awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Geometry.Manifold.VectorBundle.Hom
Goal:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_6
inst : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
E₁ : B → Type u_7
inst_1 : (x : B) → AddCommGroup (E₁ x)
inst_2 : (x : B) → 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 : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst_7 : (x : B) → AddCommGroup (E₂ x)
inst_8 : (x : B) → 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 : B) → TopologicalSpace (E₂ x)
EB : Type u_10
inst_13 : NormedAddCommGroup EB
inst_14 : NormedSpace 𝕜 EB
HB : Type u_11
inst_15 : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst_16 : TopologicalSpace B
inst_17 : ChartedSpace HB B
EM : Type u_12
inst_18 : NormedAddCommGroup EM
inst_19 : NormedSpace 𝕜 EM
HM : Type u_13
inst_20 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_21 : TopologicalSpace M
inst_22 : ChartedSpace HM M
inst_23 : FiberBundle F₁ E₁
inst_24 : VectorBundle 𝕜 F₁ E₁
inst_25 : FiberBundle F₂ E₂
inst_26 : VectorBundle 𝕜 F₂ E₂
b : M → B
v : (x : M) → E₁ (b x)
s : Set M
x : M
inst_27 : ∀ (x : B), IsTopologicalAddGroup.{u_8} (E₂ x)
inst_28 : ∀ (x : B), ContinuousSMul.{u_1, u_8} 𝕜 (E₂ x)
ϕ : (x : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b x)) (E₂ (b x))
hv : ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₁)) (M' := TotalSpace F₁ E₁) n (fun (m : M) ↦ ⟨b m, v m⟩) s x
hϕ :
  ContMDiffWithinAt.{u_1, u_12, u_13, u_6, u_10, u_11, u_2} IM IB (M' := B) n (fun (x : M) ↦ b x) s x ∧
    ContMDiffWithinAt IM (E' := F₁ →L[𝕜] F₂) 𝓘(𝕜, F₁ →L[𝕜] F₂) (M' := F₁ →L[𝕜] F₂) n
      (fun (x_1 : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b x) (b x_1) (b x) (b x_1) (ϕ x_1)) s x
⊢ ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₂)) (M' := TotalSpace F₂ E₂) n
    (fun (m : M) ↦ ⟨b m, (ϕ m : (a : E₁ (b m)) → E₂ (b m)) (v m)⟩) s x

Tactic:
hϕ.2.clm_apply_of_inCoordinates hv hϕ.1

Premises used: And.right, And.left, ContMDiffWithinAt.clm_apply_of_inCoordinates
-- Module: Mathlib.Geometry.Manifold.VectorBundle.Hom
Goal:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_6
inst : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
E₁ : B → Type u_7
inst_1 : (x : B) → AddCommGroup (E₁ x)
inst_2 : (x : B) → 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 : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst_7 : (x : B) → AddCommGroup (E₂ x)
inst_8 : (x : B) → 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 : B) → TopologicalSpace (E₂ x)
EB : Type u_10
inst_13 : NormedAddCommGroup EB
inst_14 : NormedSpace 𝕜 EB
HB : Type u_11
inst_15 : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst_16 : TopologicalSpace B
inst_17 : ChartedSpace HB B
EM : Type u_12
inst_18 : NormedAddCommGroup EM
inst_19 : NormedSpace 𝕜 EM
HM : Type u_13
inst_20 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_21 : TopologicalSpace M
inst_22 : ChartedSpace HM M
inst_23 : FiberBundle F₁ E₁
inst_24 : VectorBundle 𝕜 F₁ E₁
inst_25 : FiberBundle F₂ E₂
inst_26 : VectorBundle 𝕜 F₂ E₂
b : M → B
v : (x : M) → E₁ (b x)
s : Set M
x : M
inst_27 : ∀ (x : B), IsTopologicalAddGroup.{u_8} (E₂ x)
inst_28 : ∀ (x : B), ContinuousSMul.{u_1, u_8} 𝕜 (E₂ x)
ϕ : (x : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b x)) (E₂ (b x))
hv : ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₁)) (M' := TotalSpace F₁ E₁) n (fun (m : M) ↦ ⟨b m, v m⟩) s x
hϕ :
  ContMDiffWithinAt.{u_1, u_12, u_13, u_6, u_10, u_11, u_2} IM IB (M' := B) n (fun (x : M) ↦ b x) s x ∧
    ContMDiffWithinAt IM (E' := F₁ →L[𝕜] F₂) 𝓘(𝕜, F₁ →L[𝕜] F₂) (M' := F₁ →L[𝕜] F₂) n
      (fun (x_1 : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b x) (b x_1) (b x) (b x_1) (ϕ x_1)) s x
⊢ ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₂)) (M' := TotalSpace F₂ E₂) n
    (fun (m : M) ↦ ⟨b m, (ϕ m : (a : E₁ (b m)) → E₂ (b m)) (v m)⟩) s x

