mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness

Goal state before:
pp: case refine_2.refine_2
A : Type u
inst✝² : CommRing A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
N₁ N₂ : Submodule A M
f : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯
hf₁ : f.ker = ⊥
hf₂ : f.range = map N₁.mkQ N₂
x✝ : ∃ x, (⊥.colon {N₁.mkQ x}).IsPrime ∧ N₂ = N₁ ⊔ A ∙ x
x : M
hx : (⊥.colon {N₁.mkQ x}).IsPrime
hx' : N₂ = N₁ ⊔ A ∙ x
hxN₂ : x ∈ N₂
⊢ Function.Injective
    ⇑(mapQ { asIdeal := ⊥.colon {N₁.mkQ x}, isPrime := hx }.asIdeal (N₁.submoduleOf N₂)
        (toSpanSingleton A ↥N₂ ⟨x, hxN₂⟩) ⋯)

pp: case refine_2.refine_3
A : Type u
inst✝² : CommRing A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
N₁ N₂ : Submodule A M
f : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯
hf₁ : f.ker = ⊥
hf₂ : f.range = map N₁.mkQ N₂
x✝ : ∃ x, (⊥.colon {N₁.mkQ x}).IsPrime ∧ N₂ = N₁ ⊔ A ∙ x
x : M
hx : (⊥.colon {N₁.mkQ x}).IsPrime
hx' : N₂ = N₁ ⊔ A ∙ x
hxN₂ : x ∈ N₂
⊢ Function.Surjective
    ⇑(mapQ { asIdeal := ⊥.colon {N₁.mkQ x}, isPrime := hx }.asIdeal (N₁.submoduleOf N₂)
        (toSpanSingleton A ↥N₂ ⟨x, hxN₂⟩) ⋯)

Tactic:
·

Goal state after:
case refine_2.refine_2
A : Type u
inst✝² : CommRing A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
N₁ N₂ : Submodule A M
f : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯
hf₁ : f.ker = ⊥
hf₂ : f.range = map N₁.mkQ N₂
x✝ : ∃ x, (⊥.colon {N₁.mkQ x}).IsPrime ∧ N₂ = N₁ ⊔ A ∙ x
x : M
hx : (⊥.colon {N₁.mkQ x}).IsPrime
hx' : N₂ = N₁ ⊔ A ∙ x
hxN₂ : x ∈ N₂
⊢ Function.Injective
    ⇑(mapQ { asIdeal := ⊥.colon {N₁.mkQ x}, isPrime := hx }.asIdeal (N₁.submoduleOf N₂)
        (toSpanSingleton A ↥N₂ ⟨x, hxN₂⟩) ⋯)
Module: Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness

Goal state before:
pp: case refine_2.refine_3
A : Type u
inst✝² : CommRing A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
N₁ N₂ : Submodule A M
f : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯
hf₁ : f.ker = ⊥
hf₂ : f.range = map N₁.mkQ N₂
x✝ : ∃ x, (⊥.colon {N₁.mkQ x}).IsPrime ∧ N₂ = N₁ ⊔ A ∙ x
x : M
hx : (⊥.colon {N₁.mkQ x}).IsPrime
hx' : N₂ = N₁ ⊔ A ∙ x
hxN₂ : x ∈ N₂
⊢ Function.Surjective
    ⇑(mapQ { asIdeal := ⊥.colon {N₁.mkQ x}, isPrime := hx }.asIdeal (N₁.submoduleOf N₂)
        (toSpanSingleton A ↥N₂ ⟨x, hxN₂⟩) ⋯)

Tactic:
·

Goal state after:
case refine_2.refine_3
A : Type u
inst✝² : CommRing A
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module A M
N₁ N₂ : Submodule A M
f : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯
hf₁ : f.ker = ⊥
hf₂ : f.range = map N₁.mkQ N₂
x✝ : ∃ x, (⊥.colon {N₁.mkQ x}).IsPrime ∧ N₂ = N₁ ⊔ A ∙ x
x : M
hx : (⊥.colon {N₁.mkQ x}).IsPrime
hx' : N₂ = N₁ ⊔ A ∙ x
hxN₂ : x ∈ N₂
⊢ Function.Surjective
    ⇑(mapQ { asIdeal := ⊥.colon {N₁.mkQ x}, isPrime := hx }.asIdeal (N₁.submoduleOf N₂)
        (toSpanSingleton A ↥N₂ ⟨x, hxN₂⟩) ⋯)
Module: Mathlib.RingTheory.UniqueFactorizationDomain.ClassGroup

