Theorem: Embedding.comp_stronglyMeasurable_iff
File: Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁴ : Countable ι
f✝ g✝ : α → β
m : MeasurableSpace α
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : _root_.Embedding g
⊢ (StronglyMeasurable fun x => g (f x)) ↔ StronglyMeasurable f
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁴ : Countable ι
f✝ g✝ : α → β
m : MeasurableSpace α
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : _root_.Embedding g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
⊢ (StronglyMeasurable fun x => g (f x)) ↔ StronglyMeasurable f
tactic: letI := pseudoMetrizableSpacePseudoMetric γ
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁴ : Countable ι
f✝ g✝ : α → β
m : MeasurableSpace α
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : _root_.Embedding g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
⊢ (StronglyMeasurable fun x => g (f x)) ↔ StronglyMeasurable f
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁴ : Countable ι
f✝ g✝ : α → β
m : MeasurableSpace α
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : _root_.Embedding g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
this✝³ : MeasurableSpace β := borel β
this✝² : BorelSpace β
this✝¹ : MeasurableSpace γ := borel γ
this✝ : BorelSpace γ
⊢ (StronglyMeasurable fun x => g (f x)) ↔ StronglyMeasurable f
tactic: borelize β γ
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁴ : Countable ι
f✝ g✝ : α → β
m : MeasurableSpace α
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : _root_.Embedding g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
this✝³ : MeasurableSpace β := borel β
this✝² : BorelSpace β
this✝¹ : MeasurableSpace γ := borel γ
this✝ : BorelSpace γ
⊢ (StronglyMeasurable fun x => g (f x)) ↔ StronglyMeasurable f
state_after: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁴ : Countable ι
f✝ g✝ : α → β
m : MeasurableSpace α
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : _root_.Embedding g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
this✝³ : MeasurableSpace β := borel β
this✝² : BorelSpace β
this✝¹ : MeasurableSpace γ := borel γ
this✝ : BorelSpace γ
H : StronglyMeasurable fun x => g (f x)
⊢ Measurable f

case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁴ : Countable ι
f✝ g✝ : α → β
m : MeasurableSpace α
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : _root_.Embedding g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
this✝³ : MeasurableSpace β := borel β
this✝² : BorelSpace β
this✝¹ : MeasurableSpace γ := borel γ
this✝ : BorelSpace γ
H : StronglyMeasurable fun x => g (f x)
⊢ IsSeparable (range f)
tactic: refine
    ⟨fun H => stronglyMeasurable_iff_measurable_separable.2 ⟨?_, ?_⟩, fun H =>
      hg.continuous.comp_stronglyMeasurable H⟩