Theorems · Theorem · category theory

CategoryTheory.Abelian.isIso_of_isIso_of_mono

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Abelian C]
  {R₁ R₂ : CategoryTheory.ComposableArrows C 2} (φ : R₁ ⟶ R₂),
  R₁.Exact →
    R₂.Exact →
      CategoryTheory.Mono
          (R₁.map' 0 1 CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'._proof_9
            CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'._proof_6) →
        CategoryTheory.Mono
            (R₂.map' 0 1 CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'._proof_9
              CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'._proof_6) →
          CategoryTheory.IsIso
              (CategoryTheory.ComposableArrows.app' φ 1 CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'._proof_6) →
            CategoryTheory.Mono
                (CategoryTheory.ComposableArrows.app' φ 2 CategoryTheory.Abelian.mono_of_epi_of_epi_mono'._proof_1) →
              CategoryTheory.IsIso
                (CategoryTheory.ComposableArrows.app' φ 0 CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'._proof_2)

Defined in

Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Cited by

0 results in Mathlib

Foundations

Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Abelian