Theorem: MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded
File: Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean

Proof steps:
state_before: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a b : ι → ℝ
f : ℝ → E
I a₀ b₀ : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) volume
ha : Tendsto a l (𝓝 a₀)
hb : Tendsto b l (𝓝 b₀)
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
⊢ IntegrableOn f (Ioc a₀ b₀) volume
state_after: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a b : ι → ℝ
f : ℝ → E
I a₀ b₀ : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) volume
ha : Tendsto a l (𝓝 a₀)
hb : Tendsto b l (𝓝 b₀)
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
i : ι
hi : ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
⊢ ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ∂volume.restrict (Ioc a₀ b₀) ≤ I
tactic: refine (aecover_Ioc_of_Ioc ha hb).integrable_of_integral_norm_bounded I
    (fun i => (hfi i).restrict measurableSet_Ioc) (h.mono fun i hi ↦ ?_)
state_before: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a b : ι → ℝ
f : ℝ → E
I a₀ b₀ : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) volume
ha : Tendsto a l (𝓝 a₀)
hb : Tendsto b l (𝓝 b₀)
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
i : ι
hi : ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
⊢ ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ∂volume.restrict (Ioc a₀ b₀) ≤ I
state_after: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a b : ι → ℝ
f : ℝ → E
I a₀ b₀ : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) volume
ha : Tendsto a l (𝓝 a₀)
hb : Tendsto b l (𝓝 b₀)
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
i : ι
hi : ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
⊢ ∫ (x : ℝ) in Ioc (a i) (b i) ∩ Ioc a₀ b₀, ‖f x‖ ∂volume ≤ I
tactic: rw [Measure.restrict_restrict measurableSet_Ioc]
state_before: ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a b : ι → ℝ
f : ℝ → E
I a₀ b₀ : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) volume
ha : Tendsto a l (𝓝 a₀)
hb : Tendsto b l (𝓝 b₀)
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
i : ι
hi : ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
⊢ ∫ (x : ℝ) in Ioc (a i) (b i) ∩ Ioc a₀ b₀, ‖f x‖ ∂volume ≤ I
state_after: case refine_1.a
ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a b : ι → ℝ
f : ℝ → E
I a₀ b₀ : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) volume
ha : Tendsto a l (𝓝 a₀)
hb : Tendsto b l (𝓝 b₀)
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
i : ι
hi : ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
⊢ ∀ (a : ℝ), 0 a ≤ (fun x => ‖f x‖) a

case refine_2.a
ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.NeBot
inst✝¹ : l.IsCountablyGenerated
inst✝ : NormedAddCommGroup E
a b : ι → ℝ
f : ℝ → E
I a₀ b₀ : ℝ
hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) volume
ha : Tendsto a l (𝓝 a₀)
hb : Tendsto b l (𝓝 b₀)
h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
i : ι
hi : ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I
⊢ ∀ (a_1 : ℝ), (Ioc (a i) (b i) ∩ Ioc a₀ b₀) a_1 ≤ Ioc (a i) (b i) a_1
tactic: refine le_trans (setIntegral_mono_set (hfi i).norm ?_ ?_) hi <;> apply ae_of_all