mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
⊢ (∃ Y' s, ∃ (_ : P.monoModSerre s), f ≫ s = 0) ↔ P (Abelian.image f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Projective.Ext

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y : C
R : ProjectiveResolution X
n : ℕ
f : R.complex.X n ⟶ Y
m : ℕ
hm : n + 1 = m
hf : R.complex.d m n ≫ f = 0
X' : C
R' : ProjectiveResolution X'
g : X' ⟶ X
φ : R'.Hom R g
⊢ (Ext.mk₀ g).comp (R.extMk f m hm hf) ⋯ = R'.extMk (φ.hom.f n ≫ f) m hm ⋯

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (Abelian.image f)
⊢ P.isoModSerre (cokernel.π f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
⊢ (∃ Y' s, ∃ (_ : P.isoModSerre s), f ≫ s = 0) ↔ P (Abelian.image f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P.monoModSerre f
⊢ Abelian.factorThruImage f ≫ Abelian.image.ι f = 𝟙 X ≫ f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P.monoModSerre f
⊢ P.isoModSerre (Abelian.factorThruImage f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Subcategory

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
P : ObjectProperty C
inst✝⁴ : Abelian C
inst✝³ : P.ContainsZero
inst✝² : P.IsClosedUnderKernels
inst✝¹ : P.IsClosedUnderCokernels
X Y : P.FullSubcategory
f : X ⟶ Y
inst✝ : Mono f
this : P.ι.PreservesMonomorphisms
⊢ f ≫ homMk (cokernel.π f.hom) = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P (cokernel f)
⊢ f ≫ 𝟙 Y = Abelian.factorThruImage f ≫ kernel.ι (cokernel.π f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
hf : P.epiModSerre f
⊢ P.isoModSerre (Abelian.image.ι f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Subcategory

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
P : ObjectProperty C
inst✝⁴ : Abelian C
inst✝³ : P.ContainsZero
inst✝² : P.IsClosedUnderKernels
inst✝¹ : P.IsClosedUnderCokernels
X Y : P.FullSubcategory
f : X ⟶ Y
inst✝ : Epi f
this : P.ι.PreservesEpimorphisms
⊢ homMk (kernel.ι f.hom) ≫ f = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
D : Type u'
inst✝³ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝² : P.IsSerreClass
E : Type u''
inst✝¹ : Category.{v'', u''} E
inst✝ : Abelian E
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
hs : P.isoModSerre s
eq : s ≫ f₁ = s ≫ f₂
⊢ s ≫ (f₁ - f₂) = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Kernels

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
W : MorphismProperty C
Y✝ X₁✝ X₂✝ : C
f : X₁✝ ⟶ X₂✝
k : KernelFork f
hk : IsLimit k
hf : W f
i : k.pt ≅ Y✝
⊢ (i.inv ≫ Fork.ι k) ≫ f = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Kernels

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
W : MorphismProperty C
⊢ ∀ {X Y : C} (x : X ≅ Y), W.kernels X → W.kernels Y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Kernels

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
W : MorphismProperty C
Y✝ X₁✝ X₂✝ : C
f : X₁✝ ⟶ X₂✝
k : CokernelCofork f
hk : IsColimit k
hf : W f
i : k.pt ≅ Y✝
⊢ f ≫ Cofork.π k ≫ i.hom = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Kernels

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
W : MorphismProperty C
⊢ ∀ {X Y : C} (x : X ≅ Y), W.cokernels X → W.cokernels Y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
D : Type u'
inst✝³ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝² : P.IsSerreClass
E : Type u''
inst✝¹ : Category.{v'', u''} E
inst✝ : Abelian E
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
hs : P.isoModSerre s
eq : s ≫ f₁ = s ≫ f₂
⊢ ∃ Y' t, ∃ (_ : P.isoModSerre t), f₁ ≫ t = f₂ ≫ t

