mathlib-initiative/mathlib-tactics huggingface dataset:
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))
f :
  s.pt ⟶
    { pt := ofMulAction G ((i : ι) → F i), π := Discrete.natTrans fun i ↦ { hom := ↾fun x ↦ x i.as, comm := ⋯ } }.pt
h :
  ∀ (j : Discrete ι),
    f ≫
        { pt := ofMulAction G ((i : ι) → F i),
                π := Discrete.natTrans fun i ↦ { hom := ↾fun x ↦ x i.as, comm := ⋯ } }.π.app
          j =
      s.π.app j
⊢ f = { hom := ↾fun x i ↦ (ConcreteCategory.hom (s.π.app { as := i }).hom) x, comm := ⋯ }

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
H N : Subgroup G
inst✝¹ : Fintype (G ⧸ N)
inst✝ : N.Normal
v a b : G
h : a ≈ b
⊢ a * v⁻¹ ≈ b * v⁻¹

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
H N : Subgroup G
inst✝¹ : Fintype (G ⧸ N)
inst✝ : N.Normal
v g : G
⊢ (ofMulAction G (FintypeCat.of (G ⧸ N))).ρ g ≫ FintypeCat.homMk (Quotient.lift (fun σ ↦ ⟦σ * v⁻¹⟧) ⋯) =
    FintypeCat.homMk (Quotient.lift (fun σ ↦ ⟦σ * v⁻¹⟧) ⋯) ≫ (ofMulAction G (FintypeCat.of (G ⧸ N))).ρ g

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
H N : Subgroup G
inst✝¹ : Fintype (G ⧸ N)
inst✝ : N.Normal
⊢ { hom := FintypeCat.homMk (Quotient.lift (fun σ ↦ ⟦σ * 1⁻¹⟧) ⋯), comm := ⋯ } = 1

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
H N : Subgroup G
inst✝¹ : Fintype (G ⧸ N)
inst✝ : N.Normal
σ τ : G
⊢ { hom := FintypeCat.homMk (Quotient.lift (fun σ_1 ↦ ⟦σ_1 * (σ * τ)⁻¹⟧) ⋯), comm := ⋯ } =
    { hom := FintypeCat.homMk (Quotient.lift (fun σ_1 ↦ ⟦σ_1 * σ⁻¹⟧) ⋯), comm := ⋯ } *
      { hom := FintypeCat.homMk (Quotient.lift (fun σ ↦ ⟦σ * τ⁻¹⟧) ⋯), comm := ⋯ }

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
⊢ Epi (L.map f)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
N : Subgroup G
inst✝¹ : Fintype (G ⧸ N)
inst✝ : N.Normal
n : G
hn : n ∈ N
a✝ : G
⊢ (a✝ * n⁻¹)⁻¹ * a✝ ∈ N

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
N : Subgroup G
inst✝¹ : Fintype (G ⧸ N)
inst✝ : N.Normal
n : G
hn : n ∈ N
⊢ (toEndHom N) n = 𝟙 (ofMulAction G (FintypeCat.of (G ⧸ N)))

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
H N : Subgroup G
inst✝¹ : Fintype (G ⧸ N)
inst✝ : Fintype (G ⧸ H)
h : N ≤ H
g : G
⊢ (ofMulAction G (FintypeCat.of (G ⧸ N))).ρ g ≫ FintypeCat.homMk (Quotient.lift (fun x ↦ ⟦x⟧) ⋯) =
    FintypeCat.homMk (Quotient.lift (fun x ↦ ⟦x⟧) ⋯) ≫ (ofMulAction G (FintypeCat.of (G ⧸ H))).ρ g

Tactic:
by

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

Goal state before:
pp: V : Type (u + 1)
inst✝³ : LargeCategory V
FV : V → V → Type u_1
CV : V → Type u_2
inst✝² : (X Y : V) → FunLike (FV X Y) (CV X) (CV Y)
inst✝¹ : ConcreteCategory V FV
G : Type u_3
inst✝ : Monoid G
X : Action V G
g h : G
x : ToType X
⊢ (g * h) • x = g • h • x

