reflects_exact_of_faithful — Mathlib

English

Repeats the same equivalence as 62020 in another presentation.

Русский

Повторяет эквивалентность из 62020 в другой форме.

LaTeX

$$TFAE [Epi f, cokernel.π f = 0, (ShortComplex.mk f (0 : Z ⟶ ? ) comp_zero).Exact]$$

Lean4

theorem reflects_exact_of_faithful [F.Faithful] (S : ShortComplex C) (hS : (S.map F).Exact) : S.Exact :=
  by
  rw [ShortComplex.exact_iff_kernel_ι_comp_cokernel_π_zero] at hS ⊢
  dsimp at hS
  apply F.zero_of_map_zero
  obtain ⟨k, hk⟩ :=
    kernel.lift' (F.map S.g) (F.map (kernel.ι S.g))
      (by simp only [← F.map_comp, kernel.condition, CategoryTheory.Functor.map_zero])
  obtain ⟨l, hl⟩ :=
    cokernel.desc' (F.map S.f) (F.map (cokernel.π S.f))
      (by simp only [← F.map_comp, cokernel.condition, CategoryTheory.Functor.map_zero])
  rw [F.map_comp, ← hl, ← hk, Category.assoc, reassoc_of% hS, zero_comp, comp_zero]

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