English
Two 4-step arrows are equal if their app' components agree for indices 0..4.
Русский
Два 4-ступенчатых стрелы равны, если их компоненты app' совпадают по индексам 0..4.
LaTeX
$$$\\forall {f,g}:{\\text{ComposableArrows}}\\,C~4\\ (h_0,h_1,h_2,h_3,h_4): Eq (f.obj' i) (g.obj' i)$ для i=0..4 и соответствующих map' равенств, тогда $f=g$$$
Lean4
/-- Constructor for isomorphisms in `ComposableArrows C 4`. -/
@[simps]
def isoMk₄ {f g : ComposableArrows C 4} (app₀ : f.obj' 0 ≅ g.obj' 0) (app₁ : f.obj' 1 ≅ g.obj' 1)
(app₂ : f.obj' 2 ≅ g.obj' 2) (app₃ : f.obj' 3 ≅ g.obj' 3) (app₄ : f.obj' 4 ≅ g.obj' 4)
(w₀ : f.map' 0 1 ≫ app₁.hom = app₀.hom ≫ g.map' 0 1) (w₁ : f.map' 1 2 ≫ app₂.hom = app₁.hom ≫ g.map' 1 2)
(w₂ : f.map' 2 3 ≫ app₃.hom = app₂.hom ≫ g.map' 2 3) (w₃ : f.map' 3 4 ≫ app₄.hom = app₃.hom ≫ g.map' 3 4) : f ≅ g
where
hom := homMk₄ app₀.hom app₁.hom app₂.hom app₃.hom app₄.hom w₀ w₁ w₂ w₃
inv :=
homMk₄ app₀.inv app₁.inv app₂.inv app₃.inv app₄.inv (by rw [map'_inv_eq_inv_map' (by valid) app₀ app₁ w₀])
(by rw [map'_inv_eq_inv_map' (by valid) app₁ app₂ w₁]) (by rw [map'_inv_eq_inv_map' (by valid) app₂ app₃ w₂])
(by rw [map'_inv_eq_inv_map' (by valid) app₃ app₄ w₃])
