-- Mathlib/Probability/Kernel/CondDistrib.lean
theorem condDistrib_ae_eq_of_measure_eq_compProd (hX : Measurable X) (hY : Measurable Y)
    (κ : Kernel β Ω) [IsFiniteKernel κ] (hκ : μ.map (fun x => (X x, Y x)) = μ.map X ⊗ₘ κ) :
    ∀ᵐ x ∂μ.map X, κ x = condDistrib Y X μ x

Proof state:
case h
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
s : Set β
hs : MeasurableSet s
⊢ (Measure.map X μ) s = (Measure.map (fun x => (X x, Y x)) μ).fst s

Tactic:
rw [Measure.map_apply hX hs, Measure.fst_apply hs, Measure.map_apply]

Result:
case h
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
s : Set β
hs : MeasurableSet s
⊢ μ (X ⁻¹' s) = μ ((fun x => (X x, Y x)) ⁻¹' (Prod.fst ⁻¹' s))

case h.hf
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
s : Set β
hs : MeasurableSet s
⊢ Measurable fun x => (X x, Y x)

case h.hs
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
s : Set β
hs : MeasurableSet s
⊢ MeasurableSet (Prod.fst ⁻¹' s)

Proof state:
case h
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
s : Set β
hs : MeasurableSet s
⊢ μ (X ⁻¹' s) = μ ((fun x => (X x, Y x)) ⁻¹' (Prod.fst ⁻¹' s))

case h.hf
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
s : Set β
hs : MeasurableSet s
⊢ Measurable fun x => (X x, Y x)

case h.hs
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
s : Set β
hs : MeasurableSet s
⊢ MeasurableSet (Prod.fst ⁻¹' s)

Tactic:
exacts [rfl, Measurable.prod hX hY, measurable_fst hs]

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
⊢ ∀ᵐ (x : β) ∂Measure.map X μ, κ x = (condDistrib Y X μ) x

Tactic:
have heq : μ.map X = (μ.map (fun x ↦ (X x, Y x))).fst := by
  ext s hs
  rw [Measure.map_apply hX hs, Measure.fst_apply hs, Measure.map_apply]
  exacts [rfl, Measurable.prod hX hY, measurable_fst hs]

Result:
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
heq : Measure.map X μ = (Measure.map (fun x => (X x, Y x)) μ).fst
⊢ ∀ᵐ (x : β) ∂Measure.map X μ, κ x = (condDistrib Y X μ) x

Proof state:
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
⊢ Measure.map X μ = (Measure.map (fun x => (X x, Y x)) μ).fst

Tactic:
ext s hs

Result:
case h
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
s : Set β
hs : MeasurableSet s
⊢ (Measure.map X μ) s = (Measure.map (fun x => (X x, Y x)) μ).fst s

Proof state:
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
heq : Measure.map X μ = (Measure.map (fun x => (X x, Y x)) μ).fst
⊢ ∀ᵐ (x : β) ∂Measure.map X μ, κ x = (condDistrib Y X μ) x

Tactic:
rw [heq, condDistrib]

Result:
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
heq : Measure.map X μ = (Measure.map (fun x => (X x, Y x)) μ).fst
⊢ ∀ᵐ (x : β) ∂(Measure.map (fun x => (X x, Y x)) μ).fst, κ x = (Measure.map (fun a => (X a, Y a)) μ).condKernel x

Proof state:
case h.e'_3.h.e'_5
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
hX : Measurable X
hY : Measurable Y
κ : Kernel β Ω
inst✝ : IsFiniteKernel κ
hκ : Measure.map (fun x => (X x, Y x)) μ = Measure.map X μ ⊗ₘ κ
heq : Measure.map X μ = (Measure.map (fun x => (X x, Y x)) μ).fst
⊢ (Measure.map (fun a => (X a, Y a)) μ).fst = Measure.map X μ

Tactic:
exact heq.symm

Result:
no goals