theorem extracted_formal_statement_224.{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_original_goal : V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ ⨆ i, |V (f i)|) :
  V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ |V (f i)| := sorry

State before:
case a.H.mk
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
⊢ V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ ⨆ i, |V (f i)|

Tactic: apply le_trans _ (le_ciSup hV₁ i)

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
 ⊢ V (if (↑(s, i).1 == 0) = true then f (s, i).2 else -f (s, i).2) ≤ |V (f i)|]

Relevant lemmas:
le_trans : ∀ {α : Type u_1} [inst : Preorder α] {a b c : α}, a ≤ b → b ≤ c → a ≤ c
le_ciSup : ∀ {α : Type u_1} {ι : Sort u_4} [inst : ConditionallyCompleteLattice α] {f : ι → α},
  BddAbove (Set.range f) → ∀ (c : ι), f c ≤ iSup f