English
Extensional equality for biproduct components: ext_from_iff states f = g if and only if their components coincide on inl and inr.
Русский
Экстремальная эквивалентность для компонентов биопродукта: f = g, если и только если их составляющие совпадают на inl и inr.
LaTeX
$$$f = g \iff (\mathrm{biprod.inl} \circ f = \mathrm{biprod.inl} \circ g) \land (\mathrm{biprod.inr} \circ f = \mathrm{biprod.inr} \circ g)$$$
Lean4
@[simp]
theorem ofComponents_eq (f : X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂) :
Biprod.ofComponents (biprod.inl ≫ f ≫ biprod.fst) (biprod.inl ≫ f ≫ biprod.snd) (biprod.inr ≫ f ≫ biprod.fst)
(biprod.inr ≫ f ≫ biprod.snd) =
f :=
by
ext <;>
simp only [Category.comp_id, biprod.inr_fst, biprod.inr_snd, biprod.inl_snd, add_zero, zero_add,
Biprod.inl_ofComponents, Biprod.inr_ofComponents, Category.assoc, comp_zero, biprod.inl_fst, Preadditive.add_comp]
