mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.Morphisms.Descent

Goal state before:
pp: case inr
P P' : MorphismProperty Scheme
Q Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
H₂ : ∀ {R S : CommRingCat} {f : R ⟶ S}, P' (Spec.map f) → Q' (CommRingCat.Hom.hom f)
hP : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q)
inst✝ : MorphismProperty.DescendsAlong (@IsAffineHom) P'
hQ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
hQQ' : RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦ Q'
R S : CommRingCat
Y : Scheme
φ : R ⟶ S
g : Y ⟶ Spec R
h : P' (Spec.map φ)
hf : P (pullback.fst (Spec.map φ) g)
this✝ : IsAffine Y
this : ∀ {Y : Scheme} (g : Y ⟶ Spec R), P (pullback.fst (Spec.map φ) g) → IsAffine Y → (∃ S, Y = Spec S) → P g
hY : ¬∃ S, Y = Spec S
⊢ P (Y.isoSpec.inv ≫ g)

Tactic:
]

Goal state after:
case inr
P P' : MorphismProperty Scheme
Q Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
H₂ : ∀ {R S : CommRingCat} {f : R ⟶ S}, P' (Spec.map f) → Q' (CommRingCat.Hom.hom f)
hP : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q)
inst✝ : MorphismProperty.DescendsAlong (@IsAffineHom) P'
hQ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
hQQ' : RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦ Q'
R S : CommRingCat
Y : Scheme
φ : R ⟶ S
g : Y ⟶ Spec R
h : P' (Spec.map φ)
hf : P (pullback.fst (Spec.map φ) g)
this✝ : IsAffine Y
this : ∀ {Y : Scheme} (g : Y ⟶ Spec R), P (pullback.fst (Spec.map φ) g) → IsAffine Y → (∃ S, Y = Spec S) → P g
hY : ¬∃ S, Y = Spec S
⊢ P (Y.isoSpec.inv ≫ g)
Module: Mathlib.RingTheory.RingHom.QuasiFinite

Goal state before:
pp: R S : Type u_6
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : ↥s), (fun {R S} [CommRing R] [CommRing S] ↦ QuasiFinite) ((algebraMap S (Localization.Away ↑r)).comp f)
algInst✝ : Algebra R S := f.toAlgebra
P : Ideal R
x✝ : P.IsPrime
this : ∀ (r : ↥s), Module.Finite P.ResidueField (P.Fiber (Localization.Away ↑r))
φ : (r : ↥s) → P.Fiber S →ₐ[P.ResidueField] P.Fiber (Localization.Away ↑r) :=
  fun r ↦
    Algebra.TensorProduct.map (AlgHom.id P.ResidueField P.ResidueField)
      (IsScalarTower.toAlgHom R S (Localization.Away ↑r))
⊢ Module.Finite P.ResidueField (P.Fiber S)

Tactic:
let f : P.Fiber S →ₐ[P.ResidueField] Π r : s, (P.Fiber (Localization.Away r.1)) :=
    AlgHom.pi φ

Goal state after:
R S : Type u_6
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : ↥s), (fun {R S} [CommRing R] [CommRing S] ↦ QuasiFinite) ((algebraMap S (Localization.Away ↑r)).comp f✝)
algInst✝ : Algebra R S := f✝.toAlgebra
P : Ideal R
x✝ : P.IsPrime
this : ∀ (r : ↥s), Module.Finite P.ResidueField (P.Fiber (Localization.Away ↑r))
φ : (r : ↥s) → P.Fiber S →ₐ[P.ResidueField] P.Fiber (Localization.Away ↑r) :=
  fun r ↦
    Algebra.TensorProduct.map (AlgHom.id P.ResidueField P.ResidueField)
      (IsScalarTower.toAlgHom R S (Localization.Away ↑r))
f : P.Fiber S →ₐ[P.ResidueField] (r : ↥s) → P.Fiber (Localization.Away ↑r) := AlgHom.pi φ
⊢ Module.Finite P.ResidueField (P.Fiber S)
Module: Mathlib.RingTheory.RingHom.QuasiFinite