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
J : Ideal R
this✝ : NormalizedGCDMonoid R := Classical.arbitrary (NormalizedGCDMonoid R)
K : Ideal R
hJK0 : J * K ≠ 0
hK : Submodule.IsPrincipal (J * K)
T : Finset R
hTK : ↑T ⊆ ↑K
K' : Ideal R := span ↑T
g : R := T.gcd id
hK' : K' ≤ span {g}
y : R
hyJ : y ∈ J
hJK : J * K = R ∙ y * g
hxmemJK : y * g ∈ J * K
hxT : y * g ∈ J * K'
hxJg : y * g ∈ J * span {g}
hx0 : y * g ≠ 0
this : J = span {y}
⊢ Submodule.IsPrincipal J

pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
J : Ideal R
this✝ : NormalizedGCDMonoid R := Classical.arbitrary (NormalizedGCDMonoid R)
K : Ideal R
hJK0 : J * K ≠ 0
hK : Submodule.IsPrincipal (J * K)
T : Finset R
hTK : ↑T ⊆ ↑K
K' : Ideal R := span ↑T
g : R := T.gcd id
hK' : K' ≤ span {g}
y : R
hyJ : y ∈ J
hJK : J * K = R ∙ y * g
hxmemJK : y * g ∈ J * K
hxT : y * g ∈ J * K'
hxJg : y * g ∈ J * span {g}
hx0 : y * g ≠ 0
this : J * span {g} = span {y} * span {g}
⊢ g ≠ 0

Tactic:
·

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
J : Ideal R
this✝ : NormalizedGCDMonoid R := Classical.arbitrary (NormalizedGCDMonoid R)
K : Ideal R
hJK0 : J * K ≠ 0
hK : Submodule.IsPrincipal (J * K)
T : Finset R
hTK : ↑T ⊆ ↑K
K' : Ideal R := span ↑T
g : R := T.gcd id
hK' : K' ≤ span {g}
y : R
hyJ : y ∈ J
hJK : J * K = R ∙ y * g
hxmemJK : y * g ∈ J * K
hxT : y * g ∈ J * K'
hxJg : y * g ∈ J * span {g}
hx0 : y * g ≠ 0
this : J = span {y}
⊢ Submodule.IsPrincipal J
Module: Mathlib.RingTheory.UniqueFactorizationDomain.ClassGroup

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
J : Ideal R
this✝ : NormalizedGCDMonoid R := Classical.arbitrary (NormalizedGCDMonoid R)
K : Ideal R
hJK0 : J * K ≠ 0
hK : Submodule.IsPrincipal (J * K)
T : Finset R
hTK : ↑T ⊆ ↑K
K' : Ideal R := span ↑T
g : R := T.gcd id
hK' : K' ≤ span {g}
y : R
hyJ : y ∈ J
hJK : J * K = R ∙ y * g
hxmemJK : y * g ∈ J * K
hxT : y * g ∈ J * K'
hxJg : y * g ∈ J * span {g}
hx0 : y * g ≠ 0
this : J * span {g} = span {y} * span {g}
⊢ g ≠ 0

Tactic:
·

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
J : Ideal R
this✝ : NormalizedGCDMonoid R := Classical.arbitrary (NormalizedGCDMonoid R)
K : Ideal R
hJK0 : J * K ≠ 0
hK : Submodule.IsPrincipal (J * K)
T : Finset R
hTK : ↑T ⊆ ↑K
K' : Ideal R := span ↑T
g : R := T.gcd id
hK' : K' ≤ span {g}
y : R
hyJ : y ∈ J
hJK : J * K = R ∙ y * g
hxmemJK : y * g ∈ J * K
hxT : y * g ∈ J * K'
hxJg : y * g ∈ J * span {g}
hx0 : y * g ≠ 0
this : J * span {g} = span {y} * span {g}
⊢ g ≠ 0
Module: Mathlib.RingTheory.UniqueFactorizationDomain.ClassGroup

Goal state before:
pp: case refine_1
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
I : Ideal R
hI : IsUnit ↑I
a : R
K : Ideal R
ha0 : a ≠ 0
h : (↑I)⁻¹ = spanSingleton R⁰ ((algebraMap R (FractionRing R)) a)⁻¹ * ↑K
hIK : I * K = span {a}
⊢ I * K ≠ 0

pp: case refine_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
I : Ideal R
hI : IsUnit ↑I
a : R
K : Ideal R
ha0 : a ≠ 0
h : (↑I)⁻¹ = spanSingleton R⁰ ((algebraMap R (FractionRing R)) a)⁻¹ * ↑K
hIK : I * K = span {a}
⊢ Submodule.IsPrincipal (I * K)

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
I : Ideal R
hI : IsUnit ↑I
a : R
K : Ideal R
ha0 : a ≠ 0
h : (↑I)⁻¹ = spanSingleton R⁰ ((algebraMap R (FractionRing R)) a)⁻¹ * ↑K
hIK : I * K = span {a}
⊢ I * K ≠ 0
Module: Mathlib.RingTheory.UniqueFactorizationDomain.ClassGroup

Goal state before:
pp: case refine_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
I : Ideal R
hI : IsUnit ↑I
a : R
K : Ideal R
ha0 : a ≠ 0
h : (↑I)⁻¹ = spanSingleton R⁰ ((algebraMap R (FractionRing R)) a)⁻¹ * ↑K
hIK : I * K = span {a}
⊢ Submodule.IsPrincipal (I * K)

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsGCDMonoid R
I : Ideal R
hI : IsUnit ↑I
a : R
K : Ideal R
ha0 : a ≠ 0
h : (↑I)⁻¹ = spanSingleton R⁰ ((algebraMap R (FractionRing R)) a)⁻¹ * ↑K
hIK : I * K = span {a}
⊢ Submodule.IsPrincipal (I * K)
Module: Mathlib.RingTheory.ClassGroup.Basic

Goal state before:
pp: case mp
R : Type u_1
K : Type u_2
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
I : (FractionalIdeal R⁰ K)ˣ
x : Kˣ
hx : spanSingleton R⁰ ↑x = ↑I
⊢ ∃ x, spanSingleton R⁰ x = ↑I

pp: case mpr
R : Type u_1
K : Type u_2
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
I : (FractionalIdeal R⁰ K)ˣ
x : K
hx : spanSingleton R⁰ x = ↑I
⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I

Tactic:
·

Goal state after:
case mp
R : Type u_1
K : Type u_2
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
I : (FractionalIdeal R⁰ K)ˣ
x : Kˣ
hx : spanSingleton R⁰ ↑x = ↑I
⊢ ∃ x, spanSingleton R⁰ x = ↑I
Module: Mathlib.RingTheory.ClassGroup.Basic

Goal state before:
pp: case mpr
R : Type u_1
K : Type u_2
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
I : (FractionalIdeal R⁰ K)ˣ
x : K
hx : spanSingleton R⁰ x = ↑I
⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I

Tactic:
·

Goal state after:
case mpr
R : Type u_1
K : Type u_2
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
I : (FractionalIdeal R⁰ K)ˣ
x : K
hx : spanSingleton R⁰ x = ↑I
⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I
Module: Mathlib.RingTheory.ClassGroup.Basic

Goal state before:
pp: case mp
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I J : (FractionalIdeal R⁰ (FractionRing R))ˣ
I' J' : Ideal R
hI : ↑I = ↑I'
hJ : ↑J = ↑J'
⊢ (∃ x, I * (toPrincipalIdeal R (FractionRing R)) x = J) →
    ∃ x y, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * J'

pp: case mpr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I J : (FractionalIdeal R⁰ (FractionRing R))ˣ
I' J' : Ideal R
hI : ↑I = ↑I'
hJ : ↑J = ↑J'
⊢ (∃ x y, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * J') →
    ∃ x, I * (toPrincipalIdeal R (FractionRing R)) x = J

Tactic:
·

Goal state after:
case mp
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I J : (FractionalIdeal R⁰ (FractionRing R))ˣ
I' J' : Ideal R
hI : ↑I = ↑I'
hJ : ↑J = ↑J'
⊢ (∃ x, I * (toPrincipalIdeal R (FractionRing R)) x = J) →
    ∃ x y, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * J'
Module: Mathlib.RingTheory.ClassGroup.Basic

Goal state before:
pp: case mpr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I J : (FractionalIdeal R⁰ (FractionRing R))ˣ
I' J' : Ideal R
hI : ↑I = ↑I'
hJ : ↑J = ↑J'
⊢ (∃ x y, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * J') →
    ∃ x, I * (toPrincipalIdeal R (FractionRing R)) x = J