Tactic:
by

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

Goal state before:
pp: V : Type (u + 1)
inst✝³ : LargeCategory V
FV : V → V → Type u_1
CV : V → Type u_2
inst✝² : (X Y : V) → FunLike (FV X Y) (CV X) (CV Y)
inst✝¹ : ConcreteCategory V FV
G : Type u_3
inst✝ : Monoid G
X : Action V G
x : ToType X
⊢ 1 • x = 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
⊢ [Epi (L.map f), P.epiModSerre f, ∀ ⦃Z : C⦄ (z : Y ⟶ Z), L.map f ≫ L.map z = 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
⊢ (MorphismProperty.monomorphisms D).inverseImage L = P.monoModSerre

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
⊢ (MorphismProperty.epimorphisms D).inverseImage L = P.epiModSerre

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✝
x✝ : Mono f
⊢ 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✝
x✝ : Epi f
⊢ Epi (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 : D
f : X ⟶ Y
this✝¹ : L.PreservesMonomorphisms
this✝ : L.mapArrow.EssSurj
x✝ : Mono f
this :
  ∀ ⦃X Y : C⦄ (f : X ⟶ Y),
    Mono (L.map f) → ∃ X' Y' f', ∃ (_ : Mono f'), Nonempty (Arrow.mk (L.map f') ≅ Arrow.mk (L.map f))
⊢ ∃ X' Y' f', ∃ (_ : Mono f'), Nonempty (Arrow.mk (L.map f') ≅ Arrow.mk 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✝ : D
f✝ : X✝ ⟶ Y✝
this✝¹ : L.PreservesMonomorphisms
this✝ : L.mapArrow.EssSurj
x✝ : Mono f✝
X Y : C
f : X ⟶ Y
hf : P.monoModSerre f
this : IsIso (L.map (Abelian.factorThruImage f))
⊢ (asIso (L.map (Abelian.factorThruImage f))).hom ≫ (Arrow.mk (L.map (Abelian.image.ι f))).hom =
    (Arrow.mk (L.map f)).hom ≫ (Iso.refl (Arrow.mk (L.map f)).right).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 : D
f : X ⟶ Y
this✝ : L.PreservesMonomorphisms
this : L.mapArrow.EssSurj
X' Y' : C
f' : X' ⟶ Y'
w✝ : Mono f'
e : Arrow.mk (L.map f') ≅ Arrow.mk f
⊢ MorphismProperty.monomorphisms D (L.map f')

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
a b c : N ⋊[φ] G
⊢ (a * b * c).left = (a * (b * c)).left

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
a b c : N ⋊[φ] G
⊢ (a * b * c).right = (a * (b * c)).right

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
a : N ⋊[φ] G
⊢ (1 * a).left = a.left

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
a : N ⋊[φ] G
⊢ (a * 1).left = a.left

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
a : N ⋊[φ] G
⊢ (a⁻¹ * a).left = left 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
a : N ⋊[φ] G
⊢ (a⁻¹ * a).right = right 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
⊢ ∀ (x y : N), ⟨x * y, 1⟩ = ⟨x, 1⟩ * ⟨y, 1⟩

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 : D
f : X ⟶ Y
⊢ Mono f ↔ ∃ X' Y' f', ∃ (_ : Mono f'), Nonempty (Arrow.mk (L.map f') ≅ Arrow.mk f)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
⊢ ∀ (x y : G), ⟨1, x * y⟩ = ⟨1, x⟩ * ⟨1, y⟩

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
inst✝¹ : Group N
inst✝ : Group G
φ : G →* MulAut N
g : G
n : N
⊢ inl ((φ g) n) = inr g * inl n * inr g⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
inst✝¹ : Group N
inst✝ : Group G
φ : G →* MulAut N
g : G
n : N
⊢ inl ((φ g)⁻¹ n) = inr g⁻¹ * inl n * inr g