Tactic:
exact hϕ.2.clm_apply_of_inCoordinates hv hϕ.1

Premises used: And.right, And.left, ContMDiffWithinAt.clm_apply_of_inCoordinates
-- Module: Mathlib.Geometry.Manifold.VectorBundle.Hom
Goal:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_6
inst : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
E₁ : B → Type u_7
inst_1 : (x : B) → AddCommGroup (E₁ x)
inst_2 : (x : B) → 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 : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst_7 : (x : B) → AddCommGroup (E₂ x)
inst_8 : (x : B) → 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 : B) → TopologicalSpace (E₂ x)
EB : Type u_10
inst_13 : NormedAddCommGroup EB
inst_14 : NormedSpace 𝕜 EB
HB : Type u_11
inst_15 : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst_16 : TopologicalSpace B
inst_17 : ChartedSpace HB B
EM : Type u_12
inst_18 : NormedAddCommGroup EM
inst_19 : NormedSpace 𝕜 EM
HM : Type u_13
inst_20 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_21 : TopologicalSpace M
inst_22 : ChartedSpace HM M
inst_23 : FiberBundle F₁ E₁
inst_24 : VectorBundle 𝕜 F₁ E₁
inst_25 : FiberBundle F₂ E₂
inst_26 : VectorBundle 𝕜 F₂ E₂
b : M → B
v : (x : M) → E₁ (b x)
x : M
inst_27 : ∀ (x : B), IsTopologicalAddGroup.{u_8} (E₂ x)
inst_28 : ∀ (x : B), ContinuousSMul.{u_1, u_8} 𝕜 (E₂ x)
ϕ : (x : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b x)) (E₂ (b x))
hϕ :
  ContMDiffAt IM (E' := EB × (F₁ →L[𝕜] F₂)) (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₂)) (M' :=
    TotalSpace (F₁ →L[𝕜] F₂) fun (x : B) ↦ ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ x) (E₂ x)) n
    (fun (m : M) ↦ ⟨b m, ϕ m⟩) x
hv : ContMDiffAt IM (IB.prod 𝓘(𝕜, F₁)) (M' := TotalSpace F₁ E₁) n (fun (m : M) ↦ ⟨b m, v m⟩) x
⊢ ContMDiffAt IM (IB.prod 𝓘(𝕜, F₂)) (M' := TotalSpace F₂ E₂) n
    (fun (m : M) ↦ ⟨b m, (ϕ m : (a : E₁ (b m)) → E₂ (b m)) (v m)⟩) x

