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

Goal state before:
pp: R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPc : StableUnderComposition fun {R S} [CommRing R] [CommRing S] ↦ P
hPp : LocalizationPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
hPl : HoldsForLocalization fun {R S} [CommRing R] [CommRing S] ↦ P
M : Submonoid R
T : Submonoid S
R' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : Algebra R R'
inst✝³ : IsLocalization M R'
S' : Type u
inst✝² : CommRing S'
inst✝¹ : Algebra S S'
inst✝ : IsLocalization T S'
f : R →+* S
hy : M ≤ Submonoid.comap f T
hf : P f
hle : Submonoid.map f M ≤ T
this✝¹ : Algebra (Localization (Submonoid.map f M)) S' :=
  IsLocalization.localizationAlgebraOfSubmonoidLe (Localization (Submonoid.map f M)) S' (Submonoid.map f M) T hle
this✝ : IsScalarTower S (Localization (Submonoid.map f M)) S'
this : IsLocalization (Submonoid.map (algebraMap S (Localization (Submonoid.map f M))) T) S'
⊢ IsLocalization.map S' f hy =
    (algebraMap (Localization (Submonoid.map f M)) S').comp (IsLocalization.map (Localization (Submonoid.map f M)) f ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.EssentialFiniteness

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : f.FiniteType
⊢ f.EssFiniteType

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.EssentialFiniteness

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : CommRing T
f : R →+* S
hf : f.EssFiniteType
g₁ g₂ : S →+* T
h₁ : g₁.comp f = g₂.comp f
h₂ : ∀ x ∈ hf.finset, g₁ x = g₂ x
⊢ g₁ = g₂

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPc : StableUnderComposition fun {R S} [CommRing R] [CommRing S] ↦ P
hPp : LocalizationPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
hPl : HoldsForLocalization fun {R S} [CommRing R] [CommRing S] ↦ P
M : Submonoid R
T : Submonoid S
R' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : Algebra R R'
inst✝³ : IsLocalization M R'
S' : Type u
inst✝² : CommRing S'
inst✝¹ : Algebra S S'
inst✝ : IsLocalization T S'
f : R →+* S
hy : M ≤ Submonoid.comap f T
hf : P f
⊢ P (IsLocalization.map S' f hy)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
H : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] ↦ P
hf : Function.Bijective ⇑f
this : Algebra R S := f.toAlgebra
⊢ Submonoid.powers 1 ≤ IsUnit.submonoid R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
H : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] ↦ P
hf : Function.Bijective ⇑f
⊢ P f

Tactic:
by

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

Goal state before:
pp: R✝ : Type u_1
inst✝¹⁸ : CommSemiring R✝
S✝ : Type u_2
inst✝¹⁷ : CommSemiring S✝
inst✝¹⁶ : Algebra R✝ S✝
P✝ : Type u_3
inst✝¹⁵ : CommSemiring P✝
p✝ : Ideal R✝
inst✝¹⁴ : p✝.IsPrime
Rₚ✝ : Type u_4
inst✝¹³ : CommSemiring Rₚ✝
inst✝¹² : Algebra R✝ Rₚ✝
inst✝¹¹ : IsLocalization.AtPrime Rₚ✝ p✝
inst✝¹⁰ : IsLocalRing Rₚ✝
Sₚ : Type u_5
inst✝⁹ : CommSemiring Sₚ
inst✝⁸ : Algebra S✝ Sₚ
inst✝⁷ : IsLocalization (algebraMapSubmonoid S✝ p✝.primeCompl) Sₚ
inst✝⁶ : Algebra Rₚ✝ Sₚ
P : Ideal S✝
S : Type u_6
R : Type u_7
inst✝⁵ : CommRing R
p : Ideal R
inst✝⁴ : p.IsMaximal
Rₚ : Type u_8
inst✝³ : CommRing Rₚ
inst✝² : Algebra R Rₚ
inst✝¹ : IsLocalization.AtPrime Rₚ p
inst✝ : IsLocalRing Rₚ
n : ℕ
⊢ (((Quotient.liftₐ (p ^ n) (ofId R (Rₚ ⧸ maximalIdeal Rₚ ^ n)) ⋯).comp
              (Quotient.liftₐ (maximalIdeal Rₚ ^ n) (liftAlgHom ⋯) ⋯)).comp
          (Quotient.mkₐ R (maximalIdeal Rₚ ^ n))).comp
      (algHom R R Rₚ) =
    ((AlgHom.id R (Rₚ ⧸ maximalIdeal Rₚ ^ n)).comp (Quotient.mkₐ R (maximalIdeal Rₚ ^ n))).comp (algHom R R Rₚ)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPc : StableUnderComposition fun {R S} [CommRing R] [CommRing S] ↦ P
hPl : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] ↦ P
⊢ StableUnderCompositionWithLocalizationAway fun {R S} [CommRing R] [CommRing S] ↦ P

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] ↦ P
⊢ ContainsIdentities fun {R S} [CommRing R] [CommRing S] ↦ P

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
R S T : Type u
x✝² : CommRing R
x✝¹ : CommRing S
x✝ : CommRing T
f : R →+* S
e : S ≃+* T
hf : P f
this✝ : Algebra S T := e.toRingHom.toAlgebra
this : IsLocalization.Away 1 R
⊢ IsUnit (f 1)

Tactic:
by

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

Goal state before:
pp: R✝ : Type u_1
inst✝¹⁸ : CommSemiring R✝
S✝ : Type u_2
inst✝¹⁷ : CommSemiring S✝
inst✝¹⁶ : Algebra R✝ S✝
P✝ : Type u_3
inst✝¹⁵ : CommSemiring P✝
p✝ : Ideal R✝
inst✝¹⁴ : p✝.IsPrime
Rₚ✝ : Type u_4
inst✝¹³ : CommSemiring Rₚ✝
inst✝¹² : Algebra R✝ Rₚ✝
inst✝¹¹ : IsLocalization.AtPrime Rₚ✝ p✝
inst✝¹⁰ : IsLocalRing Rₚ✝
Sₚ : Type u_5
inst✝⁹ : CommSemiring Sₚ
inst✝⁸ : Algebra S✝ Sₚ
inst✝⁷ : IsLocalization (algebraMapSubmonoid S✝ p✝.primeCompl) Sₚ
inst✝⁶ : Algebra Rₚ✝ Sₚ
P : Ideal S✝
S : Type u_6
R : Type u_7
inst✝⁵ : CommRing R
p : Ideal R
inst✝⁴ : p.IsMaximal
Rₚ : Type u_8
inst✝³ : CommRing Rₚ
inst✝² : Algebra R Rₚ
inst✝¹ : IsLocalization.AtPrime Rₚ p
inst✝ : IsLocalRing Rₚ
n : ℕ
⊢ (R ⧸ p ^ n) ≃ₐ[R] Rₚ ⧸ maximalIdeal Rₚ ^ n

Tactic:
by

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

Goal state before:
pp: R : Type u_7
inst✝⁵ : CommRing R
p : Ideal R
inst✝⁴ : p.IsMaximal
Rₚ : Type u_8
inst✝³ : CommRing Rₚ
inst✝² : Algebra R Rₚ
inst✝¹ : IsLocalization.AtPrime Rₚ p
inst✝ : IsLocalRing Rₚ
n : ℕ
x : R
s : ↥p.primeCompl
⊢ (equivQuotMaximalIdealPow p Rₚ n).symm ((Ideal.Quotient.mk (maximalIdeal Rₚ ^ n)) (mk' Rₚ x s)) *
      (Ideal.Quotient.mk (p ^ n)) ↑s =
    (Ideal.Quotient.mk (p ^ n)) x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
R S T : Type u
x✝² : CommRing R
x✝¹ : CommRing S
x✝ : CommRing T
f : R →+* S
e : S ≃+* T
hf : P f
⊢ (fun {R S} [CommRing R] [CommRing S] ↦ P) (e.toRingHom.comp f)

Tactic:
let := e.toRingHom.toAlgebra
    have : IsLocalization.Away (1 : R) R :=
      IsLocalization.away_of_isUnit_of_bijective _ isUnit_one (Equiv.refl _).bijective
    have : IsLocalization.Away (f 1) T :=
      IsLocalization.away_of_isUnit_of_bijective _ (by simp) e.bijective
    convert! hP f 1 R T hf
    trans (IsLocalization.Away.map R T f 1).comp (algebraMap R R)
    · rw [IsLocalization.Away.map, IsLocalization.map_comp]; rfl
    · rfl

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
R S T : Type u
x✝² : CommRing R
x✝¹ : CommRing S
x✝ : CommRing T
f : R →+* S
e : S ≃+* T
hf : P f
⊢ (fun {R S} [CommRing R] [CommRing S] ↦ P) (e.toRingHom.comp f)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
R S T : Type u
x✝² : CommRing R
x✝¹ : CommRing S
x✝ : CommRing T
f : S →+* T
e : R ≃+* S
hf : P f
this✝ : Algebra S R := e.symm.toRingHom.toAlgebra
this : IsLocalization.Away 1 R
⊢ IsUnit (f 1)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
R S T : Type u
x✝² : CommRing R
x✝¹ : CommRing S
x✝ : CommRing T
f : S →+* T
e : R ≃+* S
hf : P f
⊢ (fun {R S} [CommRing R] [CommRing S] ↦ P) (f.comp e.toRingHom)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : StableUnderCompositionWithLocalizationAway fun {R S} [CommRing R] [CommRing S] ↦ P
R S T : Type u
x✝² : CommRing R
x✝¹ : CommRing S
x✝ : CommRing T
f : R →+* S
e : S ≃+* T
hf : P f
⊢ (fun {R S} [CommRing R] [CommRing S] ↦ P) (e.toRingHom.comp f)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : StableUnderCompositionWithLocalizationAway fun {R S} [CommRing R] [CommRing S] ↦ P
R S T : Type u
x✝² : CommRing R
x✝¹ : CommRing S
x✝ : CommRing T
f : S →+* T
e : R ≃+* S
hf : P f
⊢ (fun {R S} [CommRing R] [CommRing S] ↦ P) (f.comp e.toRingHom)

Tactic:
by

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

Goal state before:
pp: S : Type u_6
R : Type u_7
inst✝⁸ : CommRing R
p : Ideal R
inst✝⁷ : p.IsMaximal
Sₚ : Type u_9
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing Sₚ
inst✝³ : Algebra S Sₚ
inst✝² : Algebra R Sₚ
inst✝¹ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝ : IsScalarTower R S Sₚ
x : S
hx : x ∈ comap (algebraMap S Sₚ) (Ideal.map (algebraMap S Sₚ) pS)
⊢ ∃ α ∉ p, α • x ∈ pS

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
H : LocalizationPreserves P
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
hf : P f
⊢ IsLocalization (Submonoid.map f (Submonoid.powers r)) S'

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
H : LocalizationPreserves P
⊢ LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : PropertyIsLocal P
hPi : ContainsIdentities fun {R S} [CommRing R] [CommRing S] ↦ P
R S : Type u
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
r : R
inst✝ : IsLocalization.Away r S
⊢ algebraMap R S = (algebraMap R S).comp (id R)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : PropertyIsLocal P
hPi : ContainsIdentities fun {R S} [CommRing R] [CommRing S] ↦ P
⊢ RingHom.HoldsForLocalizationAway P

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : OfLocalizationSpanTarget P
hP' : StableUnderCompositionWithLocalizationAwaySource P
⊢ OfLocalizationSpan P

