mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
x z : R
y : S
inst✝⁵ : Away x S
T : Type u_4
inst✝⁴ : CommRing T
inst✝³ : Algebra S T
inst✝² : Algebra R T
inst✝¹ : IsScalarTower R S T
inst✝ : Away y T
h : Associated ((algebraMap R S) z) y
⊢ Away y T

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
x z : R
y : S
inst✝⁵ : Away x S
T : Type u_4
inst✝⁴ : CommRing T
inst✝³ : Algebra S T
inst✝² : Algebra R T
inst✝¹ : IsScalarTower R S T
inst✝ : Away y T
h : Associated ((algebraMap R S) z) y
⊢ Away y T
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
e : R
he : IsIdempotentElem e
inst✝ : Away e S
x : R
n : ℕ
⊢ e * x = e ^ 1 * x

Tactic:
]

Goal state after:
R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
e : R
he : IsIdempotentElem e
inst✝ : Away e S
x : R
n : ℕ
⊢ e * x = e ^ 1 * x
Module: Mathlib.RingTheory.Jacobson.Radical

Goal state before:
pp: R : Type u_1
R₂ : Type u_2
M : Type u_3
M₂ : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : Ring R₂
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup M₂
inst✝¹ : Module R₂ M₂
τ₁₂ : R →+* R₂
inst✝ : RingHomSurjective τ₁₂
f : M →ₛₗ[τ₁₂] M₂
surj : Function.Surjective ⇑f
le : f.ker ≤ ⨅ a, ↑a
⊢ jacobson R₂ M₂ ≤ map f (jacobson R M)

Tactic:
]

Goal state after:
R : Type u_1
R₂ : Type u_2
M : Type u_3
M₂ : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : Ring R₂
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup M₂
inst✝¹ : Module R₂ M₂
τ₁₂ : R →+* R₂
inst✝ : RingHomSurjective τ₁₂
f : M →ₛₗ[τ₁₂] M₂
surj : Function.Surjective ⇑f
le : f.ker ≤ ⨅ a, ↑a
⊢ jacobson R₂ M₂ ≤ map f (jacobson R M)
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
e : R
he : IsIdempotentElem e
inst✝ : Away e S
x : R
n : ℕ
⊢ e ^ (n + 1) * x = e ^ 1 * x

Tactic:
rw [he.pow_succ_eq]

Goal state after:
R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
e : R
he : IsIdempotentElem e
inst✝ : Away e S
x : R
n : ℕ
⊢ e * x = e ^ 1 * x
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
e : R
he : IsIdempotentElem e
H : ∀ (x y : R), (algebraMap R S) x = (algebraMap R S) y ↔ e * x = e * y
H' : Function.Surjective ⇑(algebraMap R S)
n : ℕ
⊢ e = e

Tactic:
]

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

Goal state before:
pp: R : Type u_1
R₂ : Type u_2
M : Type u_3
M₂ : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : Ring R₂
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup M₂
inst✝¹ : Module R₂ M₂
τ₁₂ : R →+* R₂
inst✝ : RingHomSurjective τ₁₂
f : M →ₛₗ[τ₁₂] M₂
surj : Function.Surjective ⇑f
le : f.ker ≤ jacobson R M
⊢ jacobson R M = jacobson R M

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_4
S : Type u_5
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
e : R
he : IsIdempotentElem e
H : RingHom.ker (algebraMap R S) = Ideal.span {1 - e}
H' : Function.Surjective ⇑(algebraMap R S)
x y : R
⊢ x • e = y • e ↔ e * x = e * y

Tactic:
]

Goal state after:
R : Type u_4
S : Type u_5
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
e : R
he : IsIdempotentElem e
H : RingHom.ker (algebraMap R S) = Ideal.span {1 - e}
H' : Function.Surjective ⇑(algebraMap R S)
x y : R
⊢ x • e = y • e ↔ e * x = e * y
Module: Mathlib.RingTheory.Jacobson.Radical

