-- Mathlib/MeasureTheory/Integral/Prod.lean
theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
    HasFiniteIntegral f (μ.prod ν) ↔
      (∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
        HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤
⊢ (fun x => ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν) =ᶠ[ae μ] fun a => ENNReal.ofReal (∫⁻ (a_1 : β), ‖f (a, a_1)‖ₑ ∂ν).toReal

Tactic:
filter_upwards [h2f] with x hx

Result:
case h
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤
x : α
hx : ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤
⊢ ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν = ENNReal.ofReal (∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν).toReal

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤) →
    (∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ⊤ ↔
      ∫⁻ (a : α), ENNReal.ofReal (∫⁻ (a_1 : β), ‖f (a, a_1)‖ₑ ∂ν).toReal ∂μ < ⊤)

Tactic:
intro h2f

Result:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ⊤ ↔
    ∫⁻ (a : α), ENNReal.ofReal (∫⁻ (a_1 : β), ‖f (a, a_1)‖ₑ ∂ν).toReal ∂μ < ⊤

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
x : α
⊢ ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖

Tactic:
filter_upwards with y using norm_nonneg _

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ⊤ ↔
    ∫⁻ (a : α), ENNReal.ofReal (∫⁻ (a_1 : β), ‖f (a, a_1)‖ₑ ∂ν).toReal ∂μ < ⊤

Tactic:
rw [lintegral_congr_ae]

Result:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤
⊢ (fun x => ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν) =ᶠ[ae μ] fun a => ENNReal.ofReal (∫⁻ (a_1 : β), ‖f (a, a_1)‖ₑ ∂ν).toReal

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ⊤ → ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤

Tactic:
intro h2f

Result:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
h2f : ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ⊤
⊢ ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
p q r : Prop
h1 : r → p
⊢ (r ↔ p ∧ q) ↔ p → (r ↔ q)

Tactic:
rw [← and_congr_right_iff, and_iff_right_of_imp h1]

Result:
no goals

Proof state:
case h
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤
x : α
hx : ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤
⊢ ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν = ENNReal.ofReal (∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν).toReal

Tactic:
rw [ofReal_toReal]

Result:
case h
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤
x : α
hx : ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤
⊢ ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν ≠ ⊤

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ⊤ ↔
    (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤) ∧ ∫⁻ (a : α), ‖∫ (y : β), ‖f (a, y)‖ ∂ν‖ₑ ∂μ < ⊤

Tactic:
simp_rw [integral_eq_lintegral_of_nonneg_ae (this _)
    (h1f.norm.comp_measurable measurable_prod_mk_left).aestronglyMeasurable,
  enorm_eq_ofReal toReal_nonneg, ofReal_norm_eq_enorm]

Result:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ⊤ ↔
    (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤) ∧
      ∫⁻ (a : α), ENNReal.ofReal (∫⁻ (a_1 : β), ‖f (a, a_1)‖ₑ ∂ν).toReal ∂μ < ⊤

Proof state:
case h
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤
x : α
hx : ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤
⊢ ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ⊤

Tactic:
exact hx

Result:
no goals