theorem extracted_formal_statement_3.{u_2} {ι : Type u_2} [inst : Nonempty ι] {g h : ι → ℝ} (hg : BddAbove (Set.range g))
  (hh : BddAbove (Set.range h)) (i : ι) (h_original_goal : g i + h i ≤ g i + ⨆ i, h i) : h i ≤ ⨆ i, h i := sorry

State before:
case H
ι : Type u_2
inst✝ : Nonempty ι
g h : ι → ℝ
hg : BddAbove (Set.range g)
hh : BddAbove (Set.range h)
i : ι
⊢ g i + h i ≤ g i + ⨆ i, h i

Tactic: simp only [add_le_add_iff_left]

State after:
[case H
 ι : Type u_2
 inst✝ : Nonempty ι
 g h : ι → ℝ
 hg : BddAbove (Set.range g)
 hh : BddAbove (Set.range h)
 i : ι
 ⊢ h i ≤ ⨆ i, h i]

Relevant lemmas:
add_le_add_iff_left : ∀ {α : Type u_1} [inst : Add α] [inst_1 : LE α] [inst_2 : AddLeftMono α]
  [inst_3 : AddLeftReflectLE α] (a : α) {b c : α}, a + b ≤ a + c ↔ b ≤ c