theorem extracted_formal_statement_227.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))) (s : Fin 2) (i : ι)
  (h : ¬(↑s == 0) = true)
  (h_original_goal : V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ |V (f i)|) :
  -V (f (s, i).2) ≤ |V (f i)| := sorry

State before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
s : Fin 2
i : ι
h : ¬(↑s == 0) = true
⊢ V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ |V (f i)|

Tactic: rw [if_neg h, hV₀]

State after:
[Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 inst✝ : Nonempty ι
 V : (Z → ℝ) → ℝ
 hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
 hV₁ : BddAbove (Set.range fun i => |V (f i)|)
 hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
 s : Fin 2
 i : ι
 h : ¬(↑s == 0) = true
 ⊢ -V (f (s, i).2) ≤ |V (f i)|]

Relevant lemmas:
if_neg : ∀ {c : Prop} {h : Decidable c}, ¬c → ∀ {α : Sort u} {t e : α}, (if c then t else e) = e