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
le : N ≤ jacobson R M
⊢ N ≤ jacobson R M

Tactic:
]

Goal state after:
R : Type u_1
M : Type u_3
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
le : N ≤ jacobson R M
⊢ N ≤ jacobson R M
Module: Mathlib.RingTheory.Jacobson.Radical

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ ⊥ = ⊥

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.TwoSidedIdeal.Operations

Goal state before:
pp: case refine_1
R : Type u_1
inst✝ : Ring R
s : Set R
z x y : R
hy : y ∈ univ * (s * univ)
⊢ x * y ∈ univ * (s * univ)

Tactic:
]

Goal state after:
case refine_1
R : Type u_1
inst✝ : Ring R
s : Set R
z x y : R
hy : y ∈ univ * (s * univ)
⊢ x * y ∈ univ * (s * univ)
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: case refine_2
S : Type u_4
T : Type u_5
inst✝² : CommRing S
inst✝¹ : CommRing T
inst✝ : Algebra S T
h₁ : Function.Surjective ⇑(algebraMap S T)
r : S
hr : IsUnit ((algebraMap S T) r)
n : ℕ
hn : r ^ n • RingHom.ker (algebraMap S T) ≤ ⊥
x y : S
h : (algebraMap S T) x = (algebraMap S T) y
⊢ r ^ n * (x - y) = 0

Tactic:
]

Goal state after:
case refine_2
S : Type u_4
T : Type u_5
inst✝² : CommRing S
inst✝¹ : CommRing T
inst✝ : Algebra S T
h₁ : Function.Surjective ⇑(algebraMap S T)
r : S
hr : IsUnit ((algebraMap S T) r)
n : ℕ
hn : r ^ n • RingHom.ker (algebraMap S T) ≤ ⊥
x y : S
h : (algebraMap S T) x = (algebraMap S T) y
⊢ r ^ n * (x - y) = 0
Module: Mathlib.RingTheory.Jacobson.Radical

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
m : Ideal R
inst✝ : m.IsMaximal
⊢ sInf {I | I.IsMaximal} ≤ m

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Ring R
m : Ideal R
inst✝ : m.IsMaximal
⊢ sInf {I | I.IsMaximal} ≤ m
Module: Mathlib.RingTheory.Jacobson.Radical

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
f : R ⧸ I →ₛₗ[Ideal.Quotient.mk I] R ⧸ I := { toAddHom := AddHom.id (R ⧸ I), map_smul' := ⋯ }
⊢ ↑(Submodule.map f (Module.jacobson R (R ⧸ I))) = ↑(Module.jacobson R (R ⧸ I))

pp: R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
f : R ⧸ I →ₛₗ[Ideal.Quotient.mk I] R ⧸ I := { toAddHom := AddHom.id (R ⧸ I), map_smul' := ⋯ }
⊢ f.ker ≤ Module.jacobson R (R ⧸ I)

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
f : R ⧸ I →ₛₗ[Ideal.Quotient.mk I] R ⧸ I := { toAddHom := AddHom.id (R ⧸ I), map_smul' := ⋯ }
⊢ ↑(Submodule.map f (Module.jacobson R (R ⧸ I))) = ↑(Module.jacobson R (R ⧸ I))

R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
f : R ⧸ I →ₛₗ[Ideal.Quotient.mk I] R ⧸ I := { toAddHom := AddHom.id (R ⧸ I), map_smul' := ⋯ }
⊢ f.ker ≤ Module.jacobson R (R ⧸ I)
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: case refine_1
S : Type u_4
T : Type u_5
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra S T
R : Type u_6
inst✝³ : CommRing R
inst✝² : Algebra R S
inst✝¹ : Algebra R T
inst✝ : IsScalarTower R S T
h₁ : Function.Surjective ⇑(algebraMap S T)
h₂ : Function.Surjective ⇑(algebraMap R S)
r : R
hr : IsUnit ((algebraMap R T) r)
n : ℕ
hn : r ^ n • RingHom.ker (algebraMap R T) ≤ RingHom.ker (algebraMap R S)
⊢ IsUnit ((algebraMap R T) r)

