Theorem: ProbabilityTheory.kernel.IndepFun.comp
File: Mathlib/Probability/Independence/Kernel.lean

tactic: rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩
state_before: α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
mγ : MeasurableSpace γ
mγ' : MeasurableSpace γ'
φ : β → γ
ψ : β' → γ'
hfg : IndepFun f g κ μ
hφ : Measurable φ
hψ : Measurable ψ
⊢ IndepFun (φ ∘ f) (ψ ∘ g) κ μ
state_after: case intro.intro.intro.intro
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
mγ : MeasurableSpace γ
mγ' : MeasurableSpace γ'
φ : β → γ
ψ : β' → γ'
hfg : IndepFun f g κ μ
hφ : Measurable φ
hψ : Measurable ψ
A : Set γ
hA : MeasurableSet A
B : Set γ'
hB : MeasurableSet B
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (φ ∘ f ⁻¹' A ∩ ψ ∘ g ⁻¹' B) = (κ a) (φ ∘ f ⁻¹' A) * (κ a) (ψ ∘ g ⁻¹' B)

tactic: apply hfg
state_before: case intro.intro.intro.intro
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
mγ : MeasurableSpace γ
mγ' : MeasurableSpace γ'
φ : β → γ
ψ : β' → γ'
hfg : IndepFun f g κ μ
hφ : Measurable φ
hψ : Measurable ψ
A : Set γ
hA : MeasurableSet A
B : Set γ'
hB : MeasurableSet B
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (φ ∘ f ⁻¹' A ∩ ψ ∘ g ⁻¹' B) = (κ a) (φ ∘ f ⁻¹' A) * (κ a) (ψ ∘ g ⁻¹' B)
state_after: case intro.intro.intro.intro.a
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
mγ : MeasurableSpace γ
mγ' : MeasurableSpace γ'
φ : β → γ
ψ : β' → γ'
hfg : IndepFun f g κ μ
hφ : Measurable φ
hψ : Measurable ψ
A : Set γ
hA : MeasurableSet A
B : Set γ'
hB : MeasurableSet B
⊢ φ ∘ f ⁻¹' A ∈ {s | MeasurableSet s}

case intro.intro.intro.intro.a
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
mγ : MeasurableSpace γ
mγ' : MeasurableSpace γ'
φ : β → γ
ψ : β' → γ'
hfg : IndepFun f g κ μ
hφ : Measurable φ
hψ : Measurable ψ
A : Set γ
hA : MeasurableSet A
B : Set γ'
hB : MeasurableSet B
⊢ ψ ∘ g ⁻¹' B ∈ {s | MeasurableSet s}

tactic: exact ⟨φ ⁻¹' A, hφ hA, Set.preimage_comp.symm⟩
state_before: case intro.intro.intro.intro.a
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
mγ : MeasurableSpace γ
mγ' : MeasurableSpace γ'
φ : β → γ
ψ : β' → γ'
hfg : IndepFun f g κ μ
hφ : Measurable φ
hψ : Measurable ψ
A : Set γ
hA : MeasurableSet A
B : Set γ'
hB : MeasurableSet B
⊢ φ ∘ f ⁻¹' A ∈ {s | MeasurableSet s}
state_after: no goals

tactic: exact ⟨ψ ⁻¹' B, hψ hB, Set.preimage_comp.symm⟩
state_before: case intro.intro.intro.intro.a
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
mγ : MeasurableSpace γ
mγ' : MeasurableSpace γ'
φ : β → γ
ψ : β' → γ'
hfg : IndepFun f g κ μ
hφ : Measurable φ
hψ : Measurable ψ
A : Set γ
hA : MeasurableSet A
B : Set γ'
hB : MeasurableSet B
⊢ ψ ∘ g ⁻¹' B ∈ {s | MeasurableSet s}
state_after: no goals