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Kernels

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
P : ObjectProperty C
inst✝ : P.IsClosedUnderSubobjects
⊢ (MorphismProperty.ofObjectProperty P P).kernels ≤ P

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Kernels

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderKernels
X Y : P.FullSubcategory
f : X ⟶ Y
inst✝ : HasKernel f.hom
⊢ kernel.ι f.hom ≫ (parallelPair f 0 ⋙ P.ι).map WalkingParallelPairHom.left =
    kernel.ι f.hom ≫ (parallelPair f 0 ⋙ P.ι).map WalkingParallelPairHom.right

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
D : Type u'
inst✝³ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝² : P.IsSerreClass
E : Type u''
inst✝¹ : Category.{v'', u''} E
inst✝ : Abelian E
X Y Y' : C
f₁ f₂ : X ⟶ Y
s : Y ⟶ Y'
hs : P.isoModSerre s
eq : f₁ ≫ s = f₂ ≫ s
⊢ (f₁ - f₂) ≫ s = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
D : Type u'
inst✝³ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝² : P.IsSerreClass
E : Type u''
inst✝¹ : Category.{v'', u''} E
inst✝ : Abelian E
X Y Y' : C
f₁ f₂ : X ⟶ Y
s : Y ⟶ Y'
hs : P.isoModSerre s
eq : f₁ ≫ s = f₂ ≫ s
⊢ ∃ X' t, ∃ (_ : P.isoModSerre t), t ≫ f₁ = t ≫ f₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X : C
x✝ : P X
⊢ P.isoModSerre 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Kernels

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderKernels
X Y : P.FullSubcategory
f : X ⟶ Y
inst✝ : HasKernel f.hom
⊢ CreatesLimit (parallelPair f 0) P.ι

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X : C
⊢ IsZero (L.obj X) ↔ P X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Kernels

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
P : ObjectProperty C
inst✝¹ : HasKernels C
inst✝ : P.IsClosedUnderKernels
X Y : P.FullSubcategory
f : X ⟶ Y
⊢ PreservesLimit (parallelPair f 0) P.ι

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X Y : C
f : X ⟶ Y
⊢ L.map f = 0 ↔ P (Abelian.image f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Kernels

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
P : ObjectProperty C
inst✝ : P.IsClosedUnderQuotients
⊢ (MorphismProperty.ofObjectProperty P P).cokernels ≤ P

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : Abelian C
D : Type u'
inst✝⁶ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝⁵ : P.IsSerreClass
inst✝⁴ : L.IsLocalization P.isoModSerre
inst✝³ : Preadditive D
inst✝² : L.Additive
X Z Y : C
f : X ⟶ Z
g : Z ⟶ Y
inst✝¹ : Epi f
inst✝ : Mono g
⊢ L.map f ≫ L.map g = 0 ↔ P Z

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Kernels

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderCokernels
X Y : P.FullSubcategory
f : X ⟶ Y
inst✝ : HasCokernel f.hom
⊢ (parallelPair f 0 ⋙ P.ι).map WalkingParallelPairHom.left ≫ cokernel.π f.hom =
    (parallelPair f 0 ⋙ P.ι).map WalkingParallelPairHom.right ≫ cokernel.π f.hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X Y : C
f : X ⟶ Y
this : L.EssSurj
x✝ : Mono (L.map f)
⊢ L.map (kernel.ι f) = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X Y : C
f : X ⟶ Y
this : L.EssSurj
x✝ : Mono (L.map f)
⊢ P.monoModSerre f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X Y : C
f : X ⟶ Y
this : L.EssSurj
tfae_1_to_2 : Mono (L.map f) → P.monoModSerre f
hf : P.monoModSerre f
Z : C
z : Z ⟶ X
W : C
s : Y ⟶ W
hs : P.monoModSerre s
eq : (z ≫ f) ≫ s = 0
⊢ z ≫ f ≫ s = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X Y : C
f : X ⟶ Y
this : L.EssSurj
tfae_1_to_2 : Mono (L.map f) → P.monoModSerre f
hf : P.monoModSerre f
⊢ ∀ ⦃Z : C⦄ (z : Z ⟶ X), L.map z ≫ L.map f = 0 → L.map z = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Kernels

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderCokernels
X Y : P.FullSubcategory
f : X ⟶ Y
inst✝ : HasCokernel f.hom
⊢ CreatesColimit (parallelPair f 0) P.ι

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Kernels

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
P : ObjectProperty C
inst✝¹ : HasCokernels C
inst✝ : P.IsClosedUnderCokernels
X Y : P.FullSubcategory
f : X ⟶ Y
⊢ PreservesColimit (parallelPair f 0) P.ι

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.RightDerived

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
⊢ F.toRightDerivedZero.app X =
    I.toRightDerivedZero' F ≫
      (CochainComplex.isoHomologyπ₀ ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex)).hom ≫
        (I.isoRightDerivedObj F 0).inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.RightDerived

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{v_1, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F : C ⥤ D
inst✝¹ : F.Additive
X : C
inst✝ : Injective X
⊢ IsIso (F.toRightDerivedZero.app X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.RightDerived

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{v_1, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F : C ⥤ D
inst✝¹ : F.Additive
inst✝ : PreservesFiniteLimits F
X : C
P : InjectiveResolution X
⊢ IsIso (P.toRightDerivedZero' F)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.RightDerived

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{v_1, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F : C ⥤ D
inst✝¹ : F.Additive
inst✝ : PreservesFiniteLimits F
X : C
⊢ IsIso (F.toRightDerivedZero.app X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X Y : C
f : X ⟶ Y
this : L.EssSurj
tfae_1_to_2 : Mono (L.map f) → P.monoModSerre f
tfae_2_to_3 : P.monoModSerre f → ∀ ⦃Z : C⦄ (z : Z ⟶ X), L.map z ≫ L.map f = 0 → L.map z = 0
hf : ∀ ⦃Z : C⦄ (z : Z ⟶ X), L.map z ≫ L.map f = 0 → L.map z = 0
W : D
z : W ⟶ L.obj X
hz : z ≫ L.map f = 0
φ : P.isoModSerre.RightFraction (L.objPreimage W) X
hφ : (L.objObjPreimageIso W).hom ≫ z = φ.map L ⋯
⊢ L.map φ.s ≫ (L.objObjPreimageIso W).hom ≫ z = L.map φ.f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X Y : C
f : X ⟶ Y
this : L.EssSurj
tfae_1_to_2 : Mono (L.map f) → P.monoModSerre f
tfae_2_to_3 : P.monoModSerre f → ∀ ⦃Z : C⦄ (z : Z ⟶ X), L.map z ≫ L.map f = 0 → L.map z = 0
hf : ∀ ⦃Z : C⦄ (z : Z ⟶ X), L.map z ≫ L.map f = 0 → L.map z = 0
⊢ Mono (L.map f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X Y : C
f : X ⟶ Y
⊢ [Mono (L.map f), P.monoModSerre f, ∀ ⦃Z : C⦄ (z : Z ⟶ X), L.map z ≫ L.map f = 0 → L.map z = 0].TFAE

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X Y : C
f : X ⟶ Y
this : L.EssSurj
x✝ : Epi (L.map f)
⊢ L.map (cokernel.π f) = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X Y : C
f : X ⟶ Y
this : L.EssSurj
x✝ : Epi (L.map f)
⊢ P.epiModSerre f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action

Goal state before:
pp: M : Type u_1
inst✝¹ : Monoid M
X : Type u
inst✝ : MulAction M X
x✝ : SingleObj M
⊢ (↾fun x ↦ 𝟙 x✝ • x) = 𝟙 X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action

