Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: Ω : Type u_1
β : Type u_3
β' : Type u_4
mΩ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
f : Ω → β
g : Ω → β'
γ : Type u_5
mγ : MeasurableSpace γ
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝³ : StandardBorelSpace β
inst✝² : Nonempty β
inst✝¹ : StandardBorelSpace β'
inst✝ : Nonempty β'
hf : Measurable f
hg : Measurable g
k : Ω → γ
hk : Measurable k
e : γ × β' × β ≃ᵐ (γ × β') × β := MeasurableEquiv.prodAssoc.symm
h_eq :
  ⇑(Kernel.id ×ₖ condDistrib g k μ ×ₖ condDistrib f k μ) ∘ₘ Measure.map k μ =
    ⇑(Kernel.id ×ₖ Kernel.prodMkRight β' (condDistrib f k μ)) ∘ₘ Measure.map (fun a ↦ (k a, g a)) μ
h1 : Measure.map (fun x ↦ ((k x, g x), f x)) μ = Measure.map (⇑e) (Measure.map (fun a ↦ (k a, g a, f a)) μ)
h1_symm : Measure.map (fun a ↦ (k a, g a, f a)) μ = Measure.map (⇑e.symm) (Measure.map (fun x ↦ ((k x, g x), f x)) μ)
h2 :
  ⇑(Kernel.id ×ₖ condDistrib g k μ ×ₖ condDistrib f k μ) ∘ₘ Measure.map k μ =
    Measure.map (⇑e) (⇑(Kernel.id ×ₖ (condDistrib g k μ ×ₖ condDistrib f k μ)) ∘ₘ Measure.map k μ)
⊢ Measure.map (fun ω ↦ (k ω, g ω, f ω)) μ =
      ⇑(Kernel.id ×ₖ (condDistrib g k μ ×ₖ condDistrib f k μ)) ∘ₘ Measure.map k μ ↔
    Measure.map (fun x ↦ ((k x, g x), f x)) μ =
      ⇑(Kernel.id ×ₖ condDistrib g k μ ×ₖ condDistrib f k μ) ∘ₘ Measure.map k μ

Tactic:
have h2_symm : (Kernel.id ×ₖ (condDistrib g k μ ×ₖ condDistrib f k μ)) ∘ₘ μ.map k =
      (((Kernel.id ×ₖ condDistrib g k μ) ×ₖ condDistrib f k μ) ∘ₘ μ.map k).map e.symm := by
    rw [h2, Measure.map_map (by fun_prop) (by fun_prop), MeasurableEquiv.symm_comp_self,
      Measure.map_id]

Goal state after:
Ω : Type u_1
β : Type u_3
β' : Type u_4
mΩ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
f : Ω → β
g : Ω → β'
γ : Type u_5
mγ : MeasurableSpace γ
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝³ : StandardBorelSpace β
inst✝² : Nonempty β
inst✝¹ : StandardBorelSpace β'
inst✝ : Nonempty β'
hf : Measurable f
hg : Measurable g
k : Ω → γ
hk : Measurable k
e : γ × β' × β ≃ᵐ (γ × β') × β := MeasurableEquiv.prodAssoc.symm
h_eq :
  ⇑(Kernel.id ×ₖ condDistrib g k μ ×ₖ condDistrib f k μ) ∘ₘ Measure.map k μ =
    ⇑(Kernel.id ×ₖ Kernel.prodMkRight β' (condDistrib f k μ)) ∘ₘ Measure.map (fun a ↦ (k a, g a)) μ
h1 : Measure.map (fun x ↦ ((k x, g x), f x)) μ = Measure.map (⇑e) (Measure.map (fun a ↦ (k a, g a, f a)) μ)
h1_symm : Measure.map (fun a ↦ (k a, g a, f a)) μ = Measure.map (⇑e.symm) (Measure.map (fun x ↦ ((k x, g x), f x)) μ)
h2 :
  ⇑(Kernel.id ×ₖ condDistrib g k μ ×ₖ condDistrib f k μ) ∘ₘ Measure.map k μ =
    Measure.map (⇑e) (⇑(Kernel.id ×ₖ (condDistrib g k μ ×ₖ condDistrib f k μ)) ∘ₘ Measure.map k μ)
h2_symm :
  ⇑(Kernel.id ×ₖ (condDistrib g k μ ×ₖ condDistrib f k μ)) ∘ₘ Measure.map k μ =
    Measure.map (⇑e.symm) (⇑(Kernel.id ×ₖ condDistrib g k μ ×ₖ condDistrib f k μ) ∘ₘ Measure.map k μ)
⊢ Measure.map (fun ω ↦ (k ω, g ω, f ω)) μ =
      ⇑(Kernel.id ×ₖ (condDistrib g k μ ×ₖ condDistrib f k μ)) ∘ₘ Measure.map k μ ↔
    Measure.map (fun x ↦ ((k x, g x), f x)) μ =
      ⇑(Kernel.id ×ₖ condDistrib g k μ ×ₖ condDistrib f k μ) ∘ₘ Measure.map k μ