mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y A B : C
i : A ⟶ B
hi : I i
f : X ⟶ Y
this✝¹ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this✝ : HasPushouts C
this : PreservesColimit (relativeCellComplexιObj I κ f).F (coyoneda.obj (Opposite.op A))
g : A ⟶ obj I κ f
b : B ⟶ Y
sq : CommSq g i (πObj I κ f) b
j : κ.ord.ToType
t : A ⟶ (relativeCellComplexιObj I κ f).F.obj j
ht : t ≫ (relativeCellComplexιObj I κ f).incl.app j = g
⊢ (relativeCellComplexιObj I κ f).F.map (homOfLE ⋯) ≫ (relativeCellComplexιObjFObjSuccIso I κ f j).hom =
    (Iso.refl ((relativeCellComplexιObj I κ f).F.obj j)).hom ≫
      ιFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hq : q ≠ 0
hr : r ≠ 0
this : q * r = q.num * r.num /. (↑q.den * ↑r.den)
hq' : q.num /. ↑q.den ≠ 0
⊢ r.num /. ↑r.den ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
G : C
inst✝ : Abelian C
hG : IsSeparator G
X : C
this : IsIso (largerSubobject hG ⊤).arrow
⊢ (asIso ⊤.arrow).hom ≫ (Arrow.mk (𝟙 X)).hom =
    (Arrow.mk (⊤.ofLE (largerSubobject hG ⊤) ⋯)).hom ≫ (asIso (largerSubobject hG ⊤).arrow).hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
G : C
inst✝ : Abelian C
hG : IsSeparator G
X : C
A : Subobject X
hA : ¬A = ⊤
⊢ ⋯.choose = largerSubobject hG A

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
G : C
inst✝ : Abelian C
hG : IsSeparator G
X : C
A : Subobject X
⊢ (generatingMonomorphisms G).pushouts (A.ofLE (largerSubobject hG A) ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
G : C
inst✝⁵ : Abelian C
hG : IsSeparator G
X : C
inst✝⁴ : IsGrothendieckAbelian.{w, v, u} C
A₀ : Subobject X
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hJ : HasCardinalLT (Subobject X) (Cardinal.mk J)
⊢ largerSubobject hG ⊤ = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
G : C
inst✝⁵ : Abelian C
hG : IsSeparator G
X : C
inst✝⁴ : IsGrothendieckAbelian.{w, v, u} C
A₀ : Subobject X
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hJ : HasCardinalLT (Subobject X) (Cardinal.mk J)
h : Function.Injective fun j ↦ transfiniteIterate (largerSubobject hG) j A₀
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
hG : IsSeparator G
X : C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
A₀ : Subobject X
κ : Cardinal.{w} := Order.succ (Cardinal.mk (Shrink.{w, max u v} (Subobject X)))
⊢ Nonempty κ.ord.ToType

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
hG : IsSeparator G
X : C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
A₀ : Subobject X
κ : Cardinal.{w} := Order.succ (Cardinal.mk (Shrink.{w, max u v} (Subobject X)))
this✝ : Nonempty κ.ord.ToType
this : OrderBot κ.ord.ToType
⊢ Order.succ (Cardinal.lift.{w, max u v} (Cardinal.mk (Subobject X))) ≤
    Cardinal.lift.{max u v, w} (Cardinal.mk κ.ord.ToType)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y A B : C
i : A ⟶ B
hi : I i
f : X ⟶ Y
this✝¹ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this✝ : HasPushouts C
this : PreservesColimit (relativeCellComplexιObj I κ f).F (coyoneda.obj (Opposite.op A))
g : A ⟶ obj I κ f
b : B ⟶ Y
sq : CommSq g i (πObj I κ f) b
j : κ.ord.ToType
t : A ⟶ (relativeCellComplexιObj I κ f).F.obj j
ht : t ≫ (relativeCellComplexιObj I κ f).incl.app j = g
⊢ ((relativeCellComplexιObj I κ f).incl.app (Order.succ j) ≫ πObj I κ f) ≫
      (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).symm.hom =
    (relativeCellComplexιObjFObjSuccIso I κ f j).hom ≫
      πFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
hG : IsSeparator G
X : C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
A₀ : Subobject X
⊢ ∃ o j, transfiniteIterate (largerSubobject hG) j A₀ = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
G : C
inst✝⁵ : Abelian C
hG : IsSeparator G
X : C
inst✝⁴ : IsGrothendieckAbelian.{w, v, u} C
A₀ : Subobject X
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
m : J
hm : Order.IsSuccLimit m
this : Nonempty ↑(Set.Iio m)
c : Cocone (⋯.functor ⋙ functorToMonoOver hG A₀ J ⋙ MonoOver.forget X) :=
  (Set.principalSegIio m).cocone (functorToMonoOver hG A₀ J ⋙ MonoOver.forget X)
⊢ Mono c.pt.hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y A B : C
i : A ⟶ B
hi : I i
f : X ⟶ Y
this✝¹ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this✝ : HasPushouts C
this : PreservesColimit (relativeCellComplexιObj I κ f).F (coyoneda.obj (Opposite.op A))
g : A ⟶ obj I κ f
b : B ⟶ Y
sq : CommSq g i (πObj I κ f) b
j : κ.ord.ToType
t : A ⟶ (relativeCellComplexιObj I κ f).F.obj j
ht : t ≫ (relativeCellComplexιObj I κ f).incl.app j = g
⊢ t ≫ (relativeCellComplexιObj I κ f).incl.app j ≫ πObj I κ f = I.homFamily ⟨Arrow.mk i, hi⟩ ≫ b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
G : C
inst✝⁵ : Abelian C
hG : IsSeparator G
X : C
inst✝⁴ : IsGrothendieckAbelian.{w, v, u} C
A₀ : Subobject X
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
m : J
hm : Order.IsSuccLimit m
⊢ IsColimit ((Set.principalSegIio m).cocone (functor hG A₀ J))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y A B : C
i : A ⟶ B
hi : I i
f : X ⟶ Y
this✝¹ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this✝ : HasPushouts C
this : PreservesColimit (relativeCellComplexιObj I κ f).F (coyoneda.obj (Opposite.op A))
g : A ⟶ obj I κ f
b : B ⟶ Y
sq : CommSq g i (πObj I κ f) b
j : κ.ord.ToType
t : A ⟶ (relativeCellComplexιObj I κ f).F.obj j
ht : t ≫ (relativeCellComplexιObj I κ f).incl.app j = g
l : (↑⟨Arrow.mk i, hi⟩).right ⟶ (relativeCellComplexιObj I κ f).F.obj (Order.succ j)
hl₁ : i ≫ l = t ≫ (relativeCellComplexιObj I κ f).F.map (homOfLE ⋯)
hl₂ : l ≫ (relativeCellComplexιObj I κ f).incl.app (Order.succ j) ≫ πObj I κ f = b
⊢ i ≫ l ≫ (relativeCellComplexιObj I κ f).incl.app (Order.succ j) = g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
G : C
inst✝⁵ : Abelian C
hG : IsSeparator G
X : C
inst✝⁴ : IsGrothendieckAbelian.{w, v, u} C
A₀ : Subobject X
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
j : J
k k' : ↑(Set.Iic j)
h : k ⟶ k'
⊢ (⋯.functor ⋙ functor hG A₀ J).map h ≫ (functor hG A₀ J).map (homOfLE ⋯) =
    (functor hG A₀ J).map (homOfLE ⋯) ≫ ((Functor.const ↑(Set.Iic j)).obj ((functor hG A₀ J).obj j)).map h

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y A B : C
i : A ⟶ B
hi : I i
f : X ⟶ Y
this✝¹ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this✝ : HasPushouts C
this : PreservesColimit (relativeCellComplexιObj I κ f).F (coyoneda.obj (Opposite.op A))
g : A ⟶ obj I κ f
b : B ⟶ Y
sq : CommSq g i (πObj I κ f) b
j : κ.ord.ToType
t : A ⟶ (relativeCellComplexιObj I κ f).F.obj j
ht : t ≫ (relativeCellComplexιObj I κ f).incl.app j = g
l : (↑⟨Arrow.mk i, hi⟩).right ⟶ (relativeCellComplexιObj I κ f).F.obj (Order.succ j)
hl₁ : i ≫ l = t ≫ (relativeCellComplexιObj I κ f).F.map (homOfLE ⋯)
hl₂ : l ≫ (relativeCellComplexιObj I κ f).incl.app (Order.succ j) ≫ πObj I κ f = b
⊢ (l ≫ (relativeCellComplexιObj I κ f).incl.app (Order.succ j)) ≫ πObj I κ f = b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
G : C
inst✝⁵ : Abelian C
hG : IsSeparator G
X : C
inst✝⁴ : IsGrothendieckAbelian.{w, v, u} C
A₀ : Subobject X
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
j : J
k : ↑(Set.Iic j)
hk : ¬IsMax k
⊢ (generatingMonomorphisms G).pushouts ((⋯.functor ⋙ functor hG A₀ J).map (homOfLE ⋯))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y A B : C
i : A ⟶ B
hi : I i
f : X ⟶ Y
⊢ ∀ {f_1 : A ⟶ obj I κ f} {g : B ⟶ Y} (sq : CommSq f_1 i (πObj I κ f) g), sq.HasLift

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
f : Arrow C
⊢ objMap I κ (𝟙 f) = 𝟙 (obj I κ f.hom)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
f g h : Arrow C
φ : f ⟶ g
ψ : g ⟶ h
⊢ objMap I κ (φ ≫ ψ) = objMap I κ φ ≫ objMap I κ ψ

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hq : q ≠ 0
hr : r ≠ 0
⊢ padicValRat p (q * r) = padicValRat p q + padicValRat p r

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
G : C
inst✝⁶ : Abelian C
hG : IsSeparator G
X : C
inst✝⁵ : IsGrothendieckAbelian.{w, v, u} C
A : C
f : A ⟶ X
inst✝⁴ : Mono f
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
j : J
hj : transfiniteIterate (largerSubobject hG) j (Subobject.mk f) = ⊤
⊢ (generatingMonomorphisms G).pushouts.TransfiniteCompositionOfShape (↑(Set.Iic j)) f

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
k : ℕ
⊢ padicValRat p (q ^ k) = ↑k * padicValRat p q

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
⊢ padicValRat p q⁻¹ = -padicValRat p q

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
G : C
inst✝² : Abelian C
hG : IsSeparator G
X : C
inst✝¹ : IsGrothendieckAbelian.{w, v, u} C
A : C
f : A ⟶ X
inst✝ : Mono f
⊢ ∃ J x x_1 x_2,
    ∃ (x_3 : WellFoundedLT J), Nonempty ((generatingMonomorphisms G).pushouts.TransfiniteCompositionOfShape J f)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
k : ℤ
⊢ padicValRat p (q ^ k) = k * padicValRat p q

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hq : q ≠ 0
hr : r ≠ 0
⊢ padicValRat p (q / r) = padicValRat p q - padicValRat p r

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
hG : IsSeparator G
X Y : C
p : X ⟶ Y
hp : (generatingMonomorphisms G).rlp p
A B : C
i : A ⟶ B
x✝ : Mono i
J : Type w
w✝³ : LinearOrder J
w✝² : OrderBot J
w✝¹ : SuccOrder J
w✝ : WellFoundedLT J
h : (generatingMonomorphisms G).pushouts.TransfiniteCompositionOfShape J i
⊢ (generatingMonomorphisms G).pushouts.rlp p

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
hG : IsSeparator G
⊢ (generatingMonomorphisms G).rlp = (monomorphisms C).rlp

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
κ : Cardinal.{w}
hκ' : κ.IsRegular
hκ : HasCardinalLT (Subobject G) κ
this : Fact κ.IsRegular
⊢ Nonempty κ.ord.ToType

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
f g : Arrow C
φ : f ⟶ g
⊢ objMap I κ φ ≫ πObj I κ g.hom = πObj I κ f.hom ≫ Arrow.Hom.right φ

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
κ : Cardinal.{w}
hκ' : κ.IsRegular
hκ : HasCardinalLT (Subobject G) κ
this✝¹ : Fact κ.IsRegular
this✝ : Nonempty κ.ord.ToType
this : OrderBot κ.ord.ToType
A B X Y : C
i : A ⟶ B
hi : generatingMonomorphisms G i
f : X ⟶ Y
hf : HomotopicalAlgebra.RelativeCellComplex (fun x ↦ (generatingMonomorphisms G).homFamily) f
j : κ.ord.ToType
hj : ¬IsMax j
⊢ monomorphisms C (hf.F.map (homOfLE ⋯))

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n₁ n₂ d₁ d₂ : ℤ
hn₁ : n₁ ≠ 0
hn₂ : n₂ ≠ 0
hd₁ : d₁ ≠ 0
hd₂ : d₂ ≠ 0
⊢ padicValRat p (n₁ /. d₁) ≤ padicValRat p (n₂ /. d₂) ↔ ∀ (n : ℕ), ↑p ^ n ∣ n₁ * d₂ → ↑p ^ n ∣ n₂ * d₁

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hqr : q + r ≠ 0
h : padicValRat p q ≤ padicValRat p r
hq : q = 0
⊢ padicValRat p q ≤ padicValRat p (q + r)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hqr : q + r ≠ 0
h : padicValRat p q ≤ padicValRat p r
hq : ¬q = 0
hr : r = 0
⊢ padicValRat p q ≤ padicValRat p (q + r)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y : C
f : X ⟶ Y
hf : I.rlp.llp f
⊢ ιObj I κ f ≫ πObj I κ f = f ≫ 𝟙 Y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
κ : Cardinal.{w}
hκ' : κ.IsRegular
hκ : HasCardinalLT (Subobject G) κ
this✝¹ : Fact κ.IsRegular
this✝ : Nonempty κ.ord.ToType
this : OrderBot κ.ord.ToType
A B X Y : C
i : A ⟶ B
hi : generatingMonomorphisms G i
f : X ⟶ Y
hf : HomotopicalAlgebra.RelativeCellComplex (fun x ↦ (generatingMonomorphisms G).homFamily) f
⊢ PreservesColimit hf.F (coyoneda.obj (Opposite.op A))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
⊢ (generatingMonomorphisms G).HasSmallObjectArgument

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
⊢ I.rlp.llp = ((coproducts.{w, v, u} I).pushouts.transfiniteCompositionsOfShape κ.ord.ToType).retracts

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
hG : IsSeparator G
⊢ (monomorphisms C).rlp.llp = monomorphisms C

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
I : MorphismProperty C
inst✝ : I.IsCardinalForSmallObjectArgument Cardinal.aleph0
⊢ Nonempty ((fun x1 x2 ↦ x1 < x2) ≃r fun x1 x2 ↦ x1 < x2)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
⊢ (monomorphisms C).HasFunctorialFactorization (monomorphisms C).rlp

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
I : MorphismProperty C
inst✝ : I.IsCardinalForSmallObjectArgument Cardinal.aleph0
⊢ I.rlp.llp = ((coproducts.{w, v, u} I).pushouts.transfiniteCompositionsOfShape ℕ).retracts

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
X : C
⊢ Injective (monoMapFactorizationDataRlp 0).Z

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
⊢ I.rlp.llp = (transfiniteCompositions.{w, v, u} (coproducts.{w, v, u} I).pushouts).retracts

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
q : ℚ
hq : q ≠ 0
⊢ padicNorm p q = ↑p ^ (-padicValRat p q)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
q : ℚ
hq : q = 0
⊢ 0 ≤ padicNorm p q

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
q : ℚ
hq : ¬q = 0
⊢ 0 ≤ padicNorm p q

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
⊢ padicNorm p 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
⊢ padicNorm p 1 = 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : 1 < p
⊢ padicNorm p ↑p = (↑p)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p q : ℕ
p_prime : Fact (Nat.Prime p)
q_prime : Fact (Nat.Prime q)
ne : p ≠ q
⊢ padicNorm p ↑q = 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : 1 < p
⊢ padicNorm p ↑p < 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
q : ℚ
hq : q = 0
⊢ padicNorm p (-q) = padicNorm p q

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
q : ℚ
hq : ¬q = 0
⊢ padicNorm p (-q) = padicNorm p q

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
hq : q ≠ 0
⊢ padicNorm p q ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : padicNorm p q = 0
⊢ q = 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hq : q = 0
⊢ padicNorm p (q * r) = padicNorm p q * padicNorm p r

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hq : ¬q = 0
hr : r = 0
⊢ padicNorm p (q * r) = padicNorm p q * padicNorm p r

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hq : ¬q = 0
hr : ¬r = 0
⊢ ↑p ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hq : ¬q = 0
hr : ¬r = 0
⊢ padicNorm p (q * r) = padicNorm p q * padicNorm p r

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hr : r = 0
⊢ padicNorm p (q / r) = padicNorm p q / padicNorm p r

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hr : ¬r = 0
⊢ padicNorm p (q / r) * padicNorm p r = padicNorm p q

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ
hz : z ≠ 0
⊢ -padicValRat p ↑z ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ
⊢ padicNorm p ↑z ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
h : padicValRat p q ≤ padicValRat p r
hnqp : padicNorm p q ≥ 0
hnrp : padicNorm p r ≥ 0
hq : q = 0
⊢ padicNorm p (q + r) ≤ max (padicNorm p q) (padicNorm p r)

Tactic:
by

Goal state after: