-- Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : A.IsTriangulated
X Y : A.FullSubcategory
f : X ⟶ Y
Z : C
a : Y.obj (P := A) ⟶ Z
b : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (X.obj (P := A))
h :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk f.hom a b)
hZ : (A ⊓ B) Z
mpr :
  Triangle.mor₁
        (Functor.obj (C := Triangle.{v_1, u_1} A.FullSubcategory) A.ι.mapTriangle
          (Triangle.mk (Z := FullSubcategory.mk (P := A) Z ⋯) f (homMk (P := A) a)
            (homMk (P := A) (b ≫ NatTrans.app (Iso.inv (Functor.commShiftIso A.ι 1)) X)))) ≫
      Iso.hom
        (Iso.refl
          (Triangle.obj₂
            (Functor.obj (C := Triangle.{v_1, u_1} A.FullSubcategory) A.ι.mapTriangle
              (Triangle.mk (Z := FullSubcategory.mk (P := A) Z ⋯) f (homMk (P := A) a)
                (homMk (P := A) (b ≫ NatTrans.app (Iso.inv (Functor.commShiftIso A.ι 1)) X)))))) =
    Iso.hom
        (Iso.refl
          (Triangle.obj₁
            (Functor.obj (C := Triangle.{v_1, u_1} A.FullSubcategory) A.ι.mapTriangle
              (Triangle.mk (Z := FullSubcategory.mk (P := A) Z ⋯) f (homMk (P := A) a)
                (homMk (P := A) (b ≫ NatTrans.app (Iso.inv (Functor.commShiftIso A.ι 1)) X)))))) ≫
      (Triangle.mk f.hom a b).mor₁
mpr₁ :
  Triangle.mor₂
        (Functor.obj (C := Triangle.{v_1, u_1} A.FullSubcategory) A.ι.mapTriangle
          (Triangle.mk (Z := FullSubcategory.mk (P := A) Z ⋯) f (homMk (P := A) a)
            (homMk (P := A) (b ≫ NatTrans.app (Iso.inv (Functor.commShiftIso A.ι 1)) X)))) ≫
      Iso.hom
        (Iso.refl
          (Triangle.obj₃
            (Functor.obj (C := Triangle.{v_1, u_1} A.FullSubcategory) A.ι.mapTriangle
              (Triangle.mk (Z := FullSubcategory.mk (P := A) Z ⋯) f (homMk (P := A) a)
                (homMk (P := A) (b ≫ NatTrans.app (Iso.inv (Functor.commShiftIso A.ι 1)) X)))))) =
    Iso.hom
        (Iso.refl
          (Triangle.obj₂
            (Functor.obj (C := Triangle.{v_1, u_1} A.FullSubcategory) A.ι.mapTriangle
              (Triangle.mk (Z := FullSubcategory.mk (P := A) Z ⋯) f (homMk (P := A) a)
                (homMk (P := A) (b ≫ NatTrans.app (Iso.inv (Functor.commShiftIso A.ι 1)) X)))))) ≫
      (Triangle.mk f.hom a b).mor₂
mpr₂ :
  Triangle.mor₃
        (Functor.obj (C := Triangle.{v_1, u_1} A.FullSubcategory) A.ι.mapTriangle
          (Triangle.mk (Z := FullSubcategory.mk (P := A) Z ⋯) f (homMk (P := A) a)
            (homMk (P := A) (b ≫ NatTrans.app (Iso.inv (Functor.commShiftIso A.ι 1)) X)))) ≫
      Functor.map (shiftFunctor C (A := ℤ) 1)
        (Iso.hom
          (Iso.refl
            (Triangle.obj₁
              (Functor.obj (C := Triangle.{v_1, u_1} A.FullSubcategory) A.ι.mapTriangle
                (Triangle.mk (Z := FullSubcategory.mk (P := A) Z ⋯) f (homMk (P := A) a)
                  (homMk (P := A) (b ≫ NatTrans.app (Iso.inv (Functor.commShiftIso A.ι 1)) X))))))) =
    Iso.hom
        (Iso.refl
          (Triangle.obj₃
            (Functor.obj (C := Triangle.{v_1, u_1} A.FullSubcategory) A.ι.mapTriangle
              (Triangle.mk (Z := FullSubcategory.mk (P := A) Z ⋯) f (homMk (P := A) a)
                (homMk (P := A) (b ≫ NatTrans.app (Iso.inv (Functor.commShiftIso A.ι 1)) X)))))) ≫
      (Triangle.mk f.hom a b).mor₃
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Functor.obj (C := Triangle.{v_1, u_1} A.FullSubcategory) A.ι.mapTriangle
      (Triangle.mk (Z := FullSubcategory.mk (P := A) Z ⋯) f (homMk (P := A) a)
        (homMk (P := A) (b ≫ NatTrans.app (Iso.inv (Functor.commShiftIso A.ι 1)) X))))

Tactic:
refine isomorphic_distinguished _ h _
      (Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) (Iso.refl _) ?_ ?_ ?_)

Premises used: CategoryTheory.Iso.refl, CategoryTheory.Pretriangulated.Triangle.isoMk, CategoryTheory.Pretriangulated.isomorphic_distinguished