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
inst✝¹ : Group N
inst✝ : Group G
φ : G →* MulAut N
g : G
n : N
⊢ ⟨n, g⟩ = inl n * inr g

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
inst✝¹ : Group N
inst✝ : Group G
φ : G →* MulAut N
x : N ⋊[φ] G
⊢ inl x.left * inr x.right = x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
inst✝¹ : Group N
inst✝ : Group G
φ : G →* MulAut N
⊢ rightHom.comp inl = 1

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 : D
f : X ⟶ Y
this✝¹ : L.PreservesEpimorphisms
this✝ : L.mapArrow.EssSurj
x✝ : Epi f
this :
  ∀ ⦃X Y : C⦄ (f : X ⟶ Y),
    Epi (L.map f) → ∃ X' Y' f', ∃ (_ : Epi f'), Nonempty (Arrow.mk (L.map f') ≅ Arrow.mk (L.map f))
⊢ ∃ X' Y' f', ∃ (_ : Epi f'), Nonempty (Arrow.mk (L.map f') ≅ Arrow.mk f)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
inst✝¹ : Group N
inst✝ : Group G
φ : G →* MulAut N
⊢ rightHom.comp inr = MonoidHom.id G

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
inst✝¹ : Group N
inst✝ : Group G
φ : G →* MulAut N
n : N
⊢ rightHom (inl n) = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
inst✝¹ : Group N
inst✝ : Group G
φ : G →* MulAut N
g : G
⊢ rightHom (inr g) = g

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
inst✝¹ : Group N
inst✝ : Group G
φ : G →* MulAut N
x✝ : N ⋊[φ] G
⊢ x✝ ∈ inl.range → x✝ ∈ rightHom.ker

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
inst✝¹ : Group N
inst✝ : Group G
φ : G →* MulAut N
x : N ⋊[φ] G
hx : x ∈ rightHom.ker
⊢ inl x.left = x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
fn : N →* H
fg : G →* H
h : ∀ (g : G), fn.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (fg g))).comp fn
⊢ fn (left 1) * fg (right 1) = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
fn : N →* H
fg : G →* H
h : ∀ (g : G), fn.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (fg g))).comp fn
a b : N ⋊[φ] G
⊢ fn (a * b).left * fg (a * b).right = fn a.left * fg a.right * (fn b.left * fg b.right)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
fn : N →* H
fg : G →* H
h : ∀ (g : G), fn.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (fg g))).comp fn
n : N
⊢ (lift fn fg h) (inl n) = fn n

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
fn : N →* H
fg : G →* H
h : ∀ (g : G), fn.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (fg g))).comp fn
⊢ (lift fn fg h).comp inl = fn

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
fn : N →* H
fg : G →* H
h : ∀ (g : G), fn.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (fg g))).comp fn
g : G
⊢ (lift fn fg h) (inr g) = fg g

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
fn : N →* H
fg : G →* H
h : ∀ (g : G), fn.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (fg g))).comp fn
⊢ (lift fn fg h).comp inr = fg

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✝ : D
f✝ : X✝ ⟶ Y✝
this✝¹ : L.PreservesEpimorphisms
this✝ : L.mapArrow.EssSurj
x✝ : Epi f✝
X Y : C
f : X ⟶ Y
hf : P.epiModSerre f
this : IsIso (L.map (Abelian.image.ι f))
⊢ (Iso.refl (Arrow.mk (L.map (Abelian.factorThruImage f))).left).hom ≫ (Arrow.mk (L.map f)).hom =
    (Arrow.mk (L.map (Abelian.factorThruImage f))).hom ≫ (asIso (L.map (Abelian.image.ι f))).hom