Goal state before:
pp: M : Type u_1
inst✝¹ : Monoid M
X : Type u
inst✝ : MulAction M X
X✝ Y✝ Z✝ : SingleObj M
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ (↾fun x ↦ f ≫ g • x) = (↾fun x ↦ f • x) ≫ ↾fun x ↦ g • x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action

Goal state before:
pp: M : Type u_1
inst✝¹ : Monoid M
X : Type u
inst✝ : MulAction M X
x : ActionCategory M X
⊢ ⟨(), x.back⟩ = x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action

Goal state before:
pp: M : Type u_1
inst✝³ : Monoid M
X : Type u
inst✝² : MulAction M X
x : X
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
P : ⦃a b : ActionCategory G X⦄ → (a ⟶ b) → Sort u_3
hyp : (t : X) → (g : G) → P (homOfPair t g)
a b : ActionCategory G X
f : a ⟶ b
⊢ P (homOfPair b.back ↑f) = P f

Tactic:
rcases a with ⟨⟨⟩, a : X⟩

Goal state after:
M : Type u_1
inst✝³ : Monoid M
X : Type u
inst✝² : MulAction M X
x : X
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
P : ⦃a b : ActionCategory G X⦄ → (a ⟶ b) → Sort u_3
hyp : (t : X) → (g : G) → P (homOfPair t g)
b : ActionCategory G X
a : X
f : ⟨PUnit.unit, a⟩ ⟶ b
⊢ P (homOfPair b.back ↑f) = P f
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X Y : C
f : X ⟶ Y
this : L.EssSurj
tfae_1_to_2 : Epi (L.map f) → P.epiModSerre f
hf : P.epiModSerre f
Z : C
z : Y ⟶ Z
W : C
s : W ⟶ X
hs : P.epiModSerre s
eq : s ≫ f ≫ z = 0
⊢ (s ≫ f) ≫ z = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X Y : C
f : X ⟶ Y
this : L.EssSurj
tfae_1_to_2 : Epi (L.map f) → P.epiModSerre f
hf : P.epiModSerre f
⊢ ∀ ⦃Z : C⦄ (z : Y ⟶ Z), L.map f ≫ L.map z = 0 → L.map z = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action

Goal state before:
pp: M : Type u_1
inst✝³ : Monoid M
X : Type u
inst✝² : MulAction M X
x : X
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
P : ⦃a b : ActionCategory G X⦄ → (a ⟶ b) → Sort u_3
hyp : (t : X) → (g : G) → P (homOfPair t g)
a b : ActionCategory G X
f : a ⟶ b
⊢ P f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action

Goal state before:
pp: M : Type u_1
inst✝⁴ : Monoid M
X : Type u
inst✝³ : MulAction M X
x : X
G : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G X
H : Type u_3
inst✝ : Group H
F : ActionCategory G X ⥤ SingleObj H
⊢ ∀ {a b : ActionCategory G X} {f : a ⟶ b}, F.map f = F.map (homOfPair b.back ↑f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action

Goal state before:
pp: M : Type u_1
inst✝⁴ : Monoid M
X : Type u
inst✝³ : MulAction M X
x : X
G : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G X
H : Type u_3
inst✝ : Group H
F : ActionCategory G X ⥤ SingleObj H
F_map_eq : ∀ {a b : ActionCategory G X} {f : a ⟶ b}, F.map f = F.map (homOfPair b.back ↑f)
⊢ ⟨fun b ↦ F.map (homOfPair b 1), 1⟩ = 1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action

Goal state before:
pp: M : Type u_1
inst✝⁴ : Monoid M
X : Type u
inst✝³ : MulAction M X
x : X
G : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G X
H : Type u_3
inst✝ : Group H
F : ActionCategory G X ⥤ SingleObj H
F_map_eq : ∀ {a b : ActionCategory G X} {f : a ⟶ b}, F.map f = F.map (homOfPair b.back ↑f)
⊢ ∀ (x y : G),
    ⟨fun b ↦ F.map (homOfPair b (x * y)), x * y⟩ =
      ⟨fun b ↦ F.map (homOfPair b x), x⟩ * ⟨fun b ↦ F.map (homOfPair b y), y⟩

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action

Goal state before:
pp: M : Type u_1
inst✝⁴ : Monoid M
X : Type u
inst✝³ : MulAction M X
x✝ : X
G : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G X
H : Type u_3
inst✝ : Group H
F : G →* (X → H) ⋊[mulAutArrow] G
sane : ∀ (g : G), (F g).right = g
x : ActionCategory G X
⊢ (F ↑(𝟙 x)).left x.back = 𝟙 ()

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action

Goal state before:
pp: M : Type u_1
inst✝⁴ : Monoid M
X : Type u
inst✝³ : MulAction M X
x : X
G : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G X
H : Type u_3
inst✝ : Group H
F : G →* (X → H) ⋊[mulAutArrow] G
sane : ∀ (g : G), (F g).right = g
X✝ Y✝ Z✝ : ActionCategory G X
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ (F ↑(f ≫ g)).left Z✝.back = (F ↑f).left Y✝.back ≫ (F ↑g).left Z✝.back

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action.Concrete

Goal state before:
pp: X : Type u
⊢ Function.LeftInverse (fun f ↦ ⇑(ConcreteCategory.hom f)) fun f ↦ ↾f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action.Concrete

Goal state before:
pp: X : Type u
⊢ Function.RightInverse (fun f ↦ ⇑(ConcreteCategory.hom f)) fun f ↦ ↾f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action.Concrete

Goal state before:
pp: X : Type u
⊢ ∀ (x y : Function.End X), ↾(x * y) = ↾x * ↾y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action.Concrete

Goal state before:
pp: G : Type u_1
H : Type u
inst✝¹ : Monoid G
inst✝ : MulAction G H
⊢ MulAction G H

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Action.Concrete

Goal state before:
pp: ι : Type v
G : Type (max v u)
inst✝¹ : Monoid G
F : ι → Type (max v u)
inst✝ : (i : ι) → MulAction G (F i)
s : Cone (Discrete.functor fun i ↦ ofMulAction G (F i))
g : G
⊢ (s.pt.ρ g ≫ ↾fun x i ↦ (ConcreteCategory.hom (s.π.app { as := i }).hom) x) =
    (↾fun x i ↦ (ConcreteCategory.hom (s.π.app { as := i }).hom) x) ≫
      { pt := ofMulAction G ((i : ι) → F i), π := Discrete.natTrans fun i ↦ { hom := ↾fun x ↦ x i.as, comm := ⋯ } }.pt.ρ
        g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
D : Type u'
inst✝⁴ : Category.{v', u'} D
L : C ⥤ D
P : ObjectProperty C
inst✝³ : P.IsSerreClass
inst✝² : L.IsLocalization P.isoModSerre
inst✝¹ : Preadditive D
inst✝ : L.Additive
X Y : C
f : X ⟶ Y
this : L.EssSurj
tfae_1_to_2 : Epi (L.map f) → P.epiModSerre f
tfae_2_to_3 : P.epiModSerre f → ∀ ⦃Z : C⦄ (z : Y ⟶ Z), L.map f ≫ L.map z = 0 → L.map z = 0
hf : ∀ ⦃Z : C⦄ (z : Y ⟶ Z), L.map f ≫ L.map z = 0 → L.map z = 0
W : D
z : L.obj Y ⟶ W
hz : L.map f ≫ z = 0
φ : P.isoModSerre.LeftFraction Y (L.objPreimage W)
hφ : z ≫ (L.objObjPreimageIso W).inv = φ.map L ⋯
⊢ L.map φ.f = z ≫ (L.objObjPreimageIso W).inv ≫ L.map φ.s

Tactic:
by

Goal state after: