Theorem: measurableSet_of_differentiableAt_of_isComplete_with_param
File: Mathlib/Analysis/Calculus/FDeriv/Measurable.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
⊢ MeasurableSet {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K}
state_after: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
⊢ MeasurableSet {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K}
tactic: have : {p : α × E | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K}
          = {p : α × E | p.2 ∈ D (f p.1) K} := by simp [← differentiable_set_eq_D K hK]
state_before: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
⊢ MeasurableSet {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K}
state_after: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
⊢ MeasurableSet {p | p.2 ∈ D (f p.1) K}
tactic: rw [this]
state_before: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
⊢ MeasurableSet {p | p.2 ∈ D (f p.1) K}
state_after: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
⊢ MeasurableSet
    {p | ∀ (i : ℕ), ∃ i_1, ∀ i_2 ≥ i_1, ∀ i_4 ≥ i_1, p.2 ∈ B (f p.1) K ((1 / 2) ^ i_2) ((1 / 2) ^ i_4) ((1 / 2) ^ i)}
tactic: simp only [D, mem_iInter, mem_iUnion]
state_before: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
⊢ MeasurableSet
    {p | ∀ (i : ℕ), ∃ i_1, ∀ i_2 ≥ i_1, ∀ i_4 ≥ i_1, p.2 ∈ B (f p.1) K ((1 / 2) ^ i_2) ((1 / 2) ^ i_4) ((1 / 2) ^ i)}
state_after: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
⊢ MeasurableSet
    (⋂ i,
      ⋃ i_1,
        ⋂ i_2,
          ⋂ (_ : i_2 ≥ i_1),
            ⋂ i_3, ⋂ (_ : i_3 ≥ i_1), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ i_2) ((1 / 2) ^ i_3) ((1 / 2) ^ i)})
tactic: simp only [setOf_forall, setOf_exists]
state_before: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
⊢ MeasurableSet
    (⋂ i,
      ⋃ i_1,
        ⋂ i_2,
          ⋂ (_ : i_2 ≥ i_1),
            ⋂ i_3, ⋂ (_ : i_3 ≥ i_1), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ i_2) ((1 / 2) ^ i_3) ((1 / 2) ^ i)})
state_after: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝ : ℕ
⊢ MeasurableSet
    (⋃ i,
      ⋂ i_1,
        ⋂ (_ : i_1 ≥ i), ⋂ i_2, ⋂ (_ : i_2 ≥ i), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ i_1) ((1 / 2) ^ i_2) ((1 / 2) ^ x✝)})
tactic: refine MeasurableSet.iInter (fun _ ↦ ?_)
state_before: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝ : ℕ
⊢ MeasurableSet
    (⋃ i,
      ⋂ i_1,
        ⋂ (_ : i_1 ≥ i), ⋂ i_2, ⋂ (_ : i_2 ≥ i), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ i_1) ((1 / 2) ^ i_2) ((1 / 2) ^ x✝)})
state_after: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝¹ x✝ : ℕ
⊢ MeasurableSet
    (⋂ i,
      ⋂ (_ : i ≥ x✝), ⋂ i_1, ⋂ (_ : i_1 ≥ x✝), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ i) ((1 / 2) ^ i_1) ((1 / 2) ^ x✝¹)})
tactic: refine MeasurableSet.iUnion (fun _ ↦ ?_)
state_before: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝¹ x✝ : ℕ
⊢ MeasurableSet
    (⋂ i,
      ⋂ (_ : i ≥ x✝), ⋂ i_1, ⋂ (_ : i_1 ≥ x✝), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ i) ((1 / 2) ^ i_1) ((1 / 2) ^ x✝¹)})
state_after: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝² x✝¹ x✝ : ℕ
⊢ MeasurableSet
    (⋂ (_ : x✝ ≥ x✝¹), ⋂ i, ⋂ (_ : i ≥ x✝¹), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ x✝) ((1 / 2) ^ i) ((1 / 2) ^ x✝²)})
tactic: refine MeasurableSet.iInter (fun _ ↦ ?_)
state_before: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝² x✝¹ x✝ : ℕ
⊢ MeasurableSet
    (⋂ (_ : x✝ ≥ x✝¹), ⋂ i, ⋂ (_ : i ≥ x✝¹), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ x✝) ((1 / 2) ^ i) ((1 / 2) ^ x✝²)})
state_after: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝³ x✝² x✝¹ : ℕ
x✝ : x✝¹ ≥ x✝²
⊢ MeasurableSet (⋂ i, ⋂ (_ : i ≥ x✝²), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ x✝¹) ((1 / 2) ^ i) ((1 / 2) ^ x✝³)})
tactic: refine MeasurableSet.iInter (fun _ ↦ ?_)
state_before: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝³ x✝² x✝¹ : ℕ
x✝ : x✝¹ ≥ x✝²
⊢ MeasurableSet (⋂ i, ⋂ (_ : i ≥ x✝²), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ x✝¹) ((1 / 2) ^ i) ((1 / 2) ^ x✝³)})
state_after: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝⁴ x✝³ x✝² : ℕ
x✝¹ : x✝² ≥ x✝³
x✝ : ℕ
⊢ MeasurableSet (⋂ (_ : x✝ ≥ x✝³), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ x✝²) ((1 / 2) ^ x✝) ((1 / 2) ^ x✝⁴)})
tactic: refine MeasurableSet.iInter (fun _ ↦ ?_)
state_before: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝⁴ x✝³ x✝² : ℕ
x✝¹ : x✝² ≥ x✝³
x✝ : ℕ
⊢ MeasurableSet (⋂ (_ : x✝ ≥ x✝³), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ x✝²) ((1 / 2) ^ x✝) ((1 / 2) ^ x✝⁴)})
state_after: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝⁵ x✝⁴ x✝³ : ℕ
x✝² : x✝³ ≥ x✝⁴
x✝¹ : ℕ
x✝ : x✝¹ ≥ x✝⁴
⊢ MeasurableSet {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ x✝³) ((1 / 2) ^ x✝¹) ((1 / 2) ^ x✝⁵)}
tactic: refine MeasurableSet.iInter (fun _ ↦ ?_)
state_before: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝⁵ x✝⁴ x✝³ : ℕ
x✝² : x✝³ ≥ x✝⁴
x✝¹ : ℕ
x✝ : x✝¹ ≥ x✝⁴
⊢ MeasurableSet {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ x✝³) ((1 / 2) ^ x✝¹) ((1 / 2) ^ x✝⁵)}
state_after: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this✝ : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝⁵ x✝⁴ x✝³ : ℕ
x✝² : x✝³ ≥ x✝⁴
x✝¹ : ℕ
x✝ : x✝¹ ≥ x✝⁴
this : ProperSpace E
⊢ MeasurableSet {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ x✝³) ((1 / 2) ^ x✝¹) ((1 / 2) ^ x✝⁵)}
tactic: have : ProperSpace E := .of_locallyCompactSpace 𝕜
state_before: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace E
f : α → E → F
K✝ : Set (E →L[𝕜] F)
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this✝ : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝⁵ x✝⁴ x✝³ : ℕ
x✝² : x✝³ ≥ x✝⁴
x✝¹ : ℕ
x✝ : x✝¹ ≥ x✝⁴
this : ProperSpace E
⊢ MeasurableSet {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ x✝³) ((1 / 2) ^ x✝¹) ((1 / 2) ^ x✝⁵)}
state_after: no goals
tactic: exact (isOpen_B_with_param hf K).measurableSet