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
fn : N →* H
fg : G →* H
h : ∀ (g : G), fn.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (fg g))).comp fn
F : N ⋊[φ] G →* H
x✝ : G
⊢ (F.comp inl).comp (MulEquiv.toMonoidHom (φ x✝)) =
    (MulEquiv.toMonoidHom (MulAut.conj ((F.comp inr) x✝))).comp (F.comp inl)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
F : N ⋊[φ] G →* H
⊢ F = lift (F.comp inl) (F.comp inr) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
f g : N ⋊[φ] G →* H
hl : f.comp inl = g.comp inl
hr : f.comp inr = g.comp inr
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H✝ : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H✝
φ : G →* MulAut N
fn : N →* H✝
fg : G →* H✝
h✝ : ∀ (g : G), fn.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (fg g))).comp fn
H K : Subgroup G
h : K ≤ normalizer ↑H
⊢ ∀ (g : ↥K),
    H.subtype.comp (MulEquiv.toMonoidHom ((H.normalizerMonoidHom.comp (inclusion h)) g)) =
      (MulEquiv.toMonoidHom (MulAut.conj (K.subtype g))).comp H.subtype

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝⁶ : Group N
inst✝⁵ : Group G
inst✝⁴ : Group H
φ : G →* MulAut N
N₁ : Type u_4
G₁ : Type u_5
N₂ : Type u_6
G₂ : Type u_7
inst✝³ : Group N₁
inst✝² : Group G₁
inst✝¹ : Group N₂
inst✝ : Group G₂
φ₁ : G₁ →* MulAut N₁
φ₂ : G₂ →* MulAut N₂
fn : N₁ →* N₂
fg : G₁ →* G₂
h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn
⊢ ⟨fn (left 1), fg (right 1)⟩ = 1

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 : D
f : X ⟶ Y
this✝ : L.PreservesEpimorphisms
this : L.mapArrow.EssSurj
X' Y' : C
f' : X' ⟶ Y'
w✝ : Epi f'
e : Arrow.mk (L.map f') ≅ Arrow.mk f
⊢ MorphismProperty.epimorphisms D (L.map f')

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝⁶ : Group N
inst✝⁵ : Group G
inst✝⁴ : Group H
φ : G →* MulAut N
N₁ : Type u_4
G₁ : Type u_5
N₂ : Type u_6
G₂ : Type u_7
inst✝³ : Group N₁
inst✝² : Group G₁
inst✝¹ : Group N₂
inst✝ : Group G₂
φ₁ : G₁ →* MulAut N₁
φ₂ : G₂ →* MulAut N₂
fn : N₁ →* N₂
fg : G₁ →* G₂
h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn
x y : N₁ ⋊[φ₁] G₁
⊢ ⟨fn (x * y).left, fg (x * y).right⟩ = ⟨fn x.left, fg x.right⟩ * ⟨fn y.left, fg y.right⟩

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N₁ : Type u_4
G₁ : Type u_5
N₂ : Type u_6
G₂ : Type u_7
inst✝³ : Group N₁
inst✝² : Group G₁
inst✝¹ : Group N₂
inst✝ : Group G₂
φ₁ : G₁ →* MulAut N₁
φ₂ : G₂ →* MulAut N₂
fn : N₁ →* N₂
fg : G₁ →* G₂
h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn
n : N₁
⊢ (map fn fg h) (inl n) = inl (fn n)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N₁ : Type u_4
G₁ : Type u_5
N₂ : Type u_6
G₂ : Type u_7
inst✝³ : Group N₁
inst✝² : Group G₁
inst✝¹ : Group N₂
inst✝ : Group G₂
φ₁ : G₁ →* MulAut N₁
φ₂ : G₂ →* MulAut N₂
fn : N₁ →* N₂
fg : G₁ →* G₂
h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn
⊢ (map fn fg h).comp inl = inl.comp fn