Goal state before:
pp: R S : Type u_6
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : ↥s), (fun {R S} [CommRing R] [CommRing S] ↦ QuasiFinite) ((algebraMap S (Localization.Away ↑r)).comp f✝)
algInst✝ : Algebra R S := f✝.toAlgebra
P : Ideal R
x✝ : P.IsPrime
this✝ : ∀ (r : ↥s), Module.Finite P.ResidueField (P.Fiber (Localization.Away ↑r))
φ : (r : ↥s) → P.Fiber S →ₐ[P.ResidueField] P.Fiber (Localization.Away ↑r) :=
  fun r ↦
    Algebra.TensorProduct.map (AlgHom.id P.ResidueField P.ResidueField)
      (IsScalarTower.toAlgHom R S (Localization.Away ↑r))
f : P.Fiber S →ₐ[P.ResidueField] (r : ↥s) → P.Fiber (Localization.Away ↑r) := AlgHom.pi φ
this : IsNoetherian P.ResidueField ((r : ↥s) → P.Fiber (Localization.Away ↑r))
⊢ ∀ (a : P.Fiber S), f a = 0 → a = 0

Tactic:
]

Goal state after:
R S : Type u_6
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
H : ∀ (r : ↥s), (fun {R S} [CommRing R] [CommRing S] ↦ QuasiFinite) ((algebraMap S (Localization.Away ↑r)).comp f✝)
algInst✝ : Algebra R S := f✝.toAlgebra
P : Ideal R
x✝ : P.IsPrime
this✝ : ∀ (r : ↥s), Module.Finite P.ResidueField (P.Fiber (Localization.Away ↑r))
φ : (r : ↥s) → P.Fiber S →ₐ[P.ResidueField] P.Fiber (Localization.Away ↑r) :=
  fun r ↦
    Algebra.TensorProduct.map (AlgHom.id P.ResidueField P.ResidueField)
      (IsScalarTower.toAlgHom R S (Localization.Away ↑r))
f : P.Fiber S →ₐ[P.ResidueField] (r : ↥s) → P.Fiber (Localization.Away ↑r) := AlgHom.pi φ
this : IsNoetherian P.ResidueField ((r : ↥s) → P.Fiber (Localization.Away ↑r))
⊢ ∀ (a : P.Fiber S), f a = 0 → a = 0
Module: Mathlib.RingTheory.Conductor

Goal state before:
pp: case mpr
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
L : Type u_3
inst✝⁴ : CommRing L
inst✝³ : Algebra S L
inst✝² : Algebra R L
inst✝¹ : IsScalarTower R S L
inst✝ : FaithfulSMul S L
x : S
y✝ : L
H : ∀ (z : S), ∃ x_1 ∈ R[x], (algebraMap S L) x_1 = y✝ * (algebraMap S L) z
y : S
left✝ : y ∈ R[x]
e : (algebraMap S L) y = y✝
⊢ ∃ y ∈ conductor R x, (algebraMap S L) y = y✝

Tactic:
]

Goal state after:
case mpr
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
L : Type u_3
inst✝⁴ : CommRing L
inst✝³ : Algebra S L
inst✝² : Algebra R L
inst✝¹ : IsScalarTower R S L
inst✝ : FaithfulSMul S L
x : S
y✝ : L
H : ∀ (z : S), ∃ x_1 ∈ R[x], (algebraMap S L) x_1 = y✝ * (algebraMap S L) z
y : S
left✝ : y ∈ R[x]
e : (algebraMap S L) y = y✝
⊢ ∃ y ∈ conductor R x, (algebraMap S L) y = y✝
Module: Mathlib.AlgebraicGeometry.Morphisms.Descent

