finite_coproduct_eq_finset_sup — Mathlib

English

A finite coproduct in SemilatticeSup with OrderBot is the supremum: ∐ f = sup_i f(i).

Русский

Конечный копроизвод в SemilatticeSup с OrderBot есть супремум: ∐ f = sup_i f(i).

LaTeX

$$$\coprod f = \bigvee_{i} f(i)$$$

Lean4

/-- A finite coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι] (f : ι → α) :
    ∐ f = Fintype.elems.sup f := by
  trans
  ·
    exact
      (IsColimit.coconePointUniqueUpToIso (colimit.isColimit _)
          (finiteColimitCocone (Discrete.functor f)).isColimit).to_eq
  change Finset.univ.sup (f ∘ discreteEquiv.toEmbedding) = Fintype.elems.sup f
  simp only [← Finset.sup_map, Finset.univ_map_equiv_to_embedding]
  rfl
    -- see Note [lower instance priority]

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