-- Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean
theorem stronglyMeasurable_lineDeriv_uncurry (hf : Continuous f) :
    StronglyMeasurable (fun (p : E × E) ↦ lineDeriv 𝕜 f p.1 p.2)

Proof state:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
inst✝ : SecondCountableTopology E
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E × E → 𝕜 → F := fun p t => f (p.1 + t • p.2)
⊢ StronglyMeasurable fun p => lineDeriv 𝕜 f p.1 p.2

Tactic:
have : Continuous g.uncurry :=
  hf.comp <| (continuous_fst.comp continuous_fst).add
  <| continuous_snd.smul (continuous_snd.comp continuous_fst)

Result:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
inst✝ : SecondCountableTopology E
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E × E → 𝕜 → F := fun p t => f (p.1 + t • p.2)
this : Continuous (Function.uncurry g)
⊢ StronglyMeasurable fun p => lineDeriv 𝕜 f p.1 p.2

Proof state:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
inst✝ : SecondCountableTopology E
hf : Continuous f
⊢ StronglyMeasurable fun p => lineDeriv 𝕜 f p.1 p.2

Tactic:
borelize 𝕜

Result:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
inst✝ : SecondCountableTopology E
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
⊢ StronglyMeasurable fun p => lineDeriv 𝕜 f p.1 p.2

Proof state:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
inst✝ : SecondCountableTopology E
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
⊢ StronglyMeasurable fun p => lineDeriv 𝕜 f p.1 p.2

Tactic:
let g : (E × E) → 𝕜 → F := fun p t ↦ f (p.1 + t • p.2)

Result:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
inst✝ : SecondCountableTopology E
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E × E → 𝕜 → F := fun p t => f (p.1 + t • p.2)
⊢ StronglyMeasurable fun p => lineDeriv 𝕜 f p.1 p.2

Proof state:
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
inst✝ : SecondCountableTopology E
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E × E → 𝕜 → F := fun p t => f (p.1 + t • p.2)
this : Continuous (Function.uncurry g)
⊢ StronglyMeasurable fun p => lineDeriv 𝕜 f p.1 p.2

Tactic:
exact (stronglyMeasurable_deriv_with_param this).comp_measurable measurable_prod_mk_right

Result:
no goals