Theorem: List.aestronglyMeasurable_prod'
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✝⁵ : Countable ι
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f g : α → β
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
l : List (α → M)
hl : ∀ f ∈ l, AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable l.prod μ
state_after: case nil
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁵ : Countable ι
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f g : α → β
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
hl : ∀ f ∈ [], AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable [].prod μ

case cons
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁵ : Countable ι
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g : α → β
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEStronglyMeasurable f μ) → AEStronglyMeasurable l.prod μ
hl : ∀ f_1 ∈ f :: l, AEStronglyMeasurable f_1 μ
⊢ AEStronglyMeasurable (f :: l).prod μ
tactic: induction' l with f l ihl
state_before: case cons
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁵ : Countable ι
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g : α → β
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEStronglyMeasurable f μ) → AEStronglyMeasurable l.prod μ
hl : ∀ f_1 ∈ f :: l, AEStronglyMeasurable f_1 μ
⊢ AEStronglyMeasurable (f :: l).prod μ
state_after: case cons
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁵ : Countable ι
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g : α → β
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEStronglyMeasurable f μ) → AEStronglyMeasurable l.prod μ
hl : AEStronglyMeasurable f μ ∧ ∀ x ∈ l, AEStronglyMeasurable x μ
⊢ AEStronglyMeasurable (f :: l).prod μ
tactic: rw [List.forall_mem_cons] at hl
state_before: case cons
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁵ : Countable ι
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g : α → β
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEStronglyMeasurable f μ) → AEStronglyMeasurable l.prod μ
hl : AEStronglyMeasurable f μ ∧ ∀ x ∈ l, AEStronglyMeasurable x μ
⊢ AEStronglyMeasurable (f :: l).prod μ
state_after: case cons
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁵ : Countable ι
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g : α → β
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEStronglyMeasurable f μ) → AEStronglyMeasurable l.prod μ
hl : AEStronglyMeasurable f μ ∧ ∀ x ∈ l, AEStronglyMeasurable x μ
⊢ AEStronglyMeasurable (f * l.prod) μ
tactic: rw [List.prod_cons]
state_before: case cons
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁵ : Countable ι
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g : α → β
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, AEStronglyMeasurable f μ) → AEStronglyMeasurable l.prod μ
hl : AEStronglyMeasurable f μ ∧ ∀ x ∈ l, AEStronglyMeasurable x μ
⊢ AEStronglyMeasurable (f * l.prod) μ
state_after: no goals
tactic: exact hl.1.mul (ihl hl.2)