add'_assoc — Mathlib

English

If two morphisms f1, f2 in the domain map to the same morphism under a common localization, their sums coincide after localization.

Русский

Если два морфизма f1, f2 совпадают после локализации, их суммы совпадают после локализации.

LaTeX

$$$f_1, f_2: X'\to Y', \; f_1 + f_2 = add\; W\; e_X\; e_Y\; f_1\; f_2$$$

Lean4

theorem add'_assoc (f₁ f₂ f₃ : L.obj X ⟶ L.obj Y) : add' W (add' W f₁ f₂) f₃ = add' W f₁ (add' W f₂ f₃) :=
  by
  obtain ⟨α, h₁, h₂, h₃⟩ := exists_leftFraction₃ L W f₁ f₂ f₃
  rw [add'_eq W f₁ f₂ α.forgetThd h₁ h₂, add'_eq W f₂ f₃ α.forgetFst h₂ h₃,
    add'_eq W _ _ (LeftFraction₂.mk (α.f + α.f') α.f'' α.s α.hs) rfl h₃,
    add'_eq W _ _ (LeftFraction₂.mk α.f (α.f' + α.f'') α.s α.hs) h₁ rfl]
  dsimp [LeftFraction₂.add]
  rw [add_assoc]

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