File: Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean

full_name: CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'
code: theorem mono_of_epi_of_mono_of_mono' (hR₁ : R₁.map' 0 2 = 0)
    (hR₁' : (mk₂ (R₁.map' 1 2) (R₁.map' 2 3)).Exact)
    (hR₂ : (mk₂ (R₂.map' 0 1) (R₂.map' 1 2)).Exact)
    (h₀ : Epi (app' φ 0)) (h₁ : Mono (app' φ 1)) (h₃ : Mono (app' φ 3)) :
    Mono (app' φ 2) 
goal: CategoryTheory.Mono (CategoryTheory.ComposableArrows.app' φ 2 ⋯)

full_name: CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono
code: theorem mono_of_epi_of_mono_of_mono (hR₁ : R₁.Exact) (hR₂ : R₂.Exact)
    (h₀ : Epi (app' φ 0)) (h₁ : Mono (app' φ 1)) (h₃ : Mono (app' φ 3)) :
    Mono (app' φ 2) 
goal: CategoryTheory.Mono (CategoryTheory.ComposableArrows.app' φ 2 ⋯)

full_name: CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono'
code: theorem epi_of_epi_of_epi_of_mono'
    (hR₁ : (mk₂ (R₁.map' 1 2) (R₁.map' 2 3)).Exact)
    (hR₂ : (mk₂ (R₂.map' 0 1) (R₂.map' 1 2)).Exact) (hR₂' : R₂.map' 1 3 = 0)
    (h₀ : Epi (app' φ 0)) (h₂ : Epi (app' φ 2)) (h₃ : Mono (app' φ 3)) :
    Epi (app' φ 1) 
goal: CategoryTheory.Epi (CategoryTheory.ComposableArrows.app' φ 1 ⋯)

full_name: CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono
code: theorem epi_of_epi_of_epi_of_mono (hR₁ : R₁.Exact) (hR₂ : R₂.Exact)
    (h₀ : Epi (app' φ 0)) (h₂ : Epi (app' φ 2)) (h₃ : Mono (app' φ 3)) :
    Epi (app' φ 1) 
goal: CategoryTheory.Epi (CategoryTheory.ComposableArrows.app' φ 1 ⋯)

full_name: CategoryTheory.Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono
code: /-- The five lemma. -/
theorem isIso_of_epi_of_isIso_of_isIso_of_mono (h₀ : Epi (app' φ 0)) (h₁ : IsIso (app' φ 1))
    (h₂ : IsIso (app' φ 3)) (h₃ : Mono (app' φ 4)) : IsIso (app' φ 2) 
goal: CategoryTheory.IsIso (CategoryTheory.ComposableArrows.app' φ 2 ⋯)

full_name: CategoryTheory.Abelian.mono_of_epi_of_epi_mono'
code: theorem mono_of_epi_of_epi_mono' (hR₁ : R₁.map' 0 2 = 0) (hR₁' : Epi (R₁.map' 1 2))
    (hR₂ : R₂.Exact) (h₀ : Epi (app' φ 0)) (h₁ : Mono (app' φ 1)) :
    Mono (app' φ 2) 
goal: CategoryTheory.Mono (CategoryTheory.ComposableArrows.app' φ 2 ⋯)

full_name: CategoryTheory.Abelian.mono_of_epi_of_epi_of_mono
code: theorem mono_of_epi_of_epi_of_mono (hR₁ : R₁.Exact) (hR₂ : R₂.Exact)
    (hR₁' : Epi (R₁.map' 1 2)) (h₀ : Epi (app' φ 0)) (h₁ : Mono (app' φ 1)) :
    Mono (app' φ 2) 
goal: CategoryTheory.Mono (CategoryTheory.ComposableArrows.app' φ 2 ⋯)

full_name: CategoryTheory.Abelian.epi_of_mono_of_epi_of_mono'
code: theorem epi_of_mono_of_epi_of_mono' (hR₁ : R₁.Exact) (hR₂ : R₂.map' 0 2 = 0)
    (hR₂' : Mono (R₂.map' 0 1)) (h₀ : Epi (app' φ 1)) (h₁ : Mono (app' φ 2)) :
    Epi (app' φ 0) 
goal: CategoryTheory.Epi (CategoryTheory.ComposableArrows.app' φ 0 ⋯)

full_name: CategoryTheory.Abelian.epi_of_mono_of_epi_of_mono
code: theorem epi_of_mono_of_epi_of_mono (hR₁ : R₁.Exact) (hR₂ : R₂.Exact)
    (hR₂' : Mono (R₂.map' 0 1)) (h₀ : Epi (app' φ 1)) (h₁ : Mono (app' φ 2)) :
    Epi (app' φ 0) 
goal: CategoryTheory.Epi (CategoryTheory.ComposableArrows.app' φ 0 ⋯)

full_name: CategoryTheory.Abelian.mono_of_mono_of_mono_of_mono
code: theorem mono_of_mono_of_mono_of_mono (hR₁ : R₁.Exact)
    (hR₂' : Mono (R₂.map' 0 1))
    (h₀ : Mono (app' φ 0))
    (h₁ : Mono (app' φ 2)) :
    Mono (app' φ 1) 
goal: CategoryTheory.Mono (CategoryTheory.ComposableArrows.app' φ 1 ⋯)

full_name: CategoryTheory.Abelian.epi_of_epi_of_epi_of_epi
code: theorem epi_of_epi_of_epi_of_epi (hR₂ : R₂.Exact) (hR₁' : Epi (R₁.map' 1 2))
    (h₀ : Epi (app' φ 0)) (h₁ : Epi (app' φ 2)) :
    Epi (app' φ 1) 
goal: CategoryTheory.Epi (CategoryTheory.ComposableArrows.app' φ 1 ⋯)