mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: case h
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
I J : Submodule R₁ K
aI : R₁
haI : aI ∈ R₁⁰
hI : ∀ b ∈ I, IsInteger R₁ (aI • b)
aJ : R₁
haJ : aJ ∈ R₁⁰
hJ : ∀ b ∈ J, IsInteger R₁ (aJ • b)
h : J ≠ 0
y : K
mem_J : y ∈ J
notMem_zero : y ∉ 0
y' : R₁
hy' : (algebraMap R₁ K) y' = aJ • y
⊢ aI * y' ∈ R₁⁰ ∧ ∀ b ∈ I / J, IsInteger R₁ ((aI * y') • b)

Tactic:
constructor

Goal state after:
case h.left
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
I J : Submodule R₁ K
aI : R₁
haI : aI ∈ R₁⁰
hI : ∀ b ∈ I, IsInteger R₁ (aI • b)
aJ : R₁
haJ : aJ ∈ R₁⁰
hJ : ∀ b ∈ J, IsInteger R₁ (aJ • b)
h : J ≠ 0
y : K
mem_J : y ∈ J
notMem_zero : y ∉ 0
y' : R₁
hy' : (algebraMap R₁ K) y' = aJ • y
⊢ aI * y' ∈ R₁⁰

case h.right
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
I J : Submodule R₁ K
aI : R₁
haI : aI ∈ R₁⁰
hI : ∀ b ∈ I, IsInteger R₁ (aI • b)
aJ : R₁
haJ : aJ ∈ R₁⁰
hJ : ∀ b ∈ J, IsInteger R₁ (aJ • b)
h : J ≠ 0
y : K
mem_J : y ∈ J
notMem_zero : y ∉ 0
y' : R₁
hy' : (algebraMap R₁ K) y' = aJ • y
⊢ ∀ b ∈ I / J, IsInteger R₁ ((aI * y') • b)
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
I : FractionalIdeal R₁⁰ K
⊢ ⟨↑I / ↑1, ⋯⟩ = I

Tactic:
ext

Goal state after:
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
I : FractionalIdeal R₁⁰ K
x✝ : K
⊢ x✝ ∈ ⟨↑I / ↑1, ⋯⟩ ↔ x✝ ∈ I
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
I : FractionalIdeal R₁⁰ K
x✝ : K
⊢ x✝ ∈ ⟨↑I / ↑1, ⋯⟩ ↔ x✝ ∈ I

Tactic:
constructor

Goal state after:
case mp
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
I : FractionalIdeal R₁⁰ K
x✝ : K
⊢ x✝ ∈ ⟨↑I / ↑1, ⋯⟩ → x✝ ∈ I

case mpr
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
I : FractionalIdeal R₁⁰ K
x✝ : K
⊢ x✝ ∈ I → x✝ ∈ ⟨↑I / ↑1, ⋯⟩
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: case pos
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 : ↑I * (↑I)⁻¹ = 0
⊢ (↑I * (↑I)⁻¹)⁻¹ ≤ 1

Tactic:
rw [hJ0, inv_zero']; exact zero_le _

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

Goal state before:
pp: K : Type u_4
L : Type u_5
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsFractionRing K L
I : FractionalIdeal K⁰ L
hI : ¬I = 0
x : L
⊢ x ∈ I ↔ ∃ x', (algebraMap K L) x' = x

Tactic:
constructor

Goal state after:
case mp
K : Type u_4
L : Type u_5
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsFractionRing K L
I : FractionalIdeal K⁰ L
hI : ¬I = 0
x : L
⊢ x ∈ I → ∃ x', (algebraMap K L) x' = x

case mpr
K : Type u_4
L : Type u_5
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsFractionRing K L
I : FractionalIdeal K⁰ L
hI : ¬I = 0
x : L
⊢ (∃ x', (algebraMap K L) x' = x) → x ∈ I
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
h : IsDedekindDomainInv A
this : CommGroupWithZero (FractionalIdeal A⁰ (FractionRing A)) := commGroupWithZero (FractionRing A) h
⊢ DimensionLEOne A

Tactic:
constructor

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
h : IsDedekindDomainInv A
this : CommGroupWithZero (FractionalIdeal A⁰ (FractionRing A)) := ⋯
⊢ ∀ {p : Ideal A}, p ≠ ⊥ → p.IsPrime → p.IsMaximal
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: case e'_2
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
I : FractionalIdeal S P
⊢ (algebraMap R P) ↑I.den • ↑I = I.den • ↑I

Tactic:
ext

Goal state after:
case e'_2
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
I : FractionalIdeal S P
x✝ : P
⊢ x✝ ∈ (algebraMap R P) ↑I.den • ↑I ↔ x✝ ∈ I.den • ↑I
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:

Tactic:
(1 : FractionalIdeal A⁰ (FractionRing A)) = (↑M)⁻¹ * P * ((↑P)⁻¹ * M) := by
        simp [mul_assoc, *]
      _ ≤ P * ((↑P)⁻¹ * M) := by gcongr
      _ = M := by simp [*]

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

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
⊢ spanSingleton S 0 = 0

Tactic:
ext

Goal state after:
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x✝ : P
⊢ x✝ ∈ spanSingleton S 0 ↔ x✝ ∈ 0
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
⊢ spanSingleton S 1 = 1

Tactic:
ext

Goal state after:
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x✝ : P
⊢ x✝ ∈ spanSingleton S 1 ↔ x✝ ∈ 1
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x : R
y : P
⊢ (∃ x' ∈ Ideal.span {x}, (algebraMap R P) x' = y) ↔ ∃ z, z • (algebraMap R P) x = y

Tactic:
constructor

Goal state after:
case mp
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x : R
y : P
⊢ (∃ x' ∈ Ideal.span {x}, (algebraMap R P) x' = y) → ∃ z, z • (algebraMap R P) x = y

case mpr
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x : R
y : P
⊢ (∃ z, z • (algebraMap R P) x = y) → ∃ x' ∈ Ideal.span {x}, (algebraMap R P) x' = y
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
inst✝³ : IsLocalization S P
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsLocalization S P'
x : P
⊢ (canonicalEquiv S P P') (spanSingleton S x) = spanSingleton S ((IsLocalization.map P' (RingHom.id R) ⋯) x)

Tactic:
apply SetLike.ext_iff.mpr

Goal state after:
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
inst✝³ : IsLocalization S P
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsLocalization S P'
x : P
⊢ ∀ (x_1 : P'),
    x_1 ∈ (canonicalEquiv S P P') (spanSingleton S x) ↔
      x_1 ∈ spanSingleton S ((IsLocalization.map P' (RingHom.id R) ⋯) x)
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
inst✝³ : IsLocalization S P
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsLocalization S P'
x : P
y : P'
⊢ y ∈ (canonicalEquiv S P P') (spanSingleton S x) ↔ y ∈ spanSingleton S ((IsLocalization.map P' (RingHom.id R) ⋯) x)

Tactic:
constructor

Goal state after:
case mp
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
inst✝³ : IsLocalization S P
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsLocalization S P'
x : P
y : P'
⊢ y ∈ (canonicalEquiv S P P') (spanSingleton S x) → y ∈ spanSingleton S ((IsLocalization.map P' (RingHom.id R) ⋯) x)

case mpr
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
inst✝³ : IsLocalization S P
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsLocalization S P'
x : P
y : P'
⊢ y ∈ spanSingleton S ((IsLocalization.map P' (RingHom.id R) ⋯) x) → y ∈ (canonicalEquiv S P P') (spanSingleton S x)
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x y : P
I : FractionalIdeal S P
⊢ y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y'

Tactic:
constructor

Goal state after:
case mp
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x y : P
I : FractionalIdeal S P
⊢ y ∈ spanSingleton S x * I → ∃ y' ∈ I, y = x * y'

case mpr
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x y : P
I : FractionalIdeal S P
⊢ (∃ y' ∈ I, y = x * y') → y ∈ spanSingleton S x * I
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
e : ℕ
he : 2 ≤ e
⊢ I = I ^ 1

Tactic:
rw [pow_one]

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

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x : P
I J : FractionalIdeal S P
⊢ (∀ (i : P), (∃ z, z • x = i) → ∀ j ∈ I, i * j ∈ J) ↔ ∀ z ∈ I, x * z ∈ J

Tactic:
constructor

Goal state after:
case mp
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x : P
I J : FractionalIdeal S P
⊢ (∀ (i : P), (∃ z, z • x = i) → ∀ j ∈ I, i * j ∈ J) → ∀ z ∈ I, x * z ∈ J

case mpr
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x : P
I J : FractionalIdeal S P
⊢ (∀ z ∈ I, x * z ∈ J) → ∀ (i : P), (∃ z, z • x = i) → ∀ j ∈ I, i * j ∈ J
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
I : Submodule R K
hI : IsFractional R⁰ I
⊢ IsFractional S⁰ (Submodule.map (↑(semilinearEquivOfRingEquiv K L f)) I)

Tactic:
simp only [IsFractional, mem_nonZeroDivisors_iff_ne_zero, ne_eq, Submodule.mem_map,
    forall_exists_index, and_imp, forall_apply_eq_imp_iff₂] at hI ⊢
  obtain ⟨r, hr0, hr⟩ := hI
  use f r
  refine ⟨by simp [hr0], ?_⟩
  intro x hx
  specialize hr x hx
  simp only [IsLocalization.IsInteger, RingHom.mem_rangeS] at hr ⊢
  obtain ⟨r', hr'⟩ := hr
  use f r'
  simp only [semilinearEquivOfRingEquiv, RingEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe,
    EquivLike.coe_coe, Equiv.invFun_as_coe, LinearMap.coe_mk, AddHom.coe_mk]
  rw [Algebra.smul_def, ← ringEquivOfRingEquiv_algebraMap f (K := K) (L := L) r,
    ← map_mul, ← Algebra.smul_def, ← hr', ringEquivOfRingEquiv_algebraMap]

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

Goal state before:
pp: case e'_2.e'_13
R✝ : Type u_1
inst✝¹⁷ : CommRing R✝
S✝ : Submonoid R✝
P : Type u_2
inst✝¹⁶ : CommRing P
inst✝¹⁵ : Algebra R✝ P
R₁ : Type u_3
inst✝¹⁴ : CommRing R₁
K✝ : Type u_4
inst✝¹³ : Field K✝
inst✝¹² : Algebra R₁ K✝
inst✝¹¹ : IsFractionRing R₁ K✝
inst✝¹⁰ : IsDomain R₁
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
I : FractionalIdeal R⁰ K
⊢ ↑((semilinearEquivOfRingEquiv K L f).trans (semilinearEquivOfRingEquiv L K f.symm)) = LinearMap.id

Tactic:
ext

Goal state after:
case e'_2.e'_13
R✝ : Type u_1
inst✝¹⁷ : CommRing R✝
S✝ : Submonoid R✝
P : Type u_2
inst✝¹⁶ : CommRing P
inst✝¹⁵ : Algebra R✝ P
R₁ : Type u_3
inst✝¹⁴ : CommRing R₁
K✝ : Type u_4
inst✝¹³ : Field K✝
inst✝¹² : Algebra R₁ K✝
inst✝¹¹ : IsFractionRing R₁ K✝
inst✝¹⁰ : IsDomain R₁
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
I : FractionalIdeal R⁰ K
x✝ : K
⊢ ↑((semilinearEquivOfRingEquiv K L f).trans (semilinearEquivOfRingEquiv L K f.symm)) x✝ = LinearMap.id x✝
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: case e'_2.e'_13
R✝ : Type u_1
inst✝¹⁷ : CommRing R✝
S✝ : Submonoid R✝
P : Type u_2
inst✝¹⁶ : CommRing P
inst✝¹⁵ : Algebra R✝ P
R₁ : Type u_3
inst✝¹⁴ : CommRing R₁
K✝ : Type u_4
inst✝¹³ : Field K✝
inst✝¹² : Algebra R₁ K✝
inst✝¹¹ : IsFractionRing R₁ K✝
inst✝¹⁰ : IsDomain R₁
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
I : FractionalIdeal S⁰ L
⊢ ↑((semilinearEquivOfRingEquiv L K f.symm).trans (semilinearEquivOfRingEquiv K L f)) = LinearMap.id

Tactic:
ext

Goal state after:
case e'_2.e'_13
R✝ : Type u_1
inst✝¹⁷ : CommRing R✝
S✝ : Submonoid R✝
P : Type u_2
inst✝¹⁶ : CommRing P
inst✝¹⁵ : Algebra R✝ P
R₁ : Type u_3
inst✝¹⁴ : CommRing R₁
K✝ : Type u_4
inst✝¹³ : Field K✝
inst✝¹² : Algebra R₁ K✝
inst✝¹¹ : IsFractionRing R₁ K✝
inst✝¹⁰ : IsDomain R₁
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
I : FractionalIdeal S⁰ L
x✝ : L
⊢ ↑((semilinearEquivOfRingEquiv L K f.symm).trans (semilinearEquivOfRingEquiv K L f)) x✝ = LinearMap.id x✝
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_5
K : Type u_7
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : CommRing K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
I : FractionalIdeal R⁰ K
x : K
⊢ x ∈ ↑⟨Submodule.map ↑(semilinearEquivOfRingEquiv K K (RingEquiv.refl R)) ↑I, ⋯⟩ ↔ x ∈ ↑I

Tactic:
simp [semilinearEquivOfRingEquiv]

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

Goal state before:
pp: R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
x : K
y : L
⊢ (∃ a, (semilinearEquivOfRingEquiv K L f) (a • x) = y) ↔ ∃ a, a • (IsLocalization.map L ↑f ⋯) x = y

Tactic:
constructor

Goal state after:
case mp
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
x : K
y : L
⊢ (∃ a, (semilinearEquivOfRingEquiv K L f) (a • x) = y) → ∃ a, a • (IsLocalization.map L ↑f ⋯) x = y

case mpr
R : Type u_5
S : Type u_6
K : Type u_7
L : Type u_8
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : CommRing K
inst✝⁴ : CommRing L
inst✝³ : Algebra R K
inst✝² : Algebra S L
inst✝¹ : IsFractionRing R K
inst✝ : IsFractionRing S L
f : R ≃+* S
x : K
y : L
⊢ (∃ a, a • (IsLocalization.map L ↑f ⋯) x = y) → ∃ a, (semilinearEquivOfRingEquiv K L f) (a • x) = y
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case a
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ∣ I ∧ (normalizedFactors I ∩ normalizedFactors J).prod ∣ J

Tactic:
constructor

Goal state after:
case a.left
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ∣ I

case a.right
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ∣ J
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case h
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I : Ideal T
hI : I ≠ ⊥
P : Ideal T
hpm : P.IsMaximal
⊢ P ⊔ (filter (fun x ↦ ¬P = x) (normalizedFactors I)).prod = ⊤ ∧
    I = P ^ count P (normalizedFactors I) * (filter (fun x ↦ ¬P = x) (normalizedFactors I)).prod

Tactic:
constructor

Goal state after:
case h.left
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I : Ideal T
hI : I ≠ ⊥
P : Ideal T
hpm : P.IsMaximal
⊢ P ⊔ (filter (fun x ↦ ¬P = x) (normalizedFactors I)).prod = ⊤

case h.right
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I : Ideal T
hI : I ≠ ⊥
P : Ideal T
hpm : P.IsMaximal
⊢ I = P ^ count P (normalizedFactors I) * (filter (fun x ↦ ¬P = x) (normalizedFactors I)).prod
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case h.right
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I : Ideal T
hI : I ≠ ⊥
P : Ideal T
hpm : P.IsMaximal
⊢ I = P ^ count P (normalizedFactors I) * (filter (fun x ↦ ¬P = x) (normalizedFactors I)).prod

Tactic:
← prod_normalizedFactors_eq_self hI,

Goal state after:
case h.right
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I : Ideal T
hI : I ≠ ⊥
P : Ideal T
hpm : P.IsMaximal
⊢ (normalizedFactors I).prod = P ^ count P (normalizedFactors I) * (filter (fun x ↦ ¬P = x) (normalizedFactors I)).prod
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
hR : ¬IsField R
I : Ideal R
⊢ (∃ M, M.IsMaximal ∧ I ≤ M) ↔ ∃ P, I ≤ P.asIdeal

Tactic:
constructor

Goal state after:
case mp
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
hR : ¬IsField R
I : Ideal R
⊢ (∃ M, M.IsMaximal ∧ I ≤ M) → ∃ P, I ≤ P.asIdeal

case mpr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
hR : ¬IsField R
I : Ideal R
⊢ (∃ P, I ≤ P.asIdeal) → ∃ M, M.IsMaximal ∧ I ≤ M
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: R : Type u_1
K : Type u_3
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
inst✝ : Algebra R K
hK : IsFractionRing R K
x : K
⊢ (∀ (i : HeightOneSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) ↔ x ∈ ⊥

Tactic:
constructor

Goal state after:
case mp
R : Type u_1
K : Type u_3
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
inst✝ : Algebra R K
hK : IsFractionRing R K
x : K
⊢ (∀ (i : HeightOneSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) → x ∈ ⊥

case mpr
R : Type u_1
K : Type u_3
inst✝³ : CommRing R
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
inst✝ : Algebra R K
hK : IsFractionRing R K
x : K
⊢ x ∈ ⊥ → ∀ (i : HeightOneSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯
Module: Mathlib.Algebra.Module.FinitePresentation

Goal state before:
pp: R : Type u
M : Type u_1
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
fp : FinitePresentation R M
n : ℕ
K : Submodule R (Fin n → R)
e : M ≃ₗ[R] (Fin n → R) ⧸ K
h : K.FG
m : ℕ
g' : (Fin m → R) →ₗ[R] Fin n → R
hg' : g'.range = K
⊢ Function.Exact ⇑g' ⇑K.mkQ

Tactic:
simp [LinearMap.exact_iff, hg']

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
S : Type u_4
inst✝ : CommRing S
e : R ≃+* S
x : HeightOneSpectrum R
⊢ comap ↑e ⋯ (comap ↑e.symm ⋯ x) = x

Tactic:
ext

Goal state after:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
S : Type u_4
inst✝ : CommRing S
e : R ≃+* S
x : HeightOneSpectrum R
x✝ : R
⊢ x✝ ∈ (comap ↑e ⋯ (comap ↑e.symm ⋯ x)).asIdeal ↔ x✝ ∈ x.asIdeal
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
S : Type u_4
inst✝ : CommRing S
e : R ≃+* S
x : HeightOneSpectrum S
⊢ comap ↑e.symm ⋯ (comap ↑e ⋯ x) = x

Tactic:
ext

Goal state after:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Field K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
S : Type u_4
inst✝ : CommRing S
e : R ≃+* S
x : HeightOneSpectrum S
x✝ : S
⊢ x✝ ∈ (comap ↑e.symm ⋯ (comap ↑e ⋯ x)).asIdeal ↔ x✝ ∈ x.asIdeal
Module: Mathlib.Algebra.Module.FinitePresentation

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R M
inst✝ : Module.Finite R M
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
hl' : l.ker.FG
b : Basis (Free.ChooseBasisIndex R M) R M := Free.chooseBasis R M
π : Free.ChooseBasisIndex R M → ↥⋯.toFinset := fun i ↦ ⟨l (b i), ⋯⟩
σ : ↥⋯.toFinset → Free.ChooseBasisIndex R M
hσ : ∀ (b_1 : ↥⋯.toFinset), π (σ b_1) = b_1
hπ : Subtype.val ∘ π = ⇑l ∘ ⇑b
hσ₁ : π ∘ σ = id
hσ₂ : ⇑l ∘ ⇑b ∘ σ = Subtype.val
f : M →ₗ[R] ↥⋯.toFinset →₀ R := lmapDomain R R π ∘ₗ ↑b.repr
x : ↥⋯.toFinset →₀ R
⊢ (linearCombination R Subtype.val) x = 0 ↔ ∃ y, l y = 0 ∧ f y = x

Tactic:
constructor

Goal state after:
case mp
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R M
inst✝ : Module.Finite R M
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
hl' : l.ker.FG
b : Basis (Free.ChooseBasisIndex R M) R M := ⋯
π : Free.ChooseBasisIndex R M → ↥⋯.toFinset := ⋯
σ : ↥⋯.toFinset → Free.ChooseBasisIndex R M
hσ : ∀ (b_1 : ↥⋯.toFinset), π (σ b_1) = b_1
hπ : Subtype.val ∘ π = ⇑l ∘ ⇑b
hσ₁ : π ∘ σ = id
hσ₂ : ⇑l ∘ ⇑b ∘ σ = Subtype.val
f : M →ₗ[R] ↥⋯.toFinset →₀ R := ⋯
x : ↥⋯.toFinset →₀ R
⊢ (linearCombination R Subtype.val) x = 0 → ∃ y, l y = 0 ∧ f y = x

case mpr
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R M
inst✝ : Module.Finite R M
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
hl' : l.ker.FG
b : Basis (Free.ChooseBasisIndex R M) R M := ⋯
π : Free.ChooseBasisIndex R M → ↥⋯.toFinset := ⋯
σ : ↥⋯.toFinset → Free.ChooseBasisIndex R M
hσ : ∀ (b_1 : ↥⋯.toFinset), π (σ b_1) = b_1
hπ : Subtype.val ∘ π = ⇑l ∘ ⇑b
hσ₁ : π ∘ σ = id
hσ₂ : ⇑l ∘ ⇑b ∘ σ = Subtype.val
f : M →ₗ[R] ↥⋯.toFinset →₀ R := ⋯
x : ↥⋯.toFinset →₀ R
⊢ (∃ y, l y = 0 ∧ f y = x) → (linearCombination R Subtype.val) x = 0