Tactic:
by

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

Goal state before:
pp: S : Type u_6
R : Type u_7
inst✝⁸ : CommRing R
p : Ideal R
inst✝⁷ : p.IsMaximal
Sₚ : Type u_9
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing Sₚ
inst✝³ : Algebra S Sₚ
inst✝² : Algebra R Sₚ
inst✝¹ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝ : IsScalarTower R S Sₚ
x : S
hx : x ∈ comap (algebraMap S Sₚ) (Ideal.map (algebraMap S Sₚ) pS)
α : R
hα : α ∉ p
hαx : α • x ∈ pS
⊢ ∃ β, ∃ γ ∈ p, β * α = 1 + γ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Ring.Int.Field

Goal state before:
pp: x✝ : IsField ℤ
exists_pair_ne✝ : ∃ x y, x ≠ y
mul_comm✝ : ∀ (x y : ℤ), x * y = y * x
h : ∀ {a : ℤ}, a ≠ 0 → ∃ b, a * b = 1
this : 2 ≠ 0 → ∃ b, 2 * b = 1
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P
hPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set R
hs : Ideal.span s = ⊤
hT :
  ∀ (r : ↑s),
    ∃ Rᵣ Sᵣ x x_1 x_2 x_3,
      ∃ (_ : IsLocalization.Away (↑r) Rᵣ) (_ : IsLocalization.Away (f ↑r) Sᵣ),
        ∃ fᵣ, ∃ (_ : fᵣ.comp (algebraMap R Rᵣ) = (algebraMap S Sᵣ).comp f), P fᵣ
r : ↑s
Rᵣ Sᵣ : Type u
w✝⁵ : CommRing Rᵣ
w✝⁴ : CommRing Sᵣ
w✝³ : Algebra R Rᵣ
w✝² : Algebra S Sᵣ
w✝¹ : IsLocalization.Away (↑r) Rᵣ
w✝ : IsLocalization.Away (f ↑r) Sᵣ
fᵣ : Rᵣ →+* Sᵣ
hfᵣ : fᵣ.comp (algebraMap R Rᵣ) = (algebraMap S Sᵣ).comp f
hf : P fᵣ
e₁ : Localization (Submonoid.powers ↑r) ≃+* Rᵣ := (Localization.algEquiv (Submonoid.powers ↑r) Rᵣ).toRingEquiv
e₂ : Sᵣ ≃+* Localization (Submonoid.powers (f ↑r)) :=
  (IsLocalization.algEquiv (Submonoid.powers (f ↑r)) (Localization (Submonoid.powers (f ↑r))) Sᵣ).symm.toRingEquiv
x : R
⊢ fᵣ ((algebraMap R Rᵣ) x) = (algebraMap S Sᵣ) (f x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P
hPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set R
hs : Ideal.span s = ⊤
hT :
  ∀ (r : ↑s),
    ∃ Rᵣ Sᵣ x x_1 x_2 x_3,
      ∃ (_ : IsLocalization.Away (↑r) Rᵣ) (_ : IsLocalization.Away (f ↑r) Sᵣ),
        ∃ fᵣ, ∃ (_ : fᵣ.comp (algebraMap R Rᵣ) = (algebraMap S Sᵣ).comp f), P fᵣ
r : ↑s
Rᵣ Sᵣ : Type u
w✝⁵ : CommRing Rᵣ
w✝⁴ : CommRing Sᵣ
w✝³ : Algebra R Rᵣ
w✝² : Algebra S Sᵣ
w✝¹ : IsLocalization.Away (↑r) Rᵣ
w✝ : IsLocalization.Away (f ↑r) Sᵣ
fᵣ : Rᵣ →+* Sᵣ
hfᵣ : fᵣ.comp (algebraMap R Rᵣ) = (algebraMap S Sᵣ).comp f
hf : P fᵣ
e₁ : Localization (Submonoid.powers ↑r) ≃+* Rᵣ := (Localization.algEquiv (Submonoid.powers ↑r) Rᵣ).toRingEquiv
e₂ : Sᵣ ≃+* Localization (Submonoid.powers (f ↑r)) :=
  (IsLocalization.algEquiv (Submonoid.powers (f ↑r)) (Localization (Submonoid.powers (f ↑r))) Sᵣ).symm.toRingEquiv
⊢ Localization.awayMap f ↑r = (e₂.toRingHom.comp fᵣ).comp e₁.toRingHom

Tactic:
by

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

Goal state before:
pp: S : Type u_6
R : Type u_7
inst✝⁸ : CommRing R
p : Ideal R
inst✝⁷ : p.IsMaximal
Sₚ : Type u_9
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing Sₚ
inst✝³ : Algebra S Sₚ
inst✝² : Algebra R Sₚ
inst✝¹ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝ : IsScalarTower R S Sₚ
⊢ under S (Ideal.map (algebraMap R Sₚ) p) = pS

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P
hPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set R
hs : Ideal.span s = ⊤
hT :
  ∀ (r : ↑s),
    ∃ Rᵣ Sᵣ x x_1 x_2 x_3,
      ∃ (_ : IsLocalization.Away (↑r) Rᵣ) (_ : IsLocalization.Away (f ↑r) Sᵣ),
        ∃ fᵣ, ∃ (_ : fᵣ.comp (algebraMap R Rᵣ) = (algebraMap S Sᵣ).comp f), P fᵣ
⊢ P f

Tactic:
by

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

Goal state before:
pp: R✝ : Type u_1
inst✝²⁷ : CommSemiring R✝
S✝ : Type u_2
inst✝²⁶ : CommSemiring S✝
inst✝²⁵ : Algebra R✝ S✝
P✝ : Type u_3
inst✝²⁴ : CommSemiring P✝
p✝ : Ideal R✝
inst✝²³ : p✝.IsPrime
Rₚ✝ : Type u_4
inst✝²² : CommSemiring Rₚ✝
inst✝²¹ : Algebra R✝ Rₚ✝
inst✝²⁰ : IsLocalization.AtPrime Rₚ✝ p✝
inst✝¹⁹ : IsLocalRing Rₚ✝
Sₚ✝ : Type u_5
inst✝¹⁸ : CommSemiring Sₚ✝
inst✝¹⁷ : Algebra S✝ Sₚ✝
inst✝¹⁶ : IsLocalization (algebraMapSubmonoid S✝ p✝.primeCompl) Sₚ✝
inst✝¹⁵ : Algebra Rₚ✝ Sₚ✝
P : Ideal S✝
S : Type u_6
R : Type u_7
inst✝¹⁴ : CommRing R
p : Ideal R
inst✝¹³ : p.IsMaximal
Rₚ : Type u_8
inst✝¹² : CommRing Rₚ
inst✝¹¹ : Algebra R Rₚ
inst✝¹⁰ : IsLocalization.AtPrime Rₚ p
inst✝⁹ : IsLocalRing Rₚ
Sₚ : Type u_9
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P
hPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set R
hs : Ideal.span s = ⊤
hT :
  ∀ (r : ↑s),
    ∃ Rᵣ Sᵣ x x_1 x_2 x_3,
      ∃ (x_4 : IsLocalization.Away (↑r) Rᵣ) (x_5 : IsLocalization.Away (f ↑r) Sᵣ),
        P (IsLocalization.Away.map Rᵣ Sᵣ f ↑r)
⊢ P f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : OfLocalizationSpanTarget fun {R S} [CommRing R] [CommRing S] ↦ P
hP' : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set S
hs : Ideal.span s = ⊤
hT : ∀ (r : ↑s), ∃ T x x_1, ∃ (_ : IsLocalization.Away (↑r) T), P ((algebraMap S T).comp f)
⊢ P f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : OfLocalizationSpanTarget fun {R S} [CommRing R] [CommRing S] ↦ P
hQ : OfLocalizationSpanTarget fun {R S} [CommRing R] [CommRing S] ↦ Q
⊢ OfLocalizationSpanTarget fun {R S} [CommRing R] [CommRing S] f ↦ P f ∧ Q f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P
hQ : OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ Q
⊢ OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] f ↦ P f ∧ Q f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
hQ : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ Q
⊢ LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] f ↦ P f ∧ Q f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : StableUnderCompositionWithLocalizationAwayTarget fun {R S} [CommRing R] [CommRing S] ↦ P
hQ : StableUnderCompositionWithLocalizationAwayTarget fun {R S} [CommRing R] [CommRing S] ↦ Q
⊢ StableUnderCompositionWithLocalizationAwayTarget fun {R S} [CommRing R] [CommRing S] f ↦ P f ∧ Q f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : IsStableUnderBaseChange P
R S Rᵣ Sᵣ : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing Rᵣ
inst✝⁴ : CommRing Sᵣ
inst✝³ : Algebra R Rᵣ
inst✝² : Algebra S Sᵣ
M : Submonoid R
inst✝¹ : IsLocalization M Rᵣ
f : R →+* S
inst✝ : IsLocalization (Submonoid.map f M) Sᵣ
hf : P f
⊢ P (IsLocalization.map Sᵣ f ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : IsStableUnderBaseChange P
⊢ LocalizationPreserves fun {R S} [CommRing R] [CommRing S] ↦ P