Goal state before:
pp: case inr
P P' : MorphismProperty Scheme
Q Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
H₂ : ∀ {R S : CommRingCat} {f : R ⟶ S}, P' (Spec.map f) → Q' (CommRingCat.Hom.hom f)
hP : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q)
inst✝ : MorphismProperty.DescendsAlong (@IsAffineHom) P'
hQ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
hQQ' : RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦ Q'
R S : CommRingCat
Y : Scheme
φ : R ⟶ S
g : Y ⟶ Spec R
h : P' (Spec.map φ)
hf : P (pullback.fst (Spec.map φ) g)
this✝ : IsAffine Y
this : ∀ {Y : Scheme} (g : Y ⟶ Spec R), P (pullback.fst (Spec.map φ) g) → IsAffine Y → (∃ S, Y = Spec S) → P g
hY : ¬∃ S, Y = Spec S
heq :
  pullback.fst (Spec.map φ) (Y.isoSpec.inv ≫ g) =
    pullback.map (Spec.map φ) (Y.isoSpec.inv ≫ g) (Spec.map φ) g (𝟙 (Spec S)) Y.isoSpec.inv (𝟙 (Spec R)) ⋯ ⋯ ≫
      pullback.fst (Spec.map φ) g
⊢ P (pullback.fst (Spec.map φ) g)

Tactic:
]

Goal state after:
case inr
P P' : MorphismProperty Scheme
Q Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
H₂ : ∀ {R S : CommRingCat} {f : R ⟶ S}, P' (Spec.map f) → Q' (CommRingCat.Hom.hom f)
hP : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q)
inst✝ : MorphismProperty.DescendsAlong (@IsAffineHom) P'
hQ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
hQQ' : RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦ Q'
R S : CommRingCat
Y : Scheme
φ : R ⟶ S
g : Y ⟶ Spec R
h : P' (Spec.map φ)
hf : P (pullback.fst (Spec.map φ) g)
this✝ : IsAffine Y
this : ∀ {Y : Scheme} (g : Y ⟶ Spec R), P (pullback.fst (Spec.map φ) g) → IsAffine Y → (∃ S, Y = Spec S) → P g
hY : ¬∃ S, Y = Spec S
heq :
  pullback.fst (Spec.map φ) (Y.isoSpec.inv ≫ g) =
    pullback.map (Spec.map φ) (Y.isoSpec.inv ≫ g) (Spec.map φ) g (𝟙 (Spec S)) Y.isoSpec.inv (𝟙 (Spec R)) ⋯ ⋯ ≫
      pullback.fst (Spec.map φ) g
⊢ P (pullback.fst (Spec.map φ) g)
Module: Mathlib.RingTheory.Conductor

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
L : Type u_3
inst✝⁴ : CommRing L
inst✝³ : Algebra S L
inst✝² : Algebra R L
inst✝¹ : IsScalarTower R S L
inst✝ : FaithfulSMul S L
x : S
y : L
⊢ (∀ (z : S), y * (algebraMap S L) z ∈ R[(IsScalarTower.toAlgHom R S L) x]) ↔
    ∀ (z : S), y * (algebraMap S L) z ∈ R[(algebraMap S L) x]

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
L : Type u_3
inst✝⁴ : CommRing L
inst✝³ : Algebra S L
inst✝² : Algebra R L
inst✝¹ : IsScalarTower R S L
inst✝ : FaithfulSMul S L
x : S
y : L
⊢ (∀ (z : S), y * (algebraMap S L) z ∈ R[(IsScalarTower.toAlgHom R S L) x]) ↔
    ∀ (z : S), y * (algebraMap S L) z ∈ R[(algebraMap S L) x]
Module: Mathlib.RingTheory.Algebraic.StronglyTranscendental

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
M : Submonoid R
inst✝¹ : IsLocalization M S
inst✝ : IsScalarTower R S T
x : T
h : IsStronglyTranscendental R x
t : T
p : S[X]
hp : (aeval x) p * t = 0
a : R
ha₁ : a ∈ M
ha₂ : map (algebraMap R S) (IsLocalization.integerNormalization M p) = a • p
H : (aeval x) (IsLocalization.integerNormalization M p) = a • (aeval x) p
this : map (algebraMap S T) (a • p) * C t = 0
⊢ map (algebraMap S T) p * C t = 0

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
M : Submonoid R
inst✝¹ : IsLocalization M S
inst✝ : IsScalarTower R S T
x : T
h : IsStronglyTranscendental R x
t : T
p : S[X]
hp : (aeval x) p * t = 0
a : R
ha₁ : a ∈ M
ha₂ : map (algebraMap R S) (IsLocalization.integerNormalization M p) = a • p
H : (aeval x) (IsLocalization.integerNormalization M p) = a • (aeval x) p
this : map (algebraMap S T) (a • p) * C t = 0
⊢ map (algebraMap S T) p * C t = 0
Module: Mathlib.RingTheory.Conductor

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
hz' : ∃ l ∈ Finsupp.supported S S ↑I, (Finsupp.linearCombination S ⇑(algebraMap R S)) l = z
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
hz' : ∃ l ∈ Finsupp.supported S S ↑I, (Finsupp.linearCombination S ⇑(algebraMap R S)) l = z
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)
Module: Mathlib.RingTheory.Algebraic.StronglyTranscendental

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
M : Submonoid R
inst✝¹ : IsLocalization M S
inst✝ : IsScalarTower R S T
x : T
h : IsStronglyTranscendental R x
t : T
p : S[X]
hp : (aeval x) p * t = 0
a : R
ha₁ : a ∈ M
ha₂ : map (algebraMap R S) (IsLocalization.integerNormalization M p) = a • p
H : (aeval x) (IsLocalization.integerNormalization M p) = a • (aeval x) p
this : map (algebraMap S T) (a • p) * C t = 0
⊢ C ((algebraMap S T) ((algebraMap R S) ↑⟨a, ha₁⟩)) * (map (algebraMap S T) p * C t) =
    C ((algebraMap S T) ((algebraMap R S) ↑⟨a, ha₁⟩)) * 0

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
M : Submonoid R
inst✝¹ : IsLocalization M S
inst✝ : IsScalarTower R S T
x : T
h : IsStronglyTranscendental R x
t : T
p : S[X]
hp : (aeval x) p * t = 0
a : R
ha₁ : a ∈ M
ha₂ : map (algebraMap R S) (IsLocalization.integerNormalization M p) = a • p
H : (aeval x) (IsLocalization.integerNormalization M p) = a • (aeval x) p
this : map (algebraMap S T) (a • p) * C t = 0
⊢ C ((algebraMap S T) ((algebraMap R S) ↑⟨a, ha₁⟩)) * (map (algebraMap S T) p * C t) =
    C ((algebraMap S T) ((algebraMap R S) ↑⟨a, ha₁⟩)) * 0
Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal

Goal state before:
pp: case mul
R : Type u_3
inst✝ : CommRing R
I : Ideal R
P x y : R[X]
hx✝ : x ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}
hy✝ : y ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}
hx : ∀ (i : ℕ), x.coeff i ∈ I ^ i
hy : ∀ (i : ℕ), y.coeff i ∈ I ^ i
i : ℕ
⊢ ∑ x_1 ∈ Finset.antidiagonal i, x.coeff x_1.1 * y.coeff x_1.2 ∈ I ^ i

Tactic:
]

Goal state after:
case mul
R : Type u_3
inst✝ : CommRing R
I : Ideal R
P x y : R[X]
hx✝ : x ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}
hy✝ : y ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}
hx : ∀ (i : ℕ), x.coeff i ∈ I ^ i
hy : ∀ (i : ℕ), y.coeff i ∈ I ^ i
i : ℕ
⊢ ∑ x_1 ∈ Finset.antidiagonal i, x.coeff x_1.1 * y.coeff x_1.2 ∈ I ^ i
Module: Mathlib.RingTheory.Conductor

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)
Module: Mathlib.RingTheory.Conductor

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z
⊢ (l.sum fun a c ↦ c • (algebraMap R S) a * (algebraMap R S) p) ∈
    ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z
⊢ (l.sum fun a c ↦ c • (algebraMap R S) a * (algebraMap R S) p) ∈
    ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)
Module: Mathlib.RingTheory.Algebraic.StronglyTranscendental

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : Algebra S K
inst✝² : IsScalarTower R S K
inst✝¹ : FaithfulSMul R S
inst✝ : IsFractionRing S K
x : S
this : FaithfulSMul R K
⊢ Transcendental R ((algebraMap S K) x) ↔ Transcendental R ((algebraMap S K) x)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Algebraic.StronglyTranscendental

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : Algebra S K
inst✝² : IsScalarTower R S K
inst✝¹ : FaithfulSMul R S
inst✝ : FaithfulSMul S K
x : S
H : IsStronglyTranscendental R ((algebraMap S K) x)
this : FaithfulSMul R K
⊢ IsStronglyTranscendental R x

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
K : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : Algebra S K
inst✝² : IsScalarTower R S K
inst✝¹ : FaithfulSMul R S
inst✝ : FaithfulSMul S K
x : S
H : IsStronglyTranscendental R ((algebraMap S K) x)
this : FaithfulSMul R K
⊢ IsStronglyTranscendental R x
Module: Mathlib.RingTheory.IntegralClosure.GoingDown

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p q : R[X]
hp : p.Monic
hq : q.Monic
H : q ∣ p
i : ℕ
hi : i ≠ q.natDegree
⊢ ∀ ⦃I : Ideal R⦄, I ∈ {J | Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ J ∧ J.IsPrime} → q.coeff i ∈ I

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
p q : R[X]
hp : p.Monic
hq : q.Monic
H : q ∣ p
i : ℕ
hi : i ≠ q.natDegree
⊢ ∀ ⦃I : Ideal R⦄, I ∈ {J | Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ J ∧ J.IsPrime} → q.coeff i ∈ I
Module: Mathlib.RingTheory.Conductor

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z
a : R
ha : a ∈ I
⊢ ∃ x_1 ∈ ↑(Ideal.map (algebraMap R ↥R[x]) I),
    (algebraMap (↥R[x]) S) x_1 = (algebraMap R S) a * ((algebraMap R S) p * l a)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z
a : R
ha : a ∈ I
⊢ ∃ x_1 ∈ ↑(Ideal.map (algebraMap R ↥R[x]) I),
    (algebraMap (↥R[x]) S) x_1 = (algebraMap R S) a * ((algebraMap R S) p * l a)
Module: Mathlib.RingTheory.Algebraic.StronglyTranscendental

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsReduced S
x : S
hx : IsStronglyTranscendental R x
q : Ideal S
hq : q ∈ minimalPrimes S
u : S
p : R[X]
e : (aeval x) p * u ∈ q
⊢ map (algebraMap R (S ⧸ q)) p * C ((Ideal.Quotient.mk q) u) = 0

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsReduced S
x : S
hx : IsStronglyTranscendental R x
q : Ideal S
hq : q ∈ minimalPrimes S
u : S
p : R[X]
e : (aeval x) p * u ∈ q
⊢ map (algebraMap R (S ⧸ q)) p * C ((Ideal.Quotient.mk q) u) = 0
Module: Mathlib.RingTheory.Conductor

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z
a : R
ha : a ∈ I
⊢ (algebraMap R S) p * l a ∈ R[x]

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z
a : R
ha : a ∈ I
⊢ (algebraMap R S) p * l a ∈ R[x]
Module: Mathlib.RingTheory.IntegralClosure.GoingDown

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p q : R[X]
hp : p.Monic
hq : q.Monic
H : q ∣ p
i : ℕ
hi : i ≠ q.natDegree
P : Ideal R
hPJ : Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ P
hP : P.IsPrime
this : map (Ideal.Quotient.mk P) p = X ^ p.natDegree
j : ℕ
hj : j ≤ p.natDegree
a : (R ⧸ P)[X]ˣ
r : R ⧸ P
ha : map (Ideal.Quotient.mk P) q = X ^ j * C r
hr : IsUnit r
e : C r = ↑a⁻¹
⊢ q.coeff i ∈ P

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
p q : R[X]
hp : p.Monic
hq : q.Monic
H : q ∣ p
i : ℕ
hi : i ≠ q.natDegree
P : Ideal R
hPJ : Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ P
hP : P.IsPrime
this : map (Ideal.Quotient.mk P) p = X ^ p.natDegree
j : ℕ
hj : j ≤ p.natDegree
a : (R ⧸ P)[X]ˣ
r : R ⧸ P
ha : map (Ideal.Quotient.mk P) q = X ^ j * C r
hr : IsUnit r
e : C r = ↑a⁻¹
⊢ q.coeff i ∈ P
Module: Mathlib.RingTheory.Algebraic.StronglyTranscendental

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsReduced S
x : S
hx : IsStronglyTranscendental R x
q : Ideal S
hq : q ∈ minimalPrimes S
u : S
p : R[X]
e : (aeval x) p * u ∈ q
this✝¹ : q.IsPrime
this✝ : Ring.KrullDimLE 0 (Localization.AtPrime q)
this : Field (Localization.AtPrime q) := ⋯.toField
⊢ (algebraMap S (Localization.AtPrime q)) ((aeval x) p * u) ∈ Ideal.map (algebraMap S (Localization.AtPrime q)) q

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsReduced S
x : S
hx : IsStronglyTranscendental R x
q : Ideal S
hq : q ∈ minimalPrimes S
u : S
p : R[X]
e : (aeval x) p * u ∈ q
this✝¹ : q.IsPrime
this✝ : Ring.KrullDimLE 0 (Localization.AtPrime q)
this : Field (Localization.AtPrime q) := ⋯.toField
⊢ (algebraMap S (Localization.AtPrime q)) ((aeval x) p * u) ∈ Ideal.map (algebraMap S (Localization.AtPrime q)) q
Module: Mathlib.RingTheory.Conductor

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z
a : R
ha : a ∈ I
⊢ ⟨l a * (algebraMap R S) p, ⋯⟩ * (algebraMap R ↥R[x]) a ∈ ↑(Ideal.map (algebraMap R ↥R[x]) I)

Tactic:
]

Goal state after:
case refine_1
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z
a : R
ha : a ∈ I
⊢ ⟨l a * (algebraMap R S) p, ⋯⟩ * (algebraMap R ↥R[x]) a ∈ ↑(Ideal.map (algebraMap R ↥R[x]) I)
Module: Mathlib.RingTheory.Conductor

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z✝ : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z✝
lem :
  ∀ {a : R},
    a ∈ I →
      l a • (algebraMap R S) a * (algebraMap R S) p ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)
z : ↥R[x]
hz : z ∈ ↑(Ideal.map (algebraMap R ↥R[x]) I)
y : ↥R[x]
hy : y ∈ ↑(Ideal.map (algebraMap R ↥R[x]) I)
⊢ (algebraMap (↥R[x]) S) (z + y) ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)

Tactic:
]

Goal state after:
case refine_1
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z✝ : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z✝
lem :
  ∀ {a : R},
    a ∈ I →
      l a • (algebraMap R S) a * (algebraMap R S) p ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)
z : ↥R[x]
hz : z ∈ ↑(Ideal.map (algebraMap R ↥R[x]) I)
y : ↥R[x]
hy : y ∈ ↑(Ideal.map (algebraMap R ↥R[x]) I)
⊢ (algebraMap (↥R[x]) S) (z + y) ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)
Module: Mathlib.RingTheory.IntegralClosure.GoingDown

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : IsDomain S
inst✝⁴ : FaithfulSMul R S
inst✝³ : Algebra.IsIntegral R S
inst✝² : IsIntegrallyClosed R
this✝ : IsDomain R
p : Ideal R
inst✝¹ : p.IsPrime
Q : Ideal S
inst✝ : Q.IsPrime
hpQ : p < Ideal.under R Q
SQ : Type u_2 := Localization.AtPrime Q
this : Ideal.comap (algebraMap R SQ) (Ideal.map (algebraMap R SQ) p) ≤ p
P : Ideal SQ
hP : P.IsPrime
e : Ideal.comap (algebraMap R SQ) P = p
⊢ Ideal.comap (algebraMap R SQ) P = Ideal.under R P

Tactic:
]

Goal state after: