theorem extracted_formal_statement_56.{v} {ι : Type v} [inst : Nonempty ι] {f : ι → ℝ} {a : ℝ} (hf : ∀ (i : ι), |f i| ≤ a)
  (hbdd : BddAbove (Set.range f)) (h_original_goal : |⨆ i, f i| ≤ a) : -a ≤ ⨆ i, f i ∧ ⨆ i, f i ≤ a := sorry

State before:
ι : Type v
inst✝ : Nonempty ι
f : ι → ℝ
a : ℝ
hf : ∀ (i : ι), |f i| ≤ a
hbdd : BddAbove (Set.range f)
⊢ |⨆ i, f i| ≤ a

Tactic: apply abs_le.mpr

State after:
[ι : Type v
 inst✝ : Nonempty ι
 f : ι → ℝ
 a : ℝ
 hf : ∀ (i : ι), |f i| ≤ a
 hbdd : BddAbove (Set.range f)
 ⊢ -a ≤ ⨆ i, f i ∧ ⨆ i, f i ≤ a]

Relevant lemmas: