instPreservesZeroMorphisms — Mathlib

English

Compatibility of mapTriangle with inverse rotation through an explicit isomorphism.

Русский

Согласованность mapTriangle с обратным вращением через явное изоморфизм.

LaTeX

$$$\\mathrm{mapTriangle}\\;\\mathrm{InvRotate} \\cong \\mathrm{InvRotate}\\;\\mathrm{mapTriangle}$$$

Lean4

instance (priority := 100) [F.IsTriangulated] : PreservesZeroMorphisms F where
  map_zero X
    Y := by
    have h₁ : (0 : X ⟶ Y) = 0 ≫ 𝟙 0 ≫ 0 := by simp
    have h₂ : 𝟙 (F.obj 0) = 0 := by
      rw [← IsZero.iff_id_eq_zero]
      apply Triangle.isZero₃_of_isIso₁ _ (F.map_distinguished _ (contractible_distinguished (0 : C)))
      dsimp
      infer_instance
    rw [h₁, F.map_comp, F.map_comp, F.map_id, h₂, zero_comp, comp_zero]

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