finite_product_eq_finset_inf — Mathlib

English

A finite product in the category of SemilatticeInf with OrderTop is the infimum of the factors: ∏ f = inf_i f(i).

Русский

Конечный произведение в категории SemilatticeInf с OrderTop равно infimiуму факторов: ∏ f = inf_i f(i).

LaTeX

$$$\prod^{\mathrm{c}} f = \bigwedge_{i} f(i)$$$

Lean4

/-- A finite product in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum.
-/
theorem finite_product_eq_finset_inf [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι] (f : ι → α) :
    ∏ᶜ f = Fintype.elems.inf f := by
  trans
  · exact (IsLimit.conePointUniqueUpToIso (limit.isLimit _) (finiteLimitCone (Discrete.functor f)).isLimit).to_eq
  change Finset.univ.inf (f ∘ discreteEquiv.toEmbedding) = Fintype.elems.inf f
  simp only [← Finset.inf_map, Finset.univ_map_equiv_to_embedding]
  rfl

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