace 𝕜 EB₂
HB₂ : Type u_12
inst_20 : TopologicalSpace HB₂
IB₂ : ModelWithCorners 𝕜 EB₂ HB₂
inst_21 : TopologicalSpace B₂
inst_22 : ChartedSpace HB₂ B₂
EM : Type u_13
inst_23 : NormedAddCommGroup EM
inst_24 : NormedSpace 𝕜 EM
HM : Type u_14
inst_25 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_26 : TopologicalSpace M
inst_27 : ChartedSpace HM M
inst_28 : FiberBundle F₁ E₁
inst_29 : VectorBundle 𝕜 F₁ E₁
inst_30 : FiberBundle F₂ E₂
inst_31 : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b₁ m)) (E₂ (b₂ m))
v : (m : M) → E₁ (b₁ m)
hϕ : (MDiffAt fun (m : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) m₀
hv : (MDiffAt fun (m : M) ↦ ⟨b₁ m, v m⟩) m₀
hb₂ : MDiffAt b₂ m₀
⊢ (MDiffAt fun (m : M) ↦ ⟨b₂ m, (ϕ m : (a : E₁ (b₁ m)) → E₂ (b₂ m)) (v m)⟩) m₀
have hϕ₁ : (MDiffAt[univ] fun (m : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) m₀
Tactic: rw [← mdifferentiableWithinAt_univ] at hϕ hv hb₂ ⊢
Cleared hypotheses: hϕ, hv, hb₂
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable (hypothesis)
Goal:
𝕜 : Type u_1
F₁ : Type u_2
F₂ : Type u_3
B₁ : Type u_4
B₂ : Type u_5
M : Type u_6
E₁ : B₁ → Type u_7
E₂ : B₂ → Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : (x : B₁) → AddCommGroup (E₁ x)
inst_2 : (x : B₁) → Module.{u_1, u_7} 𝕜 (E₁ x)
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
inst_5 : TopologicalSpace (TotalSpace F₁ E₁)
inst_6 : (x : B₁) → TopologicalSpace (E₁ x)
inst_7 : (x : B₂) → AddCommGroup (E₂ x)
inst_8 : (x : B₂) → Module.{u_1, u_8} 𝕜 (E₂ x)
inst_9 : NormedAddCommGroup F₂
inst_10 : NormedSpace 𝕜 F₂
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : (x : B₂) → TopologicalSpace (E₂ x)
EB₁ : Type u_9
inst_13 : NormedAddCommGroup EB₁
inst_14 : NormedSpace 𝕜 EB₁
HB₁ : Type u_10
inst_15 : TopologicalSpace HB₁
IB₁ : ModelWithCorners 𝕜 EB₁ HB₁
inst_16 : TopologicalSpace B₁
inst_17 : ChartedSpace HB₁ B₁
EB₂ : Type u_11
inst_18 : NormedAddCommGroup EB₂
inst_19 : NormedSpace 𝕜 EB₂
HB₂ : Type u_12
inst_20 : TopologicalSpace HB₂
IB₂ : ModelWithCorners 𝕜 EB₂ HB₂
inst_21 : TopologicalSpace B₂
inst_22 : ChartedSpace HB₂ B₂
EM : Type u_13
inst_23 : NormedAddCommGroup EM
inst_24 : NormedSpace 𝕜 EM
HM : Type u_14
inst_25 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_26 : TopologicalSpace M
inst_27 : ChartedSpace HM M
inst_28 : FiberBundle F₁ E₁
inst_29 : VectorBundle 𝕜 F₁ E₁
inst_30 : FiberBundle F₂ E₂
inst_31 : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b₁ m)) (E₂ (b₂ m))
v : (m : M) → E₁ (b₁ m)
hϕ : (MDiffAt fun (m : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) m₀
hv : (MDiffAt fun (m : M) ↦ ⟨b₁ m, v m⟩) m₀
hb₂ : MDiffAt b₂ m₀
hϕ₁ : (MDiffAt[univ] fun (m : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) m₀
⊢ (MDiffAt fun (m : M) ↦ ⟨b₂ m, (ϕ m : (a : E₁ (b₁ m)) → E₂ (b₂ m)) (v m)⟩) m₀
have hv₁ : (MDiffAt[univ] fun (m : M) ↦ ⟨b₁ m, v m⟩) m₀
Tactic: rw [← mdifferentiableWithinAt_univ] at hϕ hv hb₂ ⊢
Cleared hypotheses: hϕ, hv, hb₂
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable (hypothesis)
Goal:
𝕜 : Type u_1
F₁ : Type u_2
F₂ : Type u_3
B₁ : Type u_4
B₂ : Type u_5
M : Type u_6
E₁ : B₁ → Type u_7
E₂ : B₂ → Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : (x : B₁) → AddCommGroup (E₁ x)
inst_2 : (x : B₁) → Module.{u_1, u_7} 𝕜 (E₁ x)
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
inst_5 : TopologicalSpace (TotalSpace F₁ E₁)
inst_6 : (x : B₁) → TopologicalSpace (E₁ x)
inst_7 : (x : B₂) → AddCommGroup (E₂ x)
inst_8 : (x : B₂) → Module.{u_1, u_8} 𝕜 (E₂ x)
inst_9 : NormedAddCommGroup F₂
inst_10 : NormedSpace 𝕜 F₂
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : (x : B₂) → TopologicalSpace (E₂ x)
EB₁ : Type u_9
inst_13 : NormedAddCommGroup EB₁
inst_14 : NormedSpace 𝕜 EB₁
HB₁ : Type u_10
inst_15 : TopologicalSpace HB₁
IB₁ : ModelWithCorners 𝕜 EB₁ HB₁
inst_16 : TopologicalSpace B₁
inst_17 : ChartedSpace HB₁ B₁
EB₂ : Type u_11
inst_18 : NormedAddCommGroup EB₂
inst_19 : NormedSpace 𝕜 EB₂
HB₂ : Type u_12
inst_20 : TopologicalSpace HB₂
IB₂ : ModelWithCorners 𝕜 EB₂ HB₂
inst_21 : TopologicalSpace B₂
inst_22 : ChartedSpace HB₂ B₂
EM : Type u_13
inst_23 : NormedAddCommGroup EM
inst_24 : NormedSpace 𝕜 EM
HM : Type u_14
inst_25 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_26 : TopologicalSpace M
inst_27 : ChartedSpace HM M
inst_28 : FiberBundle F₁ E₁
inst_29 : VectorBundle 𝕜 F₁ E₁
inst_30 : FiberBundle F₂ E₂
inst_31 : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b₁ m)) (E₂ (b₂ m))
v : (m : M) → E₁ (b₁ m)
hϕ : (MDiffAt fun (m : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) m₀
hv : (MDiffAt fun (m : M) ↦ ⟨b₁ m, v m⟩) m₀
hb₂ : MDiffAt b₂ m₀
hϕ₁ : (MDiffAt[univ] fun (m : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) m₀
hv₁ : (MDiffAt[univ] fun (m : M) ↦ ⟨b₁ m, v m⟩) m₀
⊢ (MDiffAt fun (m : M) ↦ ⟨b₂ m, (ϕ m : (a : E₁ (b₁ m)) → E₂ (b₂ m)) (v m)⟩) m₀
have hb₃ : MDiffAt[univ] b₂ m₀
Tactic: rw [← mdifferentiableWithinAt_univ] at hϕ hv hb₂ ⊢
Cleared hypotheses: hϕ, hv, hb₂
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
V : M → Type u_6
inst_7 : TopologicalSpace (TotalSpace F V)
inst_8 : (x : M) → AddCommGroup (V x)
inst_9 : (x : M) → TopologicalSpace (V x)
inst_10 : FiberBundle F V
inst_11 : NormedSpace 𝕜 F
k : WithTop ℕ∞
inst_12 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_13 : VectorBundle 𝕜 F V
inst_14 : ContMDiffVectorBundle k F V I
x₀ : M
σ₀ : V x₀
t : Trivialization (B := M) F (Z := TotalSpace F V) (TotalSpace.proj (E := V)) := trivializationAt F V x₀
⊢ ∃ s ∈ 𝓝 x₀, ContMDiffOn I (I.prod 𝓘(𝕜, F)) (M' := TotalSpace F V) k (fun (x' : M) ↦ ⟨x', extend F (E := V) σ₀ x'⟩) s
have refine_1 : t.baseSet ∈ 𝓝 x₀
Tactic: refine ⟨t.baseSet, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
V : M → Type u_6
inst_7 : TopologicalSpace (TotalSpace F V)
inst_8 : (x : M) → AddCommGroup (V x)
inst_9 : (x : M) → TopologicalSpace (V x)
inst_10 : FiberBundle F V
inst_11 : NormedSpace 𝕜 F
k : WithTop ℕ∞
inst_12 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_13 : VectorBundle 𝕜 F V
inst_14 : ContMDiffVectorBundle k F V I
x₀ : M
σ₀ : V x₀
t : Trivialization (B := M) F (Z := TotalSpace F V) (TotalSpace.proj (E := V)) := trivializationAt F V x₀
refine_1 : t.baseSet ∈ 𝓝 x₀
⊢ ∃ s ∈ 𝓝 x₀, ContMDiffOn I (I.prod 𝓘(𝕜, F)) (M' := TotalSpace F V) k (fun (x' : M) ↦ ⟨x', extend F (E := V) σ₀ x'⟩) s
have refine_2 : ContMDiffOn I (I.prod 𝓘(𝕜, F)) (M' := TotalSpace F V) k (fun (x' : M) ↦ ⟨x', extend F (E := V) σ₀ x'⟩) t.baseSet
Tactic: refine ⟨t.baseSet, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
V : M → Type u_6
inst_7 : TopologicalSpace (TotalSpace F V)
inst_8 : (x : M) → AddCommGroup (V x)
