theorem extracted_formal_statement_274.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω)) (σ : Signs n) (a : σ ∈ Finset.univ) (i : ι)
  (h_original_goal : |(|∑ k, ↑↑(σ k) * f i (X (ω.1 k))|)| ≤ ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|) :
  |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| := sorry

State before:
case h.hf
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
σ : Signs n
a✝ : σ ∈ Finset.univ
i : ι
⊢ |(|∑ k, ↑↑(σ k) * f i (X (ω.1 k))|)| ≤ ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|

Tactic: simp only [Int.reduceNeg, abs_abs]

State after:
[case h.hf
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 σ : Signs n
 a✝ : σ ∈ Finset.univ
 i : ι
 ⊢ |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))|]

Relevant lemmas:
abs_abs : ∀ {α : Type u_1} [inst : Lattice α] [inst_4 : AddGroup α] [inst_5 : AddLeftMono α] [inst_6 : AddRightMono α]
  (a : α), |(|a|)| = |a|