theorem extracted_formal_statement_7.{u_3, u_2} {α : Type u_2} [inst : MeasurableSpace α] (μ : Measure α)
  [inst_1 : IsFiniteMeasure μ] {ι : Sort u_3} [inst_2 : Countable ι] (f : ι → α → ℝ≥0∞)
  (hfm : ∀ (i : ι), Measurable (f i)) {b : ℝ≥0∞} (hb : b ≠ ⊤) (hf : ∀ (i : ι) (x : α), f i x ≤ b)
  (hf_le : ∀ (i : ι) (x : α), f i x < ⊤) (hfle : ∀ (i : ι), ∀ᵐ (x : α) ∂μ, f i x < ⊤)
  (hfne : ∀ᵐ (x : α) ∂μ, ∀ (i : ι), f i x ≠ ⊤) (hb' : b * μ Set.univ < ⊤) (hf_nebot : ⨆ i, ∫⁻ (a : α), f i a ∂μ ≠ ⊤)
  (hf' : ∫⁻ (a : α), ⨆ i, f i a ∂μ ≠ ⊤) (hf'' : ∀ (i : ι), ∫⁻ (a : α), f i a ∂μ ≠ ⊤)
  (hfle' : ∀ᵐ (x : α) ∂μ, ⨆ i, f i x < ⊤)
  (this : ∫ (a : α), ⨆ i, (f i a).toReal ∂μ = ∫ (ω : α), (⨆ i, f i ω).toReal ∂μ) :
  (⨆ i, ∫⁻ (a : α), f i a ∂μ).toReal ≤ ∫ (ω : α), (⨆ i, f i ω).toReal ∂μ := sorry

State before:
α : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
ι : Sort u_3
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
hfm : ∀ (i : ι), Measurable (f i)
b : ℝ≥0∞
hb : b ≠ ⊤
hf : ∀ (i : ι) (x : α), f i x ≤ b
hf_le : ∀ (i : ι) (x : α), f i x < ⊤
hfle : ∀ (i : ι), ∀ᵐ (x : α) ∂μ, f i x < ⊤
hfne : ∀ᵐ (x : α) ∂μ, ∀ (i : ι), f i x ≠ ⊤
hb' : b * μ Set.univ < ⊤
hf_nebot : ⨆ i, ∫⁻ (a : α), f i a ∂μ ≠ ⊤
hf' : ∫⁻ (a : α), ⨆ i, f i a ∂μ ≠ ⊤
hf'' : ∀ (i : ι), ∫⁻ (a : α), f i a ∂μ ≠ ⊤
hfle' : ∀ᵐ (x : α) ∂μ, ⨆ i, f i x < ⊤
this : ∫ (a : α), ⨆ i, (f i a).toReal ∂μ = ∫ (ω : α), (⨆ i, f i ω).toReal ∂μ
⊢ (⨆ i, ∫⁻ (a : α), f i a ∂μ).toReal ≤ ∫ (ω : α), (⨆ i, f i ω).toReal ∂μ

Tactic: rw [MeasureTheory.integral_toReal (Measurable.iSup hfm).aemeasurable hfle']
    rw [ENNReal.toReal_le_toReal hf_nebot hf']
    apply iSup_lintegral_le

State after:

Relevant lemmas:
Measurable.iSup : ∀ {α : Type u_1} {δ : Type u_4} [inst : TopologicalSpace α] {mα : MeasurableSpace α}
  [inst_3 : BorelSpace α] {mδ : MeasurableSpace δ} [inst_4 : ConditionallyCompleteLinearOrder α]
  [inst_5 : OrderTopology α] [inst : SecondCountableTopology α] {ι : Sort u_5} [inst : Countable ι] {f : ι → δ → α},
  (∀ (i : ι), Measurable (f i)) → Measurable fun b => ⨆ i, f i b
MeasureTheory.integral_toReal : ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {f : α → ℝ≥0∞},
  AEMeasurable f μ → (∀ᵐ (x : α) ∂μ, f x < ⊤) → ∫ (a : α), (f a).toReal ∂μ = (∫⁻ (a : α), f a ∂μ).toReal
MeasureTheory.iSup_lintegral_le : ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {ι : Sort u_5}
  (f : ι → α → ℝ≥0∞), ⨆ i, ∫⁻ (a : α), f i a ∂μ ≤ ∫⁻ (a : α), ⨆ i, f i a ∂μ
ENNReal.toReal_le_toReal : ∀ {a b : ℝ≥0∞}, a ≠ ⊤ → b ≠ ⊤ → (a.toReal ≤ b.toReal ↔ a ≤ b)