inst_9 : (x : M) → TopologicalSpace (V x)
inst_10 : FiberBundle F V
inst_11 : NormedSpace 𝕜 F
k : WithTop ℕ∞
inst_12 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_13 : VectorBundle 𝕜 F V
inst_14 : ContMDiffVectorBundle k F V I
x₀ : M
σ₀ : V x₀
t : Trivialization (B := M) F (Z := TotalSpace F V) (TotalSpace.proj (E := V)) := trivializationAt F V x₀
⊢ t.baseSet ∈ 𝓝 x₀
have refine_1 : x₀ ∈ t.baseSet
Tactic: refine t.open_baseSet.mem_nhds ?_
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
V : M → Type u_6
inst_7 : TopologicalSpace (TotalSpace F V)
inst_8 : (x : M) → AddCommGroup (V x)
inst_9 : (x : M) → TopologicalSpace (V x)
inst_10 : FiberBundle F V
inst_11 : NormedSpace 𝕜 F
k : WithTop ℕ∞
inst_12 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_13 : VectorBundle 𝕜 F V
inst_14 : ContMDiffVectorBundle k F V I
x₀ : M
σ₀ : V x₀
t : Trivialization (B := M) F (Z := TotalSpace F V) (TotalSpace.proj (E := V)) := trivializationAt F V x₀
this :
  ContMDiffOn I 𝓘(𝕜, F) (M' := F) k (fun (x : M) ↦ ((↑t : TotalSpace F V → M × F) ⟨x, extend F (E := V) σ₀ x⟩).2)
    t.baseSet
x : M
hx : x ∈ t.baseSet
⊢ ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) (M' := TotalSpace F V) k (fun (x' : M) ↦ ⟨x', extend F (E := V) σ₀ x'⟩) t.baseSet
    x
have rw : ContMDiffWithinAt I 𝓘(𝕜, F) (M' := F) k (fun (x : M) ↦ ((↑t : TotalSpace F V → M × F) ⟨x, extend F (E := V) σ₀ x⟩).2)
  t.baseSet x
Tactic: rw [t.contMDiffWithinAt_section _ hx]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
V : M → Type u_6
inst_7 : TopologicalSpace (TotalSpace F V)
inst_8 : (x : M) → AddCommGroup (V x)
inst_9 : (x : M) → TopologicalSpace (V x)
inst_10 : FiberBundle F V
inst_11 : NormedSpace 𝕜 F
k : WithTop ℕ∞
inst_12 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_13 : VectorBundle 𝕜 F V
inst_14 : ContMDiffVectorBundle k F V I
x₀ : M
σ₀ : V x₀
t : Trivialization (B := M) F (Z := TotalSpace F V) (TotalSpace.proj (E := V)) := trivializationAt F V x₀
⊢ ContMDiffOn I (I.prod 𝓘(𝕜, F)) (M' := TotalSpace F V) k (fun (x' : M) ↦ ⟨x', extend F (E := V) σ₀ x'⟩) t.baseSet
have refine_2 : ContMDiffOn I 𝓘(𝕜, F) (M' := F) k (fun (x : M) ↦ ((↑t : TotalSpace F V → M × F) ⟨x, extend F (E := V) σ₀ x⟩).2)
  t.baseSet
Tactic: suffices ContMDiffOn I 𝓘(𝕜, F) k (fun x ↦ (t ⟨x, extend F σ₀ x⟩).2) t.baseSet by
    intro x hx
    rw [t.contMDiffWithinAt_section _ hx]
    exact this x hx
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable (hypothesis)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
V : M → Type u_6
inst_7 : TopologicalSpace (TotalSpace F V)
inst_8 : (x : M) → AddCommGroup (V x)
inst_9 : (x : M) → TopologicalSpace (V x)
inst_10 : FiberBundle F V
inst_11 : NormedSpace 𝕜 F
k : WithTop ℕ∞
inst_12 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_13 : VectorBundle 𝕜 F V
inst_14 : ContMDiffVectorBundle k F V I
x₀ : M
σ₀ : V x₀
t : Trivialization (B := M) F (Z := TotalSpace F V) (TotalSpace.proj (E := V)) := trivializationAt F V x₀
w : F := ((↑t : TotalSpace F V → M × F) ⟨x₀, σ₀⟩).2
⊢ ContMDiffOn I 𝓘(𝕜, F) (M' := F) k (fun (x : M) ↦ ((↑t : TotalSpace F V → M × F) ⟨x, extend F (E := V) σ₀ x⟩).2)
    t.baseSet
have this : ContMDiffOn I 𝓘(𝕜, F) (M' := F) k (fun (_x : M) ↦ w) t.baseSet
Tactic: have : ContMDiffOn I 𝓘(𝕜, F) k (fun _x ↦ w) t.baseSet := contMDiffOn_const