Theorems · Theorem · category theory

CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Abelian C]
  {R₁ R₂ : CategoryTheory.ComposableArrows C 3} (φ : R₁ ⟶ R₂),
  R₁.Exact →
    R₂.Exact →
      CategoryTheory.Epi
          (CategoryTheory.ComposableArrows.app' φ 0 CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'._proof_13) →
        CategoryTheory.Epi
            (CategoryTheory.ComposableArrows.app' φ 2 CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'._proof_4) →
          CategoryTheory.Mono
              (CategoryTheory.ComposableArrows.app' φ 3 CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'._proof_7) →
            CategoryTheory.Epi
              (CategoryTheory.ComposableArrows.app' φ 1 CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'._proof_11)

Defined in

Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Cited by

3 results in Mathlib

Foundations

Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Abelian