-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
theorem stronglyMeasurable_deriv_with_param [LocallyCompactSpace 𝕜] [MeasurableSpace 𝕜]
    [OpensMeasurableSpace 𝕜] [h : SecondCountableTopologyEither α F]
    {f : α → 𝕜 → F} (hf : Continuous f.uncurry) :
    StronglyMeasurable (fun (p : α × 𝕜) ↦ deriv (f p.1) p.2)

Proof state:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous (Function.uncurry f)
this✝² : MeasurableSpace F := borel F
this✝¹ : BorelSpace F
hα : SecondCountableTopology α
this✝ : ProperSpace 𝕜
this : (range fun p => deriv (f p.1) p.2) ⊆ closure ↑(Submodule.span 𝕜 (range (Function.uncurry f)))
⊢ IsSeparable (range fun p => deriv (f p.1) p.2)

Tactic:
exact (isSeparable_range hf).span.closure.mono this

Result:
no goals

Proof state:
case inl
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous (Function.uncurry f)
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hα : SecondCountableTopology α
⊢ StronglyMeasurable fun p => deriv (f p.1) p.2

Tactic:
have : ProperSpace 𝕜 := .of_locallyCompactSpace 𝕜

Result:
case inl
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous (Function.uncurry f)
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hα : SecondCountableTopology α
this : ProperSpace 𝕜
⊢ StronglyMeasurable fun p => deriv (f p.1) p.2

Proof state:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous (Function.uncurry f)
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hα : SecondCountableTopology α
this : ProperSpace 𝕜
p : α × 𝕜
⊢ deriv (f p.1) p.2 ∈ closure ↑(Submodule.span 𝕜 (f p.1 '' univ))

Tactic:
apply range_deriv_subset_closure_span_image _ dense_univ (mem_range_self _)

Result:
no goals

Proof state:
case inr
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous (Function.uncurry f)
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hF : SecondCountableTopology F
⊢ StronglyMeasurable fun p => deriv (f p.1) p.2

Tactic:
exact (measurable_deriv_with_param hf).stronglyMeasurable

Result:
no goals

Proof state:
case intro
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous (Function.uncurry f)
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hα : SecondCountableTopology α
this : ProperSpace 𝕜
p : α × 𝕜
A : deriv (f p.1) p.2 ∈ closure ↑(Submodule.span 𝕜 (range (f p.1)))
x : 𝕜
⊢ f p.1 x ∈ range (Function.uncurry f)

Tactic:
exact mem_range_self (p.1, x)

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous (Function.uncurry f)
⊢ StronglyMeasurable fun p => deriv (f p.1) p.2

Tactic:
borelize F

Result:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous (Function.uncurry f)
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
⊢ StronglyMeasurable fun p => deriv (f p.1) p.2

Proof state:
case inl
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous (Function.uncurry f)
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hα : SecondCountableTopology α
this : ProperSpace 𝕜
⊢ StronglyMeasurable fun p => deriv (f p.1) p.2

Tactic:
apply stronglyMeasurable_iff_measurable_separable.2 ⟨measurable_deriv_with_param hf, ?_⟩

Result:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous (Function.uncurry f)
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hα : SecondCountableTopology α
this : ProperSpace 𝕜
⊢ IsSeparable (range fun p => deriv (f p.1) p.2)

Proof state:
case intro
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : OpensMeasurableSpace α
inst✝³ : CompleteSpace F
inst✝² : LocallyCompactSpace 𝕜
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither α F
f : α → 𝕜 → F
hf : Continuous (Function.uncurry f)
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hα : SecondCountableTopology α
this : ProperSpace 𝕜
p : α × 𝕜
A : deriv (f p.1) p.2 ∈ closure ↑(Submodule.span 𝕜 (range (f p.1)))
B : range (f p.1) ⊆ range (Function.uncurry f)
⊢ (fun p => deriv (f p.1) p.2) p ∈ closure ↑(Submodule.span 𝕜 (range (Function.uncurry f)))

Tactic:
exact closure_mono (Submodule.span_mono B) A

Result:
no goals