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
p : Submonoid R
inst✝ : IsLocalization p S
I : Ideal R
⊢ Submodule.localized' S p (Algebra.linearMap R S) I = map (algebraMap R S) I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
r : R
J : Ideal R
h :
  ∀ (P : Ideal R) (x : P.IsMaximal),
    (algebraMap R (Localization.AtPrime P)) r ∈ map (algebraMap R (Localization.AtPrime P)) J
P : Ideal R
hP : P.IsMaximal
⊢ (?m.45 P) r ∈ Submodule.localized₀ P.primeCompl (?m.45 P) J

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
⊢ ⨅ p, ⨅ (x : p.IsPrime), ⨅ (_ : I ≤ p), RingHom.ker (algebraMap R (Localization.AtPrime p)) ≤ I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
h : ∀ (J : Ideal R) (x : J.IsMaximal), Ideal.map (algebraMap R (Localization.AtPrime J)) I = ⊥
P : Ideal R
hP : P.IsMaximal
⊢ Ideal.map (algebraMap R (Localization.AtPrime P)) I = Ideal.map (algebraMap R (Localization.AtPrime P)) ⊥

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Symmetric

Goal state before:
pp: α : Type u_1
n : Type u_3
inst✝² : CommSemiring α
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n α
h : A.IsSymm
k : ℕ
⊢ (A ^ k).IsSymm

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Symmetric

Goal state before:
pp: α : Type u_1
β : Type u_2
n : Type u_3
A : Matrix n n α
h : A.IsSymm
f : α → β
⊢ (A.map f).IsSymm

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Symmetric

Goal state before:
pp: α : Type u_1
β : Type u_2
n : Type u_3
A : Matrix n n α
f : α → β
hf : Function.Injective f
⊢ (A.map f).IsSymm ↔ A.IsSymm

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Symmetric

Goal state before:
pp: α : Type u_1
n : Type u_3
A : Matrix n n α
⊢ Aᵀ.IsSymm ↔ A.IsSymm

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Symmetric

Goal state before:
pp: α : Type u_1
n : Type u_3
inst✝ : InvolutiveStar α
A : Matrix n n α
⊢ Aᴴ.IsSymm ↔ A.IsSymm

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Symmetric

Goal state before:
pp: α : Type u_1
n : Type u_3
inst✝ : InvolutiveNeg α
A : Matrix n n α
⊢ (-A).IsSymm ↔ A.IsSymm

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Symmetric

Goal state before:
pp: α : Type u_1
n : Type u_3
R : Type u_5
inst✝² : Monoid R
inst✝¹ : MulAction R α
A : Matrix n n α
k : R
inst✝ : Invertible k
⊢ (k • A).IsSymm ↔ A.IsSymm

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Symmetric

Goal state before:
pp: α : Type u_1
n : Type u_3
m : Type u_4
A : Matrix n n α
h : A.IsSymm
f : n ≃ m
⊢ ((Matrix.reindex f f) A).IsSymm

Tactic:
by

Goal state after: