ab_exact_iff — Mathlib

English

A second restatement of lifting lift cycles to homology maps with refinements.

Русский

Второе повторение подъема lift cycles к гомологическим отображениям с refinements.

LaTeX

$$$\exists A', \pi, z, hz: \dots$$$

Lean4

theorem ab_exact_iff : S.Exact ↔ ∀ (x₂ : S.X₂) (_ : S.g x₂ = 0), ∃ (x₁ : S.X₁), S.f x₁ = x₂ :=
  by
  rw [exact_iff_surjective_abToCycles]
  constructor
  · intro h x₂ hx₂
    obtain ⟨x₁, hx₁⟩ := h ⟨x₂, hx₂⟩
    exact ⟨x₁, by simpa only [Subtype.ext_iff, abToCycles_apply_coe] using hx₁⟩
  · rintro h ⟨x₂, hx₂⟩
    obtain ⟨x₁, rfl⟩ := h x₂ hx₂
    exact ⟨x₁, rfl⟩

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