hasBiproduct_of_total — Mathlib

English

Given a finite index J and a bicone b for f: J → C with total condition, f has a biproduct.

Русский

При заданном конечном индексе J и биоконце b для f: J → C с условием total, f обладает билипраодктом.

LaTeX

$$HasBiproduct f$$

Lean4

/-- In a preadditive category, we can construct a biproduct for `f : J → C` from
any bicone `b` for `f` satisfying `total : ∑ j : J, b.π j ≫ b.ι j = 𝟙 b.X`.

(That is, such a bicone is a limit cone and a colimit cocone.)
-/
theorem hasBiproduct_of_total {f : J → C} (b : Bicone f) (total : ∑ j : J, b.π j ≫ b.ι j = 𝟙 b.pt) : HasBiproduct f :=
  HasBiproduct.mk
    { bicone := b
      isBilimit := isBilimitOfTotal b total }

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