Tactic:
·

Goal state after:
case mpr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I J : (FractionalIdeal R⁰ (FractionRing R))ˣ
I' J' : Ideal R
hI : ↑I = ↑I'
hJ : ↑J = ↑J'
⊢ (∃ x y, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * J') →
    ∃ x, I * (toPrincipalIdeal R (FractionRing R)) x = J
Module: Mathlib.RingTheory.Spectrum.Prime.TensorProduct

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
f : S ⊗[R] T
J : PrimeSpectrum (S ⊗[R] T)
hJ : f ∉ J.asIdeal
t : T
r : R
a : S
ht : r • t ∉ Ideal.comap g J.asIdeal
e : 1 ⊗ₜ[R] (r • t) * f = a ⊗ₜ[R] t
⊢ tensorProductTo R S T ⁻¹' (basicOpen a).carrier ×ˢ (basicOpen t).carrier ⊆ ↑(basicOpen f)

pp: case refine_2
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
f : S ⊗[R] T
J : PrimeSpectrum (S ⊗[R] T)
hJ : f ∉ J.asIdeal
t : T
r : R
a : S
ht : r • t ∉ Ideal.comap g J.asIdeal
e : 1 ⊗ₜ[R] (r • t) * f = a ⊗ₜ[R] t
⊢ J ∈ tensorProductTo R S T ⁻¹' (basicOpen a).carrier ×ˢ (basicOpen t).carrier

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
f : S ⊗[R] T
J : PrimeSpectrum (S ⊗[R] T)
hJ : f ∉ J.asIdeal
t : T
r : R
a : S
ht : r • t ∉ Ideal.comap g J.asIdeal
e : 1 ⊗ₜ[R] (r • t) * f = a ⊗ₜ[R] t
⊢ tensorProductTo R S T ⁻¹' (basicOpen a).carrier ×ˢ (basicOpen t).carrier ⊆ ↑(basicOpen f)
Module: Mathlib.RingTheory.ClassGroup.Basic

Goal state before:
pp: case mp
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
I' : Ideal R
hI : ↑I = ↑I'
⊢ (∃ x y, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * ⊤) → ∃ x, x ≠ 0 ∧ I' = Ideal.span {x}

pp: case mpr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
I' : Ideal R
hI : ↑I = ↑I'
⊢ (∃ x, x ≠ 0 ∧ I' = Ideal.span {x}) → ∃ x y, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * ⊤

Tactic:
·

Goal state after:
case mp
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
I' : Ideal R
hI : ↑I = ↑I'
⊢ (∃ x y, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * ⊤) → ∃ x, x ≠ 0 ∧ I' = Ideal.span {x}
Module: Mathlib.RingTheory.ClassGroup.Basic

Goal state before:
pp: case mpr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
I' : Ideal R
hI : ↑I = ↑I'
⊢ (∃ x, x ≠ 0 ∧ I' = Ideal.span {x}) → ∃ x y, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * ⊤

Tactic:
·

Goal state after:
case mpr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
I' : Ideal R
hI : ↑I = ↑I'
⊢ (∃ x, x ≠ 0 ∧ I' = Ideal.span {x}) → ∃ x y, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * ⊤
Module: Mathlib.RingTheory.Spectrum.Prime.TensorProduct

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
f : S ⊗[R] T
J : PrimeSpectrum (S ⊗[R] T)
hJ : f ∉ J.asIdeal
t : T
r : R
a : S
ht : r • t ∉ Ideal.comap g J.asIdeal
e : 1 ⊗ₜ[R] (r • t) * f = a ⊗ₜ[R] t
⊢ J ∈ tensorProductTo R S T ⁻¹' (basicOpen a).carrier ×ˢ (basicOpen t).carrier

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
f : S ⊗[R] T
J : PrimeSpectrum (S ⊗[R] T)
hJ : f ∉ J.asIdeal
t : T
r : R
a : S
ht : r • t ∉ Ideal.comap g J.asIdeal
e : 1 ⊗ₜ[R] (r • t) * f = a ⊗ₜ[R] t
⊢ J ∈ tensorProductTo R S T ⁻¹' (basicOpen a).carrier ×ˢ (basicOpen t).carrier
Module: Mathlib.RingTheory.ClassGroup.Basic

Goal state before:
pp: R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
P : ClassGroup R → Prop
h : ∀ (I : (FractionalIdeal R⁰ K)ˣ), P ((mk K) I)
x : ClassGroup R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
this :
  I =
    (Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv)
      ((Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv) I)
⊢ P ↑I

Tactic:
congr_arg (QuotientGroup.mk (s := (toPrincipalIdeal R (FractionRing R)).range)) this

Goal state after:
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
P : ClassGroup R → Prop
h : ∀ (I : (FractionalIdeal R⁰ K)ˣ), P ((mk K) I)
x : ClassGroup R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
this :
  I =
    (Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv)
      ((Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv) I)
⊢ P
    ↑((Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv)
        ((Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv) I))
Module: Mathlib.RingTheory.ClassGroup.Basic

Goal state before:
pp: case mp
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ K)ˣ
⊢ (∃ x,
      (∃ x_1, spanSingleton R⁰ x_1 = ↑x) ∧
        (Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv).toMonoidHom x = I) →
    ∃ x, spanSingleton R⁰ x = ↑I

pp: case mpr
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ K)ˣ
⊢ (∃ x, spanSingleton R⁰ x = ↑I) →
    ∃ x,
      (∃ x_1, spanSingleton R⁰ x_1 = ↑x) ∧
        (Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv).toMonoidHom x = I

Tactic:
·

Goal state after:
case mp
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ K)ˣ
⊢ (∃ x,
      (∃ x_1, spanSingleton R⁰ x_1 = ↑x) ∧
        (Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv).toMonoidHom x = I) →
    ∃ x, spanSingleton R⁰ x = ↑I
Module: Mathlib.RingTheory.ClassGroup.Basic

Goal state before:
pp: case mpr
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ K)ˣ
⊢ (∃ x, spanSingleton R⁰ x = ↑I) →
    ∃ x,
      (∃ x_1, spanSingleton R⁰ x_1 = ↑x) ∧
        (Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv).toMonoidHom x = I

Tactic:
·

Goal state after:
case mpr
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ K)ˣ
⊢ (∃ x, spanSingleton R⁰ x = ↑I) →
    ∃ x,
      (∃ x_1, spanSingleton R⁰ x_1 = ↑x) ∧
        (Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv).toMonoidHom x = I
Module: Mathlib.RingTheory.ClassGroup.Basic

Goal state before:
pp: case mpr.refine_1
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ K)ˣ
x : K
hx : spanSingleton R⁰ x = ↑I
⊢ spanSingleton R⁰ ((FractionRing.algEquiv R K).symm x) =
    ↑((Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv) I)

pp: case mpr.refine_2
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ K)ˣ
x : K
hx : spanSingleton R⁰ x = ↑I
⊢ ↑((Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv).toMonoidHom
        ((Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv) I)) =
    ↑I

Tactic:
·

Goal state after:
case mpr.refine_1
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ K)ˣ
x : K
hx : spanSingleton R⁰ x = ↑I
⊢ spanSingleton R⁰ ((FractionRing.algEquiv R K).symm x) =
    ↑((Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv) I)
Module: Mathlib.RingTheory.ClassGroup.Basic

Goal state before:
pp: case mpr.refine_2
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ K)ˣ
x : K
hx : spanSingleton R⁰ x = ↑I
⊢ ↑((Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv).toMonoidHom
        ((Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv) I)) =
    ↑I

Tactic:
·

Goal state after:
case mpr.refine_2
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ K)ˣ
x : K
hx : spanSingleton R⁰ x = ↑I
⊢ ↑((Units.mapEquiv (canonicalEquiv R⁰ (FractionRing R) K).toMulEquiv).toMonoidHom
        ((Units.mapEquiv (canonicalEquiv R⁰ K (FractionRing R)).toMulEquiv) I)) =
    ↑I
Module: Mathlib.RingTheory.LocalProperties.Projective

Goal state before:
pp: R : Type u_1
N : Type u_2
M : Type uM
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
f : M →ₗ[R] N
inst✝ : Module.FinitePresentation R N
H : ∀ (I : Ideal R) (x : I.IsMaximal), ∃ g, (LocalizedModule.map I.primeCompl) f ∘ₗ g = id
I : Ideal R
hI : I.IsMaximal
this :
  id ∈
    ((llcomp (Localization I.primeCompl) (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl M)
          (LocalizedModule I.primeCompl N))
        ((LocalizedModule.map I.primeCompl) f)).range
⊢ Membership.mem (Submodule.localized₀ I.primeCompl (LocalizedModule.map I.primeCompl) ((llcomp R N M N) f).range) =
    Membership.mem
      ((llcomp (Localization I.primeCompl) (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl M)
            (LocalizedModule I.primeCompl N))
          ((LocalizedModule.map I.primeCompl) f)).range

Tactic:
·

Goal state after:
R : Type u_1
N : Type u_2
M : Type uM
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
f : M →ₗ[R] N
inst✝ : Module.FinitePresentation R N
H : ∀ (I : Ideal R) (x : I.IsMaximal), ∃ g, (LocalizedModule.map I.primeCompl) f ∘ₗ g = id
I : Ideal R
hI : I.IsMaximal
this :
  id ∈
    ((llcomp (Localization I.primeCompl) (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl M)
          (LocalizedModule I.primeCompl N))
        ((LocalizedModule.map I.primeCompl) f)).range
⊢ Membership.mem (Submodule.localized₀ I.primeCompl (LocalizedModule.map I.primeCompl) ((llcomp R N M N) f).range) =
    Membership.mem
      ((llcomp (Localization I.primeCompl) (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl M)
            (LocalizedModule I.primeCompl N))
          ((LocalizedModule.map I.primeCompl) f)).range
Module: Mathlib.RingTheory.LocalProperties.Projective

Goal state before:
pp: case mp
R : Type u_1
N : Type u_2
M : Type uM
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
f✝ : M →ₗ[R] N
inst✝ : Module.FinitePresentation R N
H : ∀ (I : Ideal R) (x : I.IsMaximal), ∃ g, (LocalizedModule.map I.primeCompl) f✝ ∘ₗ g = id
I : Ideal R
hI : I.IsMaximal
this :
  id ∈
    ((llcomp (Localization I.primeCompl) (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl M)
          (LocalizedModule I.primeCompl N))
        ((LocalizedModule.map I.primeCompl) f✝)).range
f : LocalizedModule I.primeCompl N →ₗ[Localization I.primeCompl] LocalizedModule I.primeCompl N
⊢ f ∈ Submodule.localized₀ I.primeCompl (LocalizedModule.map I.primeCompl) ((llcomp R N M N) f✝).range →
    f ∈
      ((llcomp (Localization I.primeCompl) (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl M)
            (LocalizedModule I.primeCompl N))
          ((LocalizedModule.map I.primeCompl) f✝)).range

pp: case mpr
R : Type u_1
N : Type u_2
M : Type uM
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
f✝ : M →ₗ[R] N
inst✝ : Module.FinitePresentation R N
H : ∀ (I : Ideal R) (x : I.IsMaximal), ∃ g, (LocalizedModule.map I.primeCompl) f✝ ∘ₗ g = id
I : Ideal R
hI : I.IsMaximal
this :
  id ∈
    ((llcomp (Localization I.primeCompl) (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl M)
          (LocalizedModule I.primeCompl N))
        ((LocalizedModule.map I.primeCompl) f✝)).range
f : LocalizedModule I.primeCompl N →ₗ[Localization I.primeCompl] LocalizedModule I.primeCompl N
⊢ f ∈
      ((llcomp (Localization I.primeCompl) (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl M)
            (LocalizedModule I.primeCompl N))
          ((LocalizedModule.map I.primeCompl) f✝)).range →
    f ∈ Submodule.localized₀ I.primeCompl (LocalizedModule.map I.primeCompl) ((llcomp R N M N) f✝).range

Tactic:
·

Goal state after:
case mp
R : Type u_1
N : Type u_2
M : Type uM
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
f✝ : M →ₗ[R] N
inst✝ : Module.FinitePresentation R N
H : ∀ (I : Ideal R) (x : I.IsMaximal), ∃ g, (LocalizedModule.map I.primeCompl) f✝ ∘ₗ g = id
I : Ideal R
hI : I.IsMaximal
this :
  id ∈
    ((llcomp (Localization I.primeCompl) (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl M)
          (LocalizedModule I.primeCompl N))
        ((LocalizedModule.map I.primeCompl) f✝)).range
f : LocalizedModule I.primeCompl N →ₗ[Localization I.primeCompl] LocalizedModule I.primeCompl N
⊢ f ∈ Submodule.localized₀ I.primeCompl (LocalizedModule.map I.primeCompl) ((llcomp R N M N) f✝).range →
    f ∈
      ((llcomp (Localization I.primeCompl) (LocalizedModule I.primeCompl N) (LocalizedModule I.primeCompl M)
            (LocalizedModule I.primeCompl N))
          ((LocalizedModule.map I.primeCompl) f✝)).range