Tactic:
]

Goal state after:
case refine_1
S : Type u_4
T : Type u_5
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra S T
R : Type u_6
inst✝³ : CommRing R
inst✝² : Algebra R S
inst✝¹ : Algebra R T
inst✝ : IsScalarTower R S T
h₁ : Function.Surjective ⇑(algebraMap S T)
h₂ : Function.Surjective ⇑(algebraMap R S)
r : R
hr : IsUnit ((algebraMap R T) r)
n : ℕ
hn : r ^ n • RingHom.ker (algebraMap R T) ≤ RingHom.ker (algebraMap R S)
⊢ IsUnit ((algebraMap R T) r)
Module: Mathlib.RingTheory.Jacobson.Radical

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
le : (Submodule.mkQ I).ker ≤ jacobson R
⊢ (Ideal.Quotient.mk I).toSemilinearMap.ker ≤ Module.jacobson R R

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
le : (Submodule.mkQ I).ker ≤ jacobson R
⊢ (Ideal.Quotient.mk I).toSemilinearMap.ker ≤ Module.jacobson R R
Module: Mathlib.RingTheory.Jacobson.Radical

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
h : ↑(jacobson (R ⧸ I)) ⊆ ↑⊥
⊢ ↑(Module.jacobson R (R ⧸ I)) ⊆ ↑⊥

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
h : ↑(jacobson (R ⧸ I)) ⊆ ↑⊥
⊢ ↑(Module.jacobson R (R ⧸ I)) ⊆ ↑⊥
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: R : Type u
inst✝ : Ring R
I : Ideal R
x : R
hx : x ∈ I.jacobson
y : R
hxy : I ⊔ span {y * x + 1} = ⊤
p : R
hpi : p ∈ I
q : R
hq : q ∈ span {y * x + 1}
hpq : p + q = 1
r : R
hr : r * (y * x + 1) = q
⊢ -p ∈ I

Tactic:
]

Goal state after:
R : Type u
inst✝ : Ring R
I : Ideal R
x : R
hx : x ∈ I.jacobson
y : R
hxy : I ⊔ span {y * x + 1} = ⊤
p : R
hpi : p ∈ I
q : R
hq : q ∈ span {y * x + 1}
hpq : p + q = 1
r : R
hr : r * (y * x + 1) = q
⊢ -p ∈ I
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: R : Type u
inst✝ : Ring R
I : Ideal R
x : R
hx : ∀ (y : R), ∃ z, z * y * x + z - 1 ∈ I
M : Ideal R
x✝ : M ∈ {J | I ≤ J ∧ J.IsMaximal}
him : I ≤ M
hm : M.IsMaximal
hxm : x ∉ M
y i : R
hi : i ∈ M
df : y * x + i = 1
z : R
hz : z * -y * x + z - 1 ∈ I
⊢ z * i ∈ M

Tactic:
]

Goal state after:
R : Type u
inst✝ : Ring R
I : Ideal R
x : R
hx : ∀ (y : R), ∃ z, z * y * x + z - 1 ∈ I
M : Ideal R
x✝ : M ∈ {J | I ≤ J ∧ J.IsMaximal}
him : I ≤ M
hm : M.IsMaximal
hxm : x ∉ M
y i : R
hi : i ∈ M
df : y * x + i = 1
z : R
hz : z * -y * x + z - 1 ∈ I
⊢ z * i ∈ M
Module: Mathlib.RingTheory.Jacobson.Radical

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
h : ↑(jacobson (R ⧸ I)) ⊆ ↑⊥
⊢ ↑(jacobson (R ⧸ I)) ⊆ ↑⊥

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
h : ↑(jacobson (R ⧸ I)) ⊆ ↑⊥
⊢ ↑(jacobson (R ⧸ I)) ⊆ ↑⊥
Module: Mathlib.RingTheory.Jacobson.Radical

Goal state before:
pp: R : Type u_1
inst✝² : Ring R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
ne_bot : N ≠ ⊥
fg : Module.Finite R ↥N
⊢ Ring.jacobson R • N < N

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : Ring R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
ne_bot : N ≠ ⊥
fg : Module.Finite R ↥N
⊢ Ring.jacobson R • N < N
Module: Mathlib.RingTheory.Jacobson.Radical

Goal state before:
pp: R : Type u_1
inst✝² : Ring R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
ne_bot : Nontrivial ↥N
fg : Module.Finite R ↥N
⊢ Ring.jacobson R • N < N

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : Ring R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
ne_bot : Nontrivial ↥N
fg : Module.Finite R ↥N
⊢ Ring.jacobson R • N < N
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: case h
R : Type u
inst✝ : Ring R
I : Ideal R
r : R
h : r ∈ I.jacobson
s : R
hs : s * 1 * r + s - 1 ∈ I
⊢ s * r + s - 1 ∈ I

Tactic:
]

Goal state after:
case h
R : Type u
inst✝ : Ring R
I : Ideal R
r : R
h : r ∈ I.jacobson
s : R
hs : s * 1 * r + s - 1 ∈ I
⊢ s * r + s - 1 ∈ I
Module: Mathlib.RingTheory.Jacobson.Radical

Goal state before:
pp: case e'_3
R : Type u_1
inst✝² : Ring R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
ne_bot : Nontrivial ↥N
fg : Module.Finite R ↥N
⊢ Ring.jacobson R • N = Ring.jacobson R • Submodule.map N.subtype ⊤

Tactic:
]

Goal state after:
case e'_3
R : Type u_1
inst✝² : Ring R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
ne_bot : Nontrivial ↥N
fg : Module.Finite R ↥N
⊢ Ring.jacobson R • N = Ring.jacobson R • Submodule.map N.subtype ⊤
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: case refine_2
S : Type u_4
T : Type u_5
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra S T
R : Type u_6
inst✝³ : CommRing R
inst✝² : Algebra R S
inst✝¹ : Algebra R T
inst✝ : IsScalarTower R S T
h₁ : Function.Surjective ⇑(algebraMap S T)
h₂ : Function.Surjective ⇑(algebraMap R S)
r : R
hr : IsUnit ((algebraMap R T) r)
n : ℕ
hn : r ^ n • RingHom.ker (algebraMap R T) ≤ RingHom.ker (algebraMap R S)
⊢ Ideal.map (algebraMap R S) (r ^ n • RingHom.ker (algebraMap R T)) ≤ ⊥

Tactic:
]

Goal state after:
case refine_2
S : Type u_4
T : Type u_5
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra S T
R : Type u_6
inst✝³ : CommRing R
inst✝² : Algebra R S
inst✝¹ : Algebra R T
inst✝ : IsScalarTower R S T
h₁ : Function.Surjective ⇑(algebraMap S T)
h₂ : Function.Surjective ⇑(algebraMap R S)
r : R
hr : IsUnit ((algebraMap R T) r)
n : ℕ
hn : r ^ n • RingHom.ker (algebraMap R T) ≤ RingHom.ker (algebraMap R S)
⊢ Ideal.map (algebraMap R S) (r ^ n • RingHom.ker (algebraMap R T)) ≤ ⊥
Module: Mathlib.RingTheory.Jacobson.Radical

Goal state before:
pp: case e'_3
R : Type u_1
inst✝² : Ring R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
ne_bot : Nontrivial ↥N
fg : Module.Finite R ↥N
⊢ Ring.jacobson R • N = Ring.jacobson R • N

Tactic:
]

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

Goal state before:
pp: case mpr
R : Type u
inst✝ : Ring R
I : Ideal R
M : Set (Ideal R)
hM : ∀ J ∈ M, J.IsMaximal ∨ J = ⊤
hInf : I = sInf M
x : R
hx : x ∈ I.jacobson
⊢ ∀ ⦃I : Ideal R⦄, I ∈ M → x ∈ I

Tactic:
]

Goal state after:
case mpr
R : Type u
inst✝ : Ring R
I : Ideal R
M : Set (Ideal R)
hM : ∀ J ∈ M, J.IsMaximal ∨ J = ⊤
hInf : I = sInf M
x : R
hx : x ∈ I.jacobson
⊢ ∀ ⦃I : Ideal R⦄, I ∈ M → x ∈ I
Module: Mathlib.RingTheory.Jacobson.Radical

Goal state before:
pp: case e'_4
R : Type u_1
inst✝² : Ring R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
ne_bot : Nontrivial ↥N
fg : Module.Finite R ↥N
⊢ N = N

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: case refine_2
S : Type u_4
T : Type u_5
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra S T
R : Type u_6
inst✝³ : CommRing R
inst✝² : Algebra R S
inst✝¹ : Algebra R T
inst✝ : IsScalarTower R S T
h₁ : Function.Surjective ⇑(algebraMap S T)
h₂ : Function.Surjective ⇑(algebraMap R S)
r : R
hr : IsUnit ((algebraMap R T) r)
n : ℕ
hn : Ideal.map (algebraMap R S) (r ^ n • RingHom.ker (algebraMap R T)) ≤ ⊥
⊢ Ideal.map (algebraMap R S) (r ^ n • RingHom.ker (algebraMap R T)) ≤ ⊥

Tactic:
]

Goal state after:
case refine_2
S : Type u_4
T : Type u_5
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra S T
R : Type u_6
inst✝³ : CommRing R
inst✝² : Algebra R S
inst✝¹ : Algebra R T
inst✝ : IsScalarTower R S T
h₁ : Function.Surjective ⇑(algebraMap S T)
h₂ : Function.Surjective ⇑(algebraMap R S)
r : R
hr : IsUnit ((algebraMap R T) r)
n : ℕ
hn : Ideal.map (algebraMap R S) (r ^ n • RingHom.ker (algebraMap R T)) ≤ ⊥
⊢ Ideal.map (algebraMap R S) (r ^ n • RingHom.ker (algebraMap R T)) ≤ ⊥
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: case mpr
R : Type u
inst✝ : Ring R
I✝ : Ideal R
M : Set (Ideal R)
hM : ∀ J ∈ M, J.IsMaximal ∨ J = ⊤
hInf : I✝ = sInf M
x : R
hx : x ∈ I✝.jacobson
I : Ideal R
⊢ I ∈ M → x ∈ I

Tactic:
hI

Goal state after:
case mpr
R : Type u
inst✝ : Ring R
I✝ : Ideal R
M : Set (Ideal R)
hM : ∀ J ∈ M, J.IsMaximal ∨ J = ⊤
hInf : I✝ = sInf M
x : R
hx : x ∈ I✝.jacobson
I : Ideal R
hI : I ∈ M
⊢ x ∈ I
Module: Mathlib.RingTheory.LocalRing.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : IsLocalRing R
a b : R
u : Rˣ
hu : ↑u⁻¹ * a + ↑u⁻¹ * b = 1
⊢ IsUnit a ∨ IsUnit b

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : IsLocalRing R
a b : R
u : Rˣ
hu : ↑u⁻¹ * a + ↑u⁻¹ * b = 1
⊢ IsUnit a ∨ IsUnit b
Module: Mathlib.RingTheory.LocalRing.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : Ring S
inst✝ : Nontrivial S
f : R →+* S
hf : Function.Surjective ⇑f
a : R
⊢ IsUnit (1 - a) → IsUnit (f (1 - a))

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : Ring S
inst✝ : Nontrivial S
f : R →+* S
hf : Function.Surjective ⇑f
a : R
⊢ IsUnit (1 - a) → IsUnit (f (1 - a))
Module: Mathlib.RingTheory.LocalRing.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
K : Type u_3
inst✝ : Field K
a : K
h : a = 0
⊢ IsUnit 1

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
K : Type u_3
inst✝ : Field K
a : K
h : a = 0
⊢ IsUnit 1
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: case mp
R : Type u
inst✝ : Ring R
I : Ideal R
h : I.jacobson = I
x : R
hx : ¬∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsMaximal} → x ∈ I_1
⊢ ∃ M, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M

Tactic:
]

Goal state after:
case mp
R : Type u
inst✝ : Ring R
I : Ideal R
h : I.jacobson = I
x : R
hx : ¬∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsMaximal} → x ∈ I_1
⊢ ∃ M, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: case mpr
R : Type u
inst✝ : Ring R
I : Ideal R
h : ∀ x ∉ I, ∃ M, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
x : R
hx : x ∉ I
⊢ ¬∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsMaximal} → x ∈ I_1

Tactic:
]

Goal state after:
case mpr
R : Type u
inst✝ : Ring R
I : Ideal R
h : ∀ x ∉ I, ∃ M, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
x : R
hx : x ∉ I
⊢ ¬∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsMaximal} → x ∈ I_1
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommSemiring R
S : Type u_2
inst✝ : CommSemiring S
f : R →+* S
a b : R
H : ∀ (a_1 : S), ∃ b_1 m, f b_1 = f (a * b) ^ m * a_1
⊢ ∀ (a_1 : Away (f a)), ∃ b_1 m, (awayMap f a) b_1 = (awayMap f a) ((algebraMap R (Away a)) b) ^ m * a_1

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommSemiring R
S : Type u_2
inst✝ : CommSemiring S
f : R →+* S
a b : R
H : ∀ (a_1 : S), ∃ b_1 m, f b_1 = f (a * b) ^ m * a_1
⊢ ∀ (a_1 : Away (f a)), ∃ b_1 m, (awayMap f a) b_1 = (awayMap f a) ((algebraMap R (Away a)) b) ^ m * a_1
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: case refine_2
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ J.IsMaximal}, RingHom.ker f ≤ J
j : Ideal S
hj : j ∈ map f '' {J | I ≤ J ∧ J.IsMaximal}
J : Ideal R
hJ : J ∈ {J | I ≤ J ∧ J.IsMaximal} ∧ map f J = j
⊢ map f J ∈ insert ⊤ {J | map f I ≤ J ∧ J.IsMaximal}

Tactic:
]

Goal state after:
case refine_2
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ J.IsMaximal}, RingHom.ker f ≤ J
j : Ideal S
hj : j ∈ map f '' {J | I ≤ J ∧ J.IsMaximal}
J : Ideal R
hJ : J ∈ {J | I ≤ J ∧ J.IsMaximal} ∧ map f J = j
⊢ map f J ∈ insert ⊤ {J | map f I ≤ J ∧ J.IsMaximal}
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
s : Set R
span_eq : Ideal.span s = ⊤
x y : R
eq : (algebraMap R ((a : ↑s) → Away ↑a)) x = (algebraMap R ((a : ↑s) → Away ↑a)) y
⊢ Module.eqIdeal R x y = ⊤

Tactic:
by_contra ne

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
s : Set R
span_eq : Ideal.span s = ⊤
x y : R
eq : (algebraMap R ((a : ↑s) → Away ↑a)) x = (algebraMap R ((a : ↑s) → Away ↑a)) y
ne : ¬Module.eqIdeal R x y = ⊤
⊢ False
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: case refine_1
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
f : R →+* S
hf : Function.Surjective ⇑f
K : Ideal S
⊢ ⨅ I ∈ comap f '' insert ⊤ {J | K ≤ J ∧ J.IsMaximal}, I ≤ ⨅ a ∈ {J | comap f K ≤ J ∧ J.IsMaximal}, a

Tactic:
]

Goal state after:
case refine_1
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
f : R →+* S
hf : Function.Surjective ⇑f
K : Ideal S
⊢ ⨅ I ∈ comap f '' insert ⊤ {J | K ≤ J ∧ J.IsMaximal}, I ≤ ⨅ a ∈ {J | comap f K ≤ J ∧ J.IsMaximal}, a
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: R : Type u
inst✝ : Ring R
I J : Ideal R
h : I ≤ J
x : R
hx : ∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsMaximal} → x ∈ I_1
⊢ ∀ ⦃I : Ideal R⦄, I ∈ {J_1 | J ≤ J_1 ∧ J_1.IsMaximal} → x ∈ I

Tactic:
]

Goal state after:
R : Type u
inst✝ : Ring R
I J : Ideal R
h : I ≤ J
x : R
hx : ∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsMaximal} → x ∈ I_1
⊢ ∀ ⦃I : Ideal R⦄, I ∈ {J_1 | J ≤ J_1 ∧ J_1.IsMaximal} → x ∈ I
Module: Mathlib.RingTheory.Localization.Away.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
s : Finset R
f : (a : ↑↑s) → Away ↑a
h : ∀ (a b : ↑↑s), (Away.awayToAwayRight ↑a ↑b) (f a) = (Away.awayToAwayLeft ↑b ↑a) (f b)
mem : 1 ∈ Ideal.span ↑s
span_eq : Ideal.span ↑s = ⊤
n : ↑↑s → ℕ
r✝ : ↑↑s → R
eq : ∀ (a : ↑↑s), f a * (algebraMap R (Away ↑a)) ↑a ^ n a = (algebraMap R (Away ↑a)) (r✝ a)
N : ℕ := s.attach.sup n
r : ↑↑s → R := fun a ↦ ↑a ^ (N - n a) * r✝ a
a : ↑↑s
⊢ f a * (algebraMap R (Away ↑a)) (↑a ^ N) = f a * (algebraMap R (Away ↑a)) (↑a ^ s.attach.sup n)

Tactic:
]

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

Goal state before:
pp: R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
I✝ I : Ideal R
inst✝ : I.IsTwoSided
x r : R
xJ : x ∈ I.jacobson
𝔪 : Ideal R
𝔪_mem : 𝔪 ∈ {J | I ≤ J ∧ J.IsMaximal}
r𝔪 : r ∉ 𝔪
𝔪₀ : Ideal R := Submodule.comap (DistribSMul.toLinearMap R R (MulOpposite.op r)) 𝔪
I𝔪₀ : I ≤ 𝔪₀
J : Ideal R
b : R
𝔪₀J : 𝔪₀ ≤ J
b𝔪₀ : b ∉ 𝔪₀
bJ : b ∈ J
K : Ideal R := span {b * r} ⊔ 𝔪
s y : R
y𝔪 : y ∈ 𝔪
sbyr : y = r - s * (b * r)
⊢ 1 - s * b ∈ 𝔪₀

Tactic:
]

Goal state after:
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
I✝ I : Ideal R
inst✝ : I.IsTwoSided
x r : R
xJ : x ∈ I.jacobson
𝔪 : Ideal R
𝔪_mem : 𝔪 ∈ {J | I ≤ J ∧ J.IsMaximal}
r𝔪 : r ∉ 𝔪
𝔪₀ : Ideal R := Submodule.comap (DistribSMul.toLinearMap R R (MulOpposite.op r)) 𝔪
I𝔪₀ : I ≤ 𝔪₀
J : Ideal R
b : R
𝔪₀J : 𝔪₀ ≤ J
b𝔪₀ : b ∉ 𝔪₀
bJ : b ∈ J
K : Ideal R := span {b * r} ⊔ 𝔪
s y : R
y𝔪 : y ∈ 𝔪
sbyr : y = r - s * (b * r)
⊢ 1 - s * b ∈ 𝔪₀
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
I✝ I : Ideal R
inst✝ : I.IsTwoSided
x r : R
xJ : x ∈ I.jacobson
𝔪 : Ideal R
𝔪_mem : 𝔪 ∈ {J | I ≤ J ∧ J.IsMaximal}
r𝔪 : r ∉ 𝔪
𝔪₀ : Ideal R := Submodule.comap (DistribSMul.toLinearMap R R (MulOpposite.op r)) 𝔪
I𝔪₀ : I ≤ 𝔪₀
J : Ideal R
b : R
𝔪₀J : 𝔪₀ ≤ J
b𝔪₀ : b ∉ 𝔪₀
bJ : b ∈ J
K : Ideal R := span {b * r} ⊔ 𝔪
s y : R
y𝔪 : r - s * b * r ∈ 𝔪
sbyr : y = r - s * (b * r)
⊢ 1 - s * b ∈ 𝔪₀

Tactic:
]

Goal state after:
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
I✝ I : Ideal R
inst✝ : I.IsTwoSided
x r : R
xJ : x ∈ I.jacobson
𝔪 : Ideal R
𝔪_mem : 𝔪 ∈ {J | I ≤ J ∧ J.IsMaximal}
r𝔪 : r ∉ 𝔪
𝔪₀ : Ideal R := Submodule.comap (DistribSMul.toLinearMap R R (MulOpposite.op r)) 𝔪
I𝔪₀ : I ≤ 𝔪₀
J : Ideal R
b : R
𝔪₀J : 𝔪₀ ≤ J
b𝔪₀ : b ∉ 𝔪₀
bJ : b ∈ J
K : Ideal R := span {b * r} ⊔ 𝔪
s y : R
y𝔪 : r - s * b * r ∈ 𝔪
sbyr : y = r - s * (b * r)
⊢ 1 - s * b ∈ 𝔪₀
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
r : R
h : r - 1 ∈ ⊥.jacobson
s : R
hs : r * s = 1
⊢ IsUnit r

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommRing R
r : R
h : r - 1 ∈ ⊥.jacobson
s : R
hs : r * s = 1
⊢ IsUnit r
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
x : R
hx : x ∈ ⊥.jacobson
y z : R
hz : z * y * x + z - 1 ∈ ⊥
⊢ z * y * x + z - 1 = 0

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommRing R
x : R
hx : x ∈ ⊥.jacobson
y z : R
hz : z * y * x + z - 1 ∈ ⊥
⊢ z * y * x + z - 1 = 0
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
x : R
h : ∀ (y : R), IsUnit (x * y + 1)
y b : R
hb : (x * y + 1) * b = 1
⊢ b * y * x + b - (x * y + 1) * b = 0

Tactic:
by ring

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

Goal state before:
pp: case mp
R : Type u
inst✝ : CommRing R
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : (map (Quotient.mk I) I).jacobson = map (Quotient.mk I) I
⊢ ⊥.jacobson = ⊥

Tactic:
]

Goal state after:
case mp
R : Type u
inst✝ : CommRing R
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : (map (Quotient.mk I) I).jacobson = map (Quotient.mk I) I
⊢ ⊥.jacobson = ⊥
Module: Mathlib.RingTheory.Jacobson.Ideal

Goal state before:
pp: case mpr
R : Type u
inst✝ : CommRing R
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : (RingHom.ker (Quotient.mk I)).jacobson = RingHom.ker (Quotient.mk I)
⊢ I.jacobson = I

Tactic:
]

Goal state after:
case mpr
R : Type u
inst✝ : CommRing R
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : (RingHom.ker (Quotient.mk I)).jacobson = RingHom.ker (Quotient.mk I)
⊢ I.jacobson = I