homologyι_singleObjOpcyclesSelfIso_inv

English

The inverse of the homology self-iso composed with the homologyι map equals the reverse of the corresponding homology self-iso.

Русский

Обратное от гомологии_self_изоморфизма, композиция с homologyι равна обратному изоморфизму гомологии.

LaTeX

$$$(\\mathrm{singleObjHomologySelfIso} \\, c \\, j \\, A).inv \\circ (((\\mathrm{single} \\, C \\, c \\, j).\\mathrm{obj} A).homologyι \\, j) = (\\mathrm{singleObjOpcyclesSelfIso} \\; c \\; j \\; A).hom$$$

Lean4

@[reassoc (attr := simp)]
theorem homologyι_singleObjOpcyclesSelfIso_inv :
    ((single C c j).obj A).homologyι j ≫ (singleObjOpcyclesSelfIso _ _ _).inv = (singleObjHomologySelfIso _ _ _).hom :=
  by
  rw [← cancel_epi (singleObjHomologySelfIso _ _ _).inv, singleObjHomologySelfIso_inv_homologyι_assoc, Iso.hom_inv_id,
    Iso.inv_hom_id]

Похожие утверждения