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 : D
f : X ⟶ Y
⊢ Epi f ↔ ∃ X' Y' f', ∃ (_ : Epi f'), Nonempty (Arrow.mk (L.map f') ≅ Arrow.mk f)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N₁ : Type u_4
G₁ : Type u_5
N₂ : Type u_6
G₂ : Type u_7
inst✝³ : Group N₁
inst✝² : Group G₁
inst✝¹ : Group N₂
inst✝ : Group G₂
φ₁ : G₁ →* MulAut N₁
φ₂ : G₂ →* MulAut N₂
fn : N₁ →* N₂
fg : G₁ →* G₂
h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn
g : G₁
⊢ (map fn fg h) (inr g) = inr (fg g)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N₁ : Type u_4
G₁ : Type u_5
N₂ : Type u_6
G₂ : Type u_7
inst✝³ : Group N₁
inst✝² : Group G₁
inst✝¹ : Group N₂
inst✝ : Group G₂
φ₁ : G₁ →* MulAut N₁
φ₂ : G₂ →* MulAut N₂
fn : N₁ →* N₂
fg : G₁ →* G₂
h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn
⊢ (map fn fg h).comp inr = inr.comp fg

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝⁶ : Group N
inst✝⁵ : Group G
inst✝⁴ : Group H
φ : G →* MulAut N
N₁ : Type u_4
G₁ : Type u_5
N₂ : Type u_6
G₂ : Type u_7
inst✝³ : Group N₁
inst✝² : Group G₁
inst✝¹ : Group N₂
inst✝ : Group G₂
φ₁ : G₁ →* MulAut N₁
φ₂ : G₂ →* MulAut N₂
fn : N₁ ≃* N₂
fg : G₁ ≃* G₂
h : ∀ (g : G₁), MulEquiv.trans (φ₁ g) fn = fn.trans (φ₂ (fg g))
x✝ : N₁ ⋊[φ₁] G₁
⊢ (fun x ↦ ⟨fn.symm x.left, fg.symm x.right⟩) ((fun x ↦ ⟨fn x.left, fg x.right⟩) x✝) = x✝

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝⁶ : Group N
inst✝⁵ : Group G
inst✝⁴ : Group H
φ : G →* MulAut N
N₁ : Type u_4
G₁ : Type u_5
N₂ : Type u_6
G₂ : Type u_7
inst✝³ : Group N₁
inst✝² : Group G₁
inst✝¹ : Group N₂
inst✝ : Group G₂
φ₁ : G₁ →* MulAut N₁
φ₂ : G₂ →* MulAut N₂
fn : N₁ ≃* N₂
fg : G₁ ≃* G₂
h : ∀ (g : G₁), MulEquiv.trans (φ₁ g) fn = fn.trans (φ₂ (fg g))
x✝ : N₂ ⋊[φ₂] G₂
⊢ (fun x ↦ ⟨fn x.left, fg x.right⟩) ((fun x ↦ ⟨fn.symm x.left, fg.symm x.right⟩) x✝) = x✝

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SemidirectProduct

Goal state before:
pp: N : Type u_1
G : Type u_2
H : Type u_3
inst✝⁶ : Group N
inst✝⁵ : Group G
inst✝⁴ : Group H
φ : G →* MulAut N
N₁ : Type u_4
G₁ : Type u_5
N₂ : Type u_6
G₂ : Type u_7
inst✝³ : Group N₁
inst✝² : Group G₁
inst✝¹ : Group N₂
inst✝ : Group G₂
φ₁ : G₁ →* MulAut N₁
φ₂ : G₂ →* MulAut N₂
fn : N₁ ≃* N₂
fg : G₁ ≃* G₂
h : ∀ (g : G₁), MulEquiv.trans (φ₁ g) fn = fn.trans (φ₂ (fg g))
x y : N₁ ⋊[φ₁] G₁
⊢ ⟨fn (x * y).left, fg (x * y).right⟩ = ⟨fn x.left, fg x.right⟩ * ⟨fn y.left, fg y.right⟩

Tactic:
by

Goal state after: