mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Spectrum.Prime.Noetherian

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsArtinianRing R
p : Ideal R
inst✝ : p.IsPrime
r : R
hr : (toPiLocalization R) r = Pi.single { asIdeal := p, isPrime := ⋯ } 1
this✝ : (algebraMap R (Localization p.primeCompl)) r = 1
q : Ideal R
hq : q.IsPrime
e : q ≠ p
this : { asIdeal := q, isPrime := ⋯ } ≠ { asIdeal := p, isPrime := ⋯ }
⊢ (algebraMap R (Localization.AtPrime q)) r = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.Noetherian

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsArtinianRing R
p : Ideal R
inst✝ : p.IsPrime
⊢ ∃ r ∉ p, IsIdempotentElem r ∧ ∀ (q : Ideal R), q.IsPrime → q ≠ p → r ∈ q

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Finiteness.NilpotentKer

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : CommRing T
inst✝² : Algebra R S
inst✝¹ : Algebra R T
inst✝ : Module.Finite R T
f : S →ₐ[R] T
hf₁ : Function.Surjective ⇑f
hf₂ : RingHom.ker f ≤ nilradical S
hf₃ : (RingHom.ker f).FG
⊢ Module.Finite R S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝¹ : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst✝ : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
⊢ cardQuot (P ^ i) = cardQuot P ^ i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
I✝ J✝ : Ideal S
b : Basis (Free.ChooseBasisIndex ℤ S) ℤ S := Free.chooseBasis ℤ S
this : Infinite S
I J : Submodule S S
hI : IsUnit J
⊢ (I * J).cardQuot = I.cardQuot * J.cardQuot

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
I J : Ideal S
⊢ cardQuot (I * J) = cardQuot I * cardQuot J

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
⊢ cardQuot 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
⊢ cardQuot 1 = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
⊢ absNorm ⊥ = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
⊢ absNorm ⊤ = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
I : Ideal S
⊢ absNorm I = 1 ↔ I = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
I : Ideal S
hI : Irreducible (absNorm I)
h : IsUnit I
⊢ IsUnit (absNorm I)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
I : Ideal S
hI : Irreducible (absNorm I)
⊢ ∀ ⦃a b : Ideal S⦄, I = a * b → IsUnit a ∨ IsUnit b

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
I : Ideal S
⊢ ↑(absNorm I) ∈ I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
I : Ideal S
⊢ span {↑(absNorm I)} ≤ I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
I : Ideal S
hI : Nat.Prime (absNorm I)
⊢ (span {↑(absNorm I)}).IsPrime

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Jacobson.Artinian

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra.FiniteType R A
inst✝ : IsArtinianRing R
⊢ Module.Finite R A ↔ Ring.KrullDimLE 0 A

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
I : Ideal S
hI : Nat.Prime (absNorm I)
⊢ span {↑(absNorm I)} = comap (algebraMap ℤ S) I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝⁵ : CommRing S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
r : S
⊢ absNorm (span {r}) = ((Algebra.norm ℤ) r).natAbs

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
n : ℕ
⊢ absNorm (span {↑n}) = n ^ finrank ℤ S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
I : Ideal S
x : S
h : x ∈ I
⊢ ↑(absNorm I) ∣ (Algebra.norm ℤ) x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
r : S
s : Set S
⊢ absNorm (span {r}) ∣ ((Algebra.norm ℤ) r).natAbs

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
I : Ideal S
⊢ absNorm I = 0 ↔ I = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
I : Ideal S
⊢ absNorm I ≠ 0 ↔ I ∈ (Ideal S)⁰

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
I : Ideal S
⊢ 0 < absNorm I ↔ I ∈ (Ideal S)⁰

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
I : Ideal S
hI : I ≠ ⊥
⊢ (toAddSubgroup I).FiniteIndex

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
I : Ideal S
hI : I ≠ ⊥
J : Ideal S
⊢ (toAddSubgroup I).IsFiniteRelIndex (toAddSubgroup J)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDedekindDomain S
inst✝² : Free ℤ S
inst✝¹ : Module.Finite ℤ S
P : Ideal S
inst✝ : P.IsMaximal
⊢ ∃ p n, 0 < n ∧ ↑p ∈ P ∧ Nat.Prime p ∧ absNorm P = p ^ n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
p : ℤ
hp : Prime p
I : Ideal S
hI : p ∣ ↑(absNorm I)
this✝ : IsAddTorsionFree S
this : CharZero S
⊢ span {p} ≠ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.InvSubmonoid

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Submonoid R
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
⊢ Submonoid.map (algebraMap R S) M ≤ IsUnit.submonoid S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
p : ℤ
hp : Prime p
this✝ : IsAddTorsionFree S
this : CharZero S
hpMax : (span {p}).IsMaximal
hI : p ∣ ↑(absNorm 0)
x : ℤ
⊢ x ∈ RingHom.ker (algebraMap ℤ S) → x ∈ span {p}

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
a b : ↥(IsUnit.submonoid M)
⊢ a * b = b * a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
p : ℤ
hp : Prime p
this✝ : IsAddTorsionFree S
this : CharZero S
hpMax : (span {p}).IsMaximal
I : Ideal S
hI' : IsUnit I
hI : p ∣ ↑(absNorm I)
⊢ I = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : Monoid M
S : Submonoid M
a _b : M
x✝¹ : a ∈ {x | ∃ y, x * ↑y = 1}
x✝ : _b ∈ {x | ∃ y, x * ↑y = 1}
a' : ↥S
ha : a * ↑a' = 1
b' : ↥S
hb : _b * ↑b' = 1
⊢ a * _b * ↑(b' * a') = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
p : ℤ
hp : Prime p
this✝ : IsAddTorsionFree S
this : CharZero S
hpMax : (span {p}).IsMaximal
hI' : IsUnit ⊤
hI : p ∣ ↑(absNorm ⊤)
⊢ p ∣ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.InvSubmonoid

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Submonoid R
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
m : ↥M
⊢ m • ↑((toInvSubmonoid M S) m) = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : Monoid M
S : Submonoid M
⊢ S.leftInv.leftInv ≤ S

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : Monoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
x : M
hx : x ∈ S
⊢ x = ↑(IsUnit.unit ⋯)⁻¹⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : Monoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
⊢ S.leftInv.leftInv = S

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
x : ↥S.leftInv
⊢ ↑(S.fromLeftInv x) * ↑x = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
p : ℤ
hp : Prime p
this✝¹ : IsAddTorsionFree S
this✝ : CharZero S
hpMax : (span {p}).IsMaximal
I P : Ideal S
hI' : I ≠ 0
hP : Prime P
IH : p ∣ ↑(absNorm I) → ∃ P, P.IsMaximal ∧ under ℤ P = span {p} ∧ P ∣ I
hI : p ∣ ↑(absNorm P)
this : P.IsMaximal
⊢ under ℤ P ≠ ⊤

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
a : ↥S.leftInv
b : M
⊢ ↑(S.fromLeftInv a) = b ↔ ↑a * b = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopkinsLevitzki

Goal state before:
pp: R₀ : Type u_1
R : Type u_2
M : Type u
inst✝⁷ : Ring R₀
inst✝⁶ : Ring R
inst✝⁵ : Module R₀ R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R₀ M
inst✝² : Module R M
inst✝¹ : IsScalarTower R₀ R M
inst✝ : IsSemiprimaryRing R
P : (M : Type u) → [inst : AddCommGroup M] → [Module R₀ M] → [Module R M] → Prop
h0 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M]
    [IsSemisimpleModule R M], Module.IsTorsionBySet R M ↑(Ring.jacobson R) → P M
h1 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [inst_3 : IsScalarTower R₀ R M],
    let N := Ring.jacobson R • ⊤;
    P ↥N → P (M ⧸ N) → P M
ss : IsSemisimpleRing (R ⧸ Ring.jacobson R)
n : ℕ
hn : Ring.jacobson R ^ n = 0
⊢ P M

Tactic:
set Jac := Ring.jacobson R

Goal state after:
R₀ : Type u_1
R : Type u_2
M : Type u
inst✝⁷ : Ring R₀
inst✝⁶ : Ring R
inst✝⁵ : Module R₀ R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R₀ M
inst✝² : Module R M
inst✝¹ : IsScalarTower R₀ R M
inst✝ : IsSemiprimaryRing R
P : (M : Type u) → [inst : AddCommGroup M] → [Module R₀ M] → [Module R M] → Prop
ss : IsSemisimpleRing (R ⧸ Ring.jacobson R)
n : ℕ
Jac : Ideal R := Ring.jacobson R
h0 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M]
    [IsSemisimpleModule R M], Module.IsTorsionBySet R M ↑Jac → P M
h1 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [inst_3 : IsScalarTower R₀ R M],
    let N := Jac • ⊤;
    P ↥N → P (M ⧸ N) → P M
hn : Jac ^ n = 0
⊢ P M
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
x y : ↥S.leftInv
⊢ ↑(S.fromLeftInv (x * y)) = ↑(S.fromLeftInv x * S.fromLeftInv y)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.InvSubmonoid

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Submonoid R
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
z : S
⊢ ∃ r m, z = r • ↑((toInvSubmonoid M S) m)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
x : ↥S
⊢ ↑(IsUnit.unit ⋯)⁻¹ * ↑x = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
⊢ Function.LeftInverse (fun x ↦ ⟨↑(IsUnit.unit ⋯)⁻¹, ⋯⟩) (↑S.fromCommLeftInv).toFun

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.InvSubmonoid

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Submonoid R
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
x : ↥M
⊢ ↑((toInvSubmonoid M S) x) = mk' S 1 x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
⊢ Function.RightInverse (fun x ↦ ⟨↑(IsUnit.unit ⋯)⁻¹, ⋯⟩) (↑S.fromCommLeftInv).toFun

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
x : ↥S.leftInv
⊢ ↑((S.leftInvEquiv hS) x) * ↑x = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
x : ↥S.leftInv
⊢ ↑x * ↑((S.leftInvEquiv hS) x) = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.InvSubmonoid

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Submonoid R
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
x : S
⊢ x ∈ invSubmonoid M S ↔ ∃ m, mk' S 1 m = x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
x : ↥S
⊢ ↑((S.leftInvEquiv hS).symm x) * ↑x = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
x : ↥S
⊢ ↑x * ↑((S.leftInvEquiv hS).symm x) = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : Group M
S : Submonoid M
x : ↥S.leftInv
⊢ ↑(S.fromLeftInv x) = (↑x)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.InvSubmonoid

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Submonoid R
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
⊢ Submodule.span R ↑(invSubmonoid M S) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
p : ℤ
hp✝ : Prime p
this✝¹ : IsAddTorsionFree S
this✝ : CharZero S
hpMax : (span {p}).IsMaximal
I P : Ideal S
hI' : I ≠ 0
hP : Prime P
IH : p ∣ ↑(absNorm I) → ∃ P, P.IsMaximal ∧ under ℤ P = span {p} ∧ P ∣ I
this : P.IsMaximal
q n : ℕ
hn : 0 < n
hqP : ↑q ∈ P
hq : Nat.Prime q
H : absNorm P = q ^ n
m : ℕ
hmn : m ≤ n
hp : Associated p (↑q ^ m)
h : m = 0
⊢ IsUnit p

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : CommGroup M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
x : ↥S
⊢ ↑((S.leftInvEquiv hS).symm x) = (↑x)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
p : ℤ
hp✝ : Prime p
this✝² : IsAddTorsionFree S
this✝¹ : CharZero S
hpMax : (span {p}).IsMaximal
I P : Ideal S
hI' : I ≠ 0
hP : Prime P
IH : p ∣ ↑(absNorm I) → ∃ P, P.IsMaximal ∧ under ℤ P = span {p} ∧ P ∣ I
this✝ : P.IsMaximal
q n : ℕ
hn : 0 < n
hqP : ↑q ∈ P
hq : Nat.Prime q
H : absNorm P = q ^ n
m : ℕ
hmn : m ≤ n
hp : Associated p (↑q ^ m)
this : m ≠ 0
⊢ ↑q ∈ under ℤ P

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopkinsLevitzki

Goal state before:
pp: R₀ : Type u_1
R : Type u_2
M✝ : Type u
inst✝¹¹ : Ring R₀
inst✝¹⁰ : Ring R
inst✝⁹ : Module R₀ R
inst✝⁸ : AddCommGroup M✝
inst✝⁷ : Module R₀ M✝
inst✝⁶ : Module R M✝
inst✝⁵ : IsScalarTower R₀ R M✝
inst✝⁴ : IsSemiprimaryRing R
P : (M : Type u) → [inst : AddCommGroup M] → [Module R₀ M] → [Module R M] → Prop
ss : IsSemisimpleRing (R ⧸ Ring.jacobson R)
n : ℕ
Jac : Ideal R := Ring.jacobson R
h0 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M]
    [IsSemisimpleModule R M], Module.IsTorsionBySet R M ↑Jac → P M
h1 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [inst_3 : IsScalarTower R₀ R M],
    let N := Jac • ⊤;
    P ↥N → P (M ⧸ N) → P M
hn : Jac ^ n ≤ Module.annihilator R M✝
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R₀ M
inst✝¹ : Module R M
inst✝ : IsScalarTower R₀ R M
⊢ Jac ≤ Module.annihilator R M → P M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Localization.InvSubmonoid

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsLocalization M S
inst✝ : Monoid.FG ↥M
⊢ Algebra.FiniteType R S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
p : ℤ
hp : Prime p
I : Ideal S
hI : p ∣ ↑(absNorm I)
⊢ ∃ P, P.IsMaximal ∧ under ℤ P = span {p} ∧ P ∣ I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
p : ℕ
hp : Nat.Prime p
I : Ideal S
hI : p ∣ absNorm I
⊢ Nat.Prime (↑p).natAbs

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
p : ℕ
hp : Nat.Prime p
I : Ideal S
hI : p ∣ absNorm I
⊢ ↑p ∣ ↑(absNorm I)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDedekindDomain S
inst✝² : Free ℤ S
inst✝¹ : Module.Finite ℤ S
inst✝ : CharZero S
n : ℕ
hn : n > 0
f : Ideal S → Ideal (S ⧸ span {↑n}) := fun I ↦ map (Quotient.mk (span {↑n})) I
this : Finite (S ⧸ span {↑n})
⊢ (f '' {I | absNorm I = n}).Finite

Tactic:
by

Goal state after:
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : Function.Injective f
hf' : Continuous[inst✝¹, inst✝] f
⊢ f ⁻¹' closedPoints Y ⊆ closedPoints X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsClosedEmbedding f
⊢ f ⁻¹' closedPoints Y = closedPoints X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
hs : s.Finite
hs' : s ⊆ closedPoints X
x : ↑s
⊢ IsClosed[instTopologicalSpaceSubtype] {x}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
hs : s.Finite
hs' : s ⊆ closedPoints X
⊢ IsDiscrete s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
⊢ closure[inst✝¹] (closedPoints X) = Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDedekindDomain S
inst✝² : Free ℤ S
inst✝¹ : Module.Finite ℤ S
inst✝ : CharZero S
n : ℕ
⊢ {I | absNorm I = n}.Finite

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDedekindDomain S
inst✝² : Free ℤ S
inst✝¹ : Module.Finite ℤ S
inst✝ : CharZero S
n : ℕ
⊢ {I | absNorm I ≤ n}.Finite

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.Jacobson

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsJacobsonRing R
s : Set (PrimeSpectrum R)
hs : IsOpen s
hs' : s.Nonempty
⊢ ∃ x ∈ s, IsClosed {x}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDedekindDomain S
inst✝² : Free ℤ S
inst✝¹ : Module.Finite ℤ S
inst✝ : CharZero S
n : ℕ
⊢ {I | absNorm ↑I ≤ n}.Finite

Tactic:
by

Goal state after:
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
H : ∀ {Z : Set X}, IsClosed[inst✝] Z → closure[inst✝] (Z ∩ closedPoints X) = Z
Z : Set X
hZ : Z ≠ ∅
hZ' : IsClosed[inst✝] (closure[inst✝] Z \ Z)
e : Z ∩ closedPoints X = ∅
⊢ Z ⊆ closure[inst✝] Z \ Z

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopkinsLevitzki

Goal state before:
pp: R₀ : Type u_1
R : Type u_2
M : Type u
inst✝⁷ : Ring R₀
inst✝⁶ : Ring R
inst✝⁵ : Module R₀ R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R₀ M
inst✝² : Module R M
inst✝¹ : IsScalarTower R₀ R M
inst✝ : IsSemiprimaryRing R
P : (M : Type u) → [inst : AddCommGroup M] → [Module R₀ M] → [Module R M] → Prop
h0 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M]
    [IsSemisimpleModule R M], Module.IsTorsionBySet R M ↑(Ring.jacobson R) → P M
h1 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [inst_3 : IsScalarTower R₀ R M],
    let N := Ring.jacobson R • ⊤;
    P ↥N → P (M ⧸ N) → P M
⊢ P M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.Jacobson

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsJacobsonRing R
⊢ JacobsonSpace (PrimeSpectrum R)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.Jacobson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ IsJacobsonRing R ↔ JacobsonSpace (PrimeSpectrum R)

Tactic:
by

Goal state after: