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

tactic: borelize F
state_before: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f✝ : E → F
K : Set (E →L[𝕜] F)
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : CompleteSpace F
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither 𝕜 F
f : 𝕜 → F
⊢ StronglyMeasurable (deriv f)
state_after: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f✝ : E → F
K : Set (E →L[𝕜] F)
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : CompleteSpace F
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither 𝕜 F
f : 𝕜 → F
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
⊢ StronglyMeasurable (deriv f)

tactic: rcases h.out with h𝕜|hF
state_before: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f✝ : E → F
K : Set (E →L[𝕜] F)
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : CompleteSpace F
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither 𝕜 F
f : 𝕜 → F
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
⊢ StronglyMeasurable (deriv f)
state_after: case inl
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f✝ : E → F
K : Set (E →L[𝕜] F)
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : CompleteSpace F
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither 𝕜 F
f : 𝕜 → F
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
h𝕜 : SecondCountableTopology 𝕜
⊢ StronglyMeasurable (deriv f)

case inr
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f✝ : E → F
K : Set (E →L[𝕜] F)
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : CompleteSpace F
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither 𝕜 F
f : 𝕜 → F
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hF : SecondCountableTopology F
⊢ StronglyMeasurable (deriv f)

tactic: exact stronglyMeasurable_iff_measurable_separable.2
  ⟨measurable_deriv f, isSeparable_range_deriv _⟩
state_before: case inl
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f✝ : E → F
K : Set (E →L[𝕜] F)
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : CompleteSpace F
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither 𝕜 F
f : 𝕜 → F
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
h𝕜 : SecondCountableTopology 𝕜
⊢ StronglyMeasurable (deriv f)
state_after: no goals

tactic: exact (measurable_deriv f).stronglyMeasurable
state_before: case inr
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f✝ : E → F
K : Set (E →L[𝕜] F)
inst✝⁴ : MeasurableSpace E
inst✝³ : OpensMeasurableSpace E
inst✝² : CompleteSpace F
inst✝¹ : MeasurableSpace 𝕜
inst✝ : OpensMeasurableSpace 𝕜
h : SecondCountableTopologyEither 𝕜 F
f : 𝕜 → F
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
hF : SecondCountableTopology F
⊢ StronglyMeasurable (deriv f)
state_after: no goals