theorem extracted_formal_statement_2.{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 ≤ ⨆ j, g i + h j) :
  g i + h i ≤ g 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 ≤ ⨆ j, g i + h j

Tactic: rw [← add_ciSup hh]

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
 ⊢ g i + h i ≤ g i + ⨆ i, h i]

Relevant lemmas:
add_ciSup : ∀ {ι : Type u_1} {G : Type u_2} [inst : AddGroup G] [inst_1 : ConditionallyCompleteLattice G]
  [inst_2 : Nonempty ι] {f : ι → G} [inst_3 : AddLeftMono G],
  BddAbove (Set.range f) → ∀ (a : G), a + ⨆ i, f i = ⨆ i, a + f i