Tactic:
ContMDiffWithinAt.clm_bundle_apply hϕ hv

Premises used: ContMDiffWithinAt.clm_bundle_apply, ContMDiffAt
-- Module: Mathlib.Geometry.Manifold.VectorBundle.Hom
Goal:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_6
inst : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
E₁ : B → Type u_7
inst_1 : (x : B) → AddCommGroup (E₁ x)
inst_2 : (x : B) → 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 : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst_7 : (x : B) → AddCommGroup (E₂ x)
inst_8 : (x : B) → 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 : B) → TopologicalSpace (E₂ x)
EB : Type u_10
inst_13 : NormedAddCommGroup EB
inst_14 : NormedSpace 𝕜 EB
HB : Type u_11
inst_15 : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst_16 : TopologicalSpace B
inst_17 : ChartedSpace HB B
EM : Type u_12
inst_18 : NormedAddCommGroup EM
inst_19 : NormedSpace 𝕜 EM
HM : Type u_13
inst_20 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_21 : TopologicalSpace M
inst_22 : ChartedSpace HM M
inst_23 : FiberBundle F₁ E₁
inst_24 : VectorBundle 𝕜 F₁ E₁
inst_25 : FiberBundle F₂ E₂
inst_26 : VectorBundle 𝕜 F₂ E₂
b : M → B
v : (x : M) → E₁ (b x)
s : Set M
inst_27 : ∀ (x : B), IsTopologicalAddGroup.{u_8} (E₂ x)
inst_28 : ∀ (x : B), ContinuousSMul.{u_1, u_8} 𝕜 (E₂ x)
ϕ : (x : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b x)) (E₂ (b x))
hϕ :
  ContMDiffOn IM (E' := EB × (F₁ →L[𝕜] F₂)) (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₂)) (M' :=
    TotalSpace (F₁ →L[𝕜] F₂) fun (x : B) ↦ ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ x) (E₂ x)) n
    (fun (m : M) ↦ ⟨b m, ϕ m⟩) s
hv : ContMDiffOn IM (IB.prod 𝓘(𝕜, F₁)) (M' := TotalSpace F₁ E₁) n (fun (m : M) ↦ ⟨b m, v m⟩) s
⊢ ContMDiffOn IM (IB.prod 𝓘(𝕜, F₂)) (M' := TotalSpace F₂ E₂) n
    (fun (m : M) ↦ ⟨b m, (ϕ m : (a : E₁ (b m)) → E₂ (b m)) (v m)⟩) s

Tactic:
fun x hx ↦ (hϕ x hx).clm_bundle_apply (hv x hx)

Premises used: ContMDiffWithinAt.clm_bundle_apply, ContMDiffOn
-- Module: Mathlib.Geometry.Manifold.VectorBundle.Hom
Goal:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_6
inst : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
E₁ : B → Type u_7
inst_1 : (x : B) → AddCommGroup (E₁ x)
inst_2 : (x : B) → 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 : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst_7 : (x : B) → AddCommGroup (E₂ x)
inst_8 : (x : B) → 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 : B) → TopologicalSpace (E₂ x)
EB : Type u_10
inst_13 : NormedAddCommGroup EB
inst_14 : NormedSpace 𝕜 EB
HB : Type u_11
inst_15 : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst_16 : TopologicalSpace B
inst_17 : ChartedSpace HB B
EM : Type u_12
inst_18 : NormedAddCommGroup EM
inst_19 : NormedSpace 𝕜 EM
HM : Type u_13
inst_20 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_21 : TopologicalSpace M
inst_22 : ChartedSpace HM M
inst_23 : FiberBundle F₁ E₁
inst_24 : VectorBundle 𝕜 F₁ E₁
inst_25 : FiberBundle F₂ E₂
inst_26 : VectorBundle 𝕜 F₂ E₂
b : M → B
v : (x : M) → E₁ (b x)
inst_27 : ∀ (x : B), IsTopologicalAddGroup.{u_8} (E₂ x)
inst_28 : ∀ (x : B), ContinuousSMul.{u_1, u_8} 𝕜 (E₂ x)
ϕ : (x : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b x)) (E₂ (b x))
hϕ :
  ContMDiff IM (E' := EB × (F₁ →L[𝕜] F₂)) (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₂)) (M' :=
    TotalSpace (F₁ →L[𝕜] F₂) fun (x : B) ↦ ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ x) (E₂ x)) n
    fun (m : M) ↦ ⟨b m, ϕ m⟩
hv : ContMDiff IM (IB.prod 𝓘(𝕜, F₁)) (M' := TotalSpace F₁ E₁) n fun (m : M) ↦ ⟨b m, v m⟩
⊢ ContMDiff IM (IB.prod 𝓘(𝕜, F₂)) (M' := TotalSpace F₂ E₂) n fun (m : M) ↦
    ⟨b m, (ϕ m : (a : E₁ (b m)) → E₂ (b m)) (v m)⟩

Tactic:
fun x ↦ (hϕ x).clm_bundle_apply (hv x)

Premises used: ContMDiff, ContMDiffAt.clm_bundle_apply
-- Module: Mathlib.Geometry.Manifold.VectorBundle.Hom
Goal:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_6
inst : NontriviallyNormedField 𝕜
E₁ : B → Type u_7
inst_1 : (x : B) → AddCommGroup (E₁ x)
inst_2 : (x : B) → 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 : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst_7 : (x : B) → AddCommGroup (E₂ x)
inst_8 : (x : B) → 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 : B) → TopologicalSpace (E₂ x)
EB : Type u_10
inst_13 : NormedAddCommGroup EB
inst_14 : NormedSpace 𝕜 EB
HB : Type u_11
inst_15 : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst_16 : TopologicalSpace B
inst_17 : ChartedSpace HB B
EM : Type u_12
inst_18 : NormedAddCommGroup EM
inst_19 : NormedSpace 𝕜 EM
HM : Type u_13
inst_20 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_21 : TopologicalSpace M
inst_22 : ChartedSpace HM M
inst_23 : FiberBundle F₁ E₁
inst_24 : VectorBundle 𝕜 F₁ E₁
inst_25 : FiberBundle F₂ E₂
inst_26 : VectorBundle 𝕜 F₂ E₂
b : M → B
v : (x : M) → E₁ (b x)
s : Set M
x : M
inst_27 : ∀ (x : B), IsTopologicalAddGroup.{u_8} (E₂ x)
inst_28 : ∀ (x : B), ContinuousSMul.{u_1, u_8} 𝕜 (E₂ x)
ϕ : (x : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b x)) (E₂ (b x))
hϕ : MDiffAt[s] (T% ϕ) x
hv : MDiffAt[s] (T% v) x
⊢ (MDiffAt[s] fun (x : M) ↦ b x) x ∧
    (MDiffAt[s] fun (x_1 : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b x) (b x_1) (b x) (b x_1) (ϕ x_1)) x

Tactic:
simp only [mdifferentiableWithinAt_hom_bundle] at hϕ

Premises used: mdifferentiableWithinAt_hom_bundle
-- Module: Mathlib.Geometry.Manifold.VectorBundle.Hom
Goal:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_6
inst : NontriviallyNormedField 𝕜
E₁ : B → Type u_7
inst_1 : (x : B) → AddCommGroup (E₁ x)
inst_2 : (x : B) → 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 : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst_7 : (x : B) → AddCommGroup (E₂ x)
inst_8 : (x : B) → 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 : B) → TopologicalSpace (E₂ x)
EB : Type u_10
inst_13 : NormedAddCommGroup EB
inst_14 : NormedSpace 𝕜 EB
HB : Type u_11
inst_15 : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst_16 : TopologicalSpace B
inst_17 : ChartedSpace HB B
EM : Type u_12
inst_18 : NormedAddCommGroup EM
inst_19 : NormedSpace 𝕜 EM
HM : Type u_13
inst_20 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_21 : TopologicalSpace M
inst_22 : ChartedSpace HM M
inst_23 : FiberBundle F₁ E₁
inst_24 : VectorBundle 𝕜 F₁ E₁
inst_25 : FiberBundle F₂ E₂
inst_26 : VectorBundle 𝕜 F₂ E₂
b : M → B
v : (x : M) → E₁ (b x)
s : Set M
x : M
inst_27 : ∀ (x : B), IsTopologicalAddGroup.{u_8} (E₂ x)
inst_28 : ∀ (x : B), ContinuousSMul.{u_1, u_8} 𝕜 (E₂ x)
ϕ : (x : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b x)) (E₂ (b x))
hv : MDiffAt[s] (T% v) x
hϕ :
  (MDiffAt[s] fun (x : M) ↦ b x) x ∧
    (MDiffAt[s] fun (x_1 : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b x) (b x_1) (b x) (b x_1) (ϕ x_1)) x
⊢ (MDiffAt[s] fun (m : M) ↦ ⟨b m, (ϕ m : (a : E₁ (b m)) → E₂ (b m)) (v m)⟩) x

Tactic:
hϕ.2.clm_apply_of_inCoordinates hv hϕ.1

Premises used: MDifferentiableWithinAt.clm_apply_of_inCoordinates, And.right, And.left
-- Module: Mathlib.Geometry.Manifold.VectorBundle.Hom
Goal:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_6
inst : NontriviallyNormedField 𝕜
E₁ : B → Type u_7
inst_1 : (x : B) → AddCommGroup (E₁ x)
inst_2 : (x : B) → 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 : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst_7 : (x : B) → AddCommGroup (E₂ x)
inst_8 : (x : B) → 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 : B) → TopologicalSpace (E₂ x)
EB : Type u_10
inst_13 : NormedAddCommGroup EB
inst_14 : NormedSpace 𝕜 EB
HB : Type u_11
inst_15 : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst_16 : TopologicalSpace B
inst_17 : ChartedSpace HB B
EM : Type u_12
inst_18 : NormedAddCommGroup EM
inst_19 : NormedSpace 𝕜 EM
HM : Type u_13
inst_20 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_21 : TopologicalSpace M
inst_22 : ChartedSpace HM M
inst_23 : FiberBundle F₁ E₁
inst_24 : VectorBundle 𝕜 F₁ E₁
inst_25 : FiberBundle F₂ E₂
inst_26 : VectorBundle 𝕜 F₂ E₂
b : M → B
v : (x : M) → E₁ (b x)
s : Set M
x : M
inst_27 : ∀ (x : B), IsTopologicalAddGroup.{u_8} (E₂ x)
inst_28 : ∀ (x : B), ContinuousSMul.{u_1, u_8} 𝕜 (E₂ x)
ϕ : (x : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b x)) (E₂ (b x))
hv : MDiffAt[s] (T% v) x
hϕ :
  (MDiffAt[s] fun (x : M) ↦ b x) x ∧
    (MDiffAt[s] fun (x_1 : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b x) (b x_1) (b x) (b x_1) (ϕ x_1)) x
⊢ (MDiffAt[s] fun (m : M) ↦ ⟨b m, (ϕ m : (a : E₁ (b m)) → E₂ (b m)) (v m)⟩) x

Tactic:
exact hϕ.2.clm_apply_of_inCoordinates hv hϕ.1

Premises used: MDifferentiableWithinAt.clm_apply_of_inCoordinates, And.right, And.left
-- Module: Mathlib.Geometry.Manifold.VectorBundle.Hom
Goal:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_6
inst : NontriviallyNormedField 𝕜
E₁ : B → Type u_7
inst_1 : (x : B) → AddCommGroup (E₁ x)
inst_2 : (x : B) → 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 : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst_7 : (x : B) → AddCommGroup (E₂ x)
inst_8 : (x : B) → 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 : B) → TopologicalSpace (E₂ x)
EB : Type u_10
inst_13 : NormedAddCommGroup EB
inst_14 : NormedSpace 𝕜 EB
HB : Type u_11
inst_15 : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst_16 : TopologicalSpace B
inst_17 : ChartedSpace HB B
EM : Type u_12
inst_18 : NormedAddCommGroup EM
inst_19 : NormedSpace 𝕜 EM
HM : Type u_13
inst_20 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_21 : TopologicalSpace M
inst_22 : ChartedSpace HM M
inst_23 : FiberBundle F₁ E₁
inst_24 : VectorBundle 𝕜 F₁ E₁
inst_25 : FiberBundle F₂ E₂
inst_26 : VectorBundle 𝕜 F₂ E₂
b : M → B
v : (x : M) → E₁ (b x)
x : M
inst_27 : ∀ (x : B), IsTopologicalAddGroup.{u_8} (E₂ x)
inst_28 : ∀ (x : B), ContinuousSMul.{u_1, u_8} 𝕜 (E₂ x)
ϕ : (x : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b x)) (E₂ (b x))
hϕ : MDiffAt (T% ϕ) x
hv : MDiffAt (T% v) x
⊢ (MDiffAt fun (m : M) ↦ ⟨b m, (ϕ m : (a : E₁ (b m)) → E₂ (b m)) (v m)⟩) x

Tactic:
MDifferentiableWithinAt.clm_bundle_apply hϕ hv

Premises used: MDifferentiableAt, MDifferentiableWithinAt.clm_bundle_apply, ChartedSpace.LiftPropAt
-- Module: Mathlib.Geometry.Manifold.VectorBundle.Hom
Goal:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_6
inst : NontriviallyNormedField 𝕜
E₁ : B → Type u_7
inst_1 : (x : B) → AddCommGroup (E₁ x)
inst_2 : (x : B) → 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 : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst_7 : (x : B) → AddCommGroup (E₂ x)
inst_8 : (x : B) → 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 : B) → TopologicalSpace (E₂ x)
EB : Type u_10
inst_13 : NormedAddCommGroup EB
inst_14 : NormedSpace 𝕜 EB
HB : Type u_11
inst_15 : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst_16 : TopologicalSpace B
inst_17 : ChartedSpace HB B
EM : Type u_12
inst_18 : NormedAddCommGroup EM
inst_19 : NormedSpace 𝕜 EM
HM : Type u_13
inst_20 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_21 : TopologicalSpace M
inst_22 : ChartedSpace HM M
inst_23 : FiberBundle F₁ E₁
inst_24 : VectorBundle 𝕜 F₁ E₁
inst_25 : FiberBundle F₂ E₂
inst_26 : VectorBundle 𝕜 F₂ E₂
b : M → B
v : (x : M) → E₁ (b x)
s : Set M
inst_27 : ∀ (x : B), IsTopologicalAddGroup.{u_8} (E₂ x)
inst_28 : ∀ (x : B), ContinuousSMul.{u_1, u_8} 𝕜 (E₂ x)
ϕ : (x : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b x)) (E₂ (b x))
hϕ : MDiff[s] (T% ϕ)
hv : MDiff[s] (T% v)
⊢ MDiff[s] fun (m : M) ↦ ⟨b m, (ϕ m : (a : E₁ (b m)) → E₂ (b m)) (v m)⟩

Tactic:
fun x hx ↦ (hϕ x hx).clm_bundle_apply (hv x hx)

Premises used: MDifferentiableOn, MDifferentiableWithinAt.clm_bundle_apply
-- Module: Mathlib.Geometry.Manifold.VectorBundle.Hom
Goal:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
M : Type u_6
inst : NontriviallyNormedField 𝕜
E₁ : B → Type u_7
inst_1 : (x : B) → AddCommGroup (E₁ x)
inst_2 : (x : B) → 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 : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst_7 : (x : B) → AddCommGroup (E₂ x)
inst_8 : (x : B) → 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 : B) → TopologicalSpace (E₂ x)
EB : Type u_10
inst_13 : NormedAddCommGroup EB
inst_14 : NormedSpace 𝕜 EB
HB : Type u_11
inst_15 : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst_16 : TopologicalSpace B
inst_17 : ChartedSpace HB B
EM : Type u_12
inst_18 : NormedAddCommGroup EM
inst_19 : NormedSpace 𝕜 EM
HM : Type u_13
inst_20 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_21 : TopologicalSpace M
inst_22 : ChartedSpace HM M
inst_23 : FiberBundle F₁ E₁
inst_24 : VectorBundle 𝕜 F₁ E₁
inst_25 : FiberBundle F₂ E₂
inst_26 : VectorBundle 𝕜 F₂ E₂
b : M → B
v : (x : M) → E₁ (b x)
inst_27 : ∀ (x : B), IsTopologicalAddGroup.{u_8} (E₂ x)
inst_28 : ∀ (x : B), ContinuousSMul.{u_1, u_8} 𝕜 (E₂ x)
ϕ : (x : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b x)) (E₂ (b x))
hϕ : MDiff (T% ϕ)
hv : MDiff (T% v)
⊢ MDiff fun (m : M) ↦ ⟨b m, (ϕ m : (a : E₁ (b m)) → E₂ (b m)) (v m)⟩

Tactic:
fun x ↦ (hϕ x).clm_bundle_apply (hv x)

Premises used: MDifferentiableAt.clm_bundle_apply, MDifferentiable
-- Module: Mathlib.Geometry.Manifold.VectorBundle.Hom
Goal:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
F₂ : Type u_4
F₃ : Type u_5
M : Type u_6
inst : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
E₁ : B → Type u_7
inst_1 : (x : B) → AddCommGroup (E₁ x)
inst_2 : (x : B) → 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 : B) → TopologicalSpace (E₁ x)
E₂ : B → Type u_8
inst_7 : (x : B) → AddCommGroup (E₂ x)
inst_8 : (x : B) → 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 : B) → TopologicalSpace (E₂ x)
E₃ : B → Type u_9
inst_13 : (x : B) → AddCommGroup (E₃ x)
inst_14 : (x : B) → Module.{u_1, u_9} 𝕜 (E₃ x)
inst_15 : NormedAddCommGroup F₃
inst_16 : NormedSpace 𝕜 F₃
inst_17 : TopologicalSpace (TotalSpace F₃ E₃)
inst_18 : (x : B) → TopologicalSpace (E₃ x)
EB : Type u_10
inst_19 : NormedAddCommGroup EB
inst_20 : NormedSpace 𝕜 EB
HB : Type u_11
inst_21 : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst_22 : TopologicalSpace B
inst_23 : ChartedSpace HB B
EM : Type u_12
inst_24 : NormedAddCommGroup EM
inst_25 : NormedSpace 𝕜 EM
HM : Type u_13
inst_26 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_27 : TopologicalSpace M
inst_28 : ChartedSpace HM M
inst_29 : FiberBundle F₁ E₁
inst_30 : VectorBundle 𝕜 F₁ E₁
inst_31 : FiberBundle F₂ E₂
inst_32 : VectorBundle 𝕜 F₂ E₂
inst_33 : FiberBundle F₃ E₃
inst_34 : VectorBundle 𝕜 F₃ E₃
b : M → B
v : (x : M) → E₁ (b x)
s : Set M
x : M
inst_35 : ∀ (x : B), IsTopologicalAddGroup.{u_9} (E₃ x)
inst_36 : ∀ (x : B), ContinuousSMul.{u_1, u_9} 𝕜 (E₃ x)
ψ : (x : M) → E₁ (b x) →L[𝕜] ContinuousLinearMap.{u_1, u_1, u_8, u_9} (RingHom.id 𝕜) (E₂ (b x)) (E₃ (b x))
w : (x : M) → E₂ (b x)
hψ :
  ContMDiffWithinAt IM (E' := EB × (F₁ →L[𝕜] F₂ →L[𝕜] F₃)) (IB.prod 𝓘(𝕜, F₁ →L[𝕜] F₂ →L[𝕜] F₃)) (M' :=
    TotalSpace (F₁ →L[𝕜] F₂ →L[𝕜] F₃) fun (x : B) ↦
      E₁ x →L[𝕜] ContinuousLinearMap.{u_1, u_1, u_8, u_9} (RingHom.id 𝕜) (E₂ x) (E₃ x))
    n (fun (m : M) ↦ ⟨b m, ψ m⟩) s x
hv : ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₁)) (M' := TotalSpace F₁ E₁) n (fun (m : M) ↦ ⟨b m, v m⟩) s x
hw : ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₂)) (M' := TotalSpace F₂ E₂) n (fun (m : M) ↦ ⟨b m, w m⟩) s x
⊢ ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F₃)) (M' := TotalSpace F₃ E₃) n
    (fun (m : M) ↦
      ⟨b m,
        (((ψ m : (a : E₁ (b m)) → E₂ (b m) →L[𝕜] E₃ (b m)) (v m) :
              ContinuousLinearMap.{u_1, u_1, u_8, u_9} (RingHom.id 𝕜) (E₂ (b m)) (E₃ (b m))) :
            (a : E₂ (b m)) → E₃ (b m))
          (w m)⟩)
    s x

Tactic:
hψ.clm_bundle_apply hv |>.clm_bundle_apply hw

Premises used: ContMDiffWithinAt.clm_bundle_apply