-- Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean
theorem measurable_lineDeriv [MeasurableSpace F] [BorelSpace F]
    (hf : Continuous f) : Measurable (fun x ↦ lineDeriv 𝕜 f x v)

Proof state:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace F
f : E → F
v : E
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E → 𝕜 → F := fun x t => f (x + t • v)
⊢ Measurable fun x => lineDeriv 𝕜 f x v

Tactic:
have hg : Continuous g.uncurry := by fun_prop

Result:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace F
f : E → F
v : E
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E → 𝕜 → F := fun x t => f (x + t • v)
hg : Continuous (Function.uncurry g)
⊢ Measurable fun x => lineDeriv 𝕜 f x v

Proof state:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace F
f : E → F
v : E
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
⊢ Measurable fun x => lineDeriv 𝕜 f x v

Tactic:
let g : E → 𝕜 → F := fun x t ↦ f (x + t • v)

Result:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace F
f : E → F
v : E
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E → 𝕜 → F := fun x t => f (x + t • v)
⊢ Measurable fun x => lineDeriv 𝕜 f x v

Proof state:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace F
f : E → F
v : E
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E → 𝕜 → F := fun x t => f (x + t • v)
hg : Continuous (Function.uncurry g)
⊢ Measurable fun x => lineDeriv 𝕜 f x v

Tactic:
exact (measurable_deriv_with_param hg).comp measurable_prod_mk_right

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace F
f : E → F
v : E
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
hf : Continuous f
⊢ Measurable fun x => lineDeriv 𝕜 f x v

Tactic:
borelize 𝕜

Result:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace F
f : E → F
v : E
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
⊢ Measurable fun x => lineDeriv 𝕜 f x v

Proof state:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace F
f : E → F
v : E
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
hf : Continuous f
this✝¹ : MeasurableSpace 𝕜 := borel 𝕜
this✝ : BorelSpace 𝕜
g : E → 𝕜 → F := fun x t => f (x + t • v)
⊢ Continuous (Function.uncurry g)

Tactic:
fun_prop

Result:
no goals