mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point

Goal state before:
pp: R : Type r
S : Type s
F : Type u
K : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Field F
inst✝⁷ : Field K
W' : Projective R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R F
inst✝⁴ : Algebra S F
inst✝³ : IsScalarTower R S F
inst✝² : Algebra R K
inst✝¹ : Algebra S K
inst✝ : IsScalarTower R S K
f : F →ₐ[S] K
P Q : Fin 3 → F
hP : (W'⁄F).Nonsingular P
hQ : (W'⁄F).Nonsingular Q
⊢ (W'⁄K).add (⇑↑f ∘ P) (⇑↑f ∘ Q) = (W'⁄K).add (⇑↑f ∘ P) (⇑↑f ∘ Q)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.RingHom.Smooth

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
⊢ Algebra.FormallySmooth R S ↔ Algebra.FormallySmooth R S

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.RingHom.Smooth

Goal state before:
pp: R S T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
H : Algebra.FormallySmooth R T
⊢ Algebra.FormallySmooth S (S ⊗[R] T)

Tactic:
]

Goal state after:
R S T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
H : Algebra.FormallySmooth R T
⊢ Algebra.FormallySmooth S (S ⊗[R] T)
Module: Mathlib.RingTheory.RingHom.Smooth

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
⊢ Algebra.Smooth R S ↔ Algebra.Smooth R S

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.RingHom.Smooth

Goal state before:
pp: R : Type u_3
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : f.FormallySmooth ∧ f.FinitePresentation
⊢ f.FormallySmooth

Tactic:
]

Goal state after:
R : Type u_3
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : f.FormallySmooth ∧ f.FinitePresentation
⊢ f.FormallySmooth
Module: Mathlib.RingTheory.RingHom.Smooth

Goal state before:
pp: R : Type u_3
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : f.FormallySmooth ∧ f.FinitePresentation
⊢ f.FinitePresentation

Tactic:
]

Goal state after:
R : Type u_3
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : f.FormallySmooth ∧ f.FinitePresentation
⊢ f.FinitePresentation
Module: Mathlib.RingTheory.RingHom.Smooth

Goal state before:
pp: x✝⁴ x✝³ : Type u_6
x✝² : CommRing x✝⁴
x✝¹ : CommRing x✝³
x✝ : x✝⁴ →+* x✝³
⊢ x✝.FormallySmooth ∧ x✝.FinitePresentation ↔ x✝.FormallySmooth ∧ x✝.FinitePresentation

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.RingHom.Smooth

Goal state before:
pp: ⊢ IsStableUnderBaseChange fun {R S} [CommRing R] [CommRing S] ↦ Smooth

Tactic:
by
  convert!
    RingHom.FormallySmooth.isStableUnderBaseChange.and
      RingHom.finitePresentation_isStableUnderBaseChange
  rw [smooth_def]

Goal state after:
Module: Mathlib.RingTheory.RingHom.Smooth

Goal state before:
pp: R S : Type u_6
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : R
h : IsLocalization.Away r S
⊢ Algebra.Smooth R S

Tactic:
]

Goal state after:
R S : Type u_6
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
r : R
h : IsLocalization.Away r S
⊢ Algebra.Smooth R S
Module: Mathlib.RingTheory.RingHom.Smooth

Goal state before:
pp: R : Type u_3
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : Function.Bijective ⇑f
⊢ f.FormallySmooth ∧ f.FinitePresentation

Tactic:
]

Goal state after:
R : Type u_3
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : Function.Bijective ⇑f
⊢ f.FormallySmooth ∧ f.FinitePresentation
Module: Mathlib.RingTheory.RingHom.Unramified

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
⊢ Algebra.FormallyUnramified R S ↔ Algebra.FormallyUnramified R S

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.RingHom.Smooth

Goal state before:
pp: R S : Type u_6
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set S
hs : Ideal.span s = ⊤
hf : ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] ↦ Smooth) ((algebraMap S (Localization.Away ↑r)).comp f)
this : f.FinitePresentation
algInst✝ : Algebra R S := f.toAlgebra
algebraizeInst✝ : Algebra.FinitePresentation R S
⊢ ↑(⨆ i ∈ s, PrimeSpectrum.basicOpen i) ⊆ Algebra.smoothLocus R S

Tactic:
]

Goal state after:
R S : Type u_6
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set S
hs : Ideal.span s = ⊤
hf : ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] ↦ Smooth) ((algebraMap S (Localization.Away ↑r)).comp f)
this : f.FinitePresentation
algInst✝ : Algebra R S := f.toAlgebra
algebraizeInst✝ : Algebra.FinitePresentation R S
⊢ ↑(⨆ i ∈ s, PrimeSpectrum.basicOpen i) ⊆ Algebra.smoothLocus R S
Module: Mathlib.RingTheory.RingHom.Unramified

Goal state before:
pp: R S T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
H : Algebra.FormallyUnramified R T
⊢ Algebra.FormallyUnramified S (TensorProduct R S T)

Tactic:
]

Goal state after:
R S T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
H : Algebra.FormallyUnramified R T
⊢ Algebra.FormallyUnramified S (TensorProduct R S T)
Module: Mathlib.RingTheory.RingHom.Unramified

Goal state before:
pp: R S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
M : Submonoid R
inst✝ : IsLocalization M S
⊢ Algebra.FormallyUnramified R S

Tactic:
]

Goal state after:
R S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
M : Submonoid R
inst✝ : IsLocalization M S
⊢ Algebra.FormallyUnramified R S
Module: Mathlib.RingTheory.RingHom.Unramified

Goal state before:
pp: R S : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
H :
  ∀ (J : Ideal S) (x : J.IsPrime),
    (fun {R S} [CommRing R] [CommRing S] ↦ FormallyUnramified) (Localization.localRingHom (Ideal.comap f J) J f ⋯)
algInst✝ : Algebra R S := f.toAlgebra
⊢ ∀ (q : PrimeSpectrum S), Algebra.IsUnramifiedAt R q.asIdeal

Tactic:
]

Goal state after:
R S : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
H :
  ∀ (J : Ideal S) (x : J.IsPrime),
    (fun {R S} [CommRing R] [CommRing S] ↦ FormallyUnramified) (Localization.localRingHom (Ideal.comap f J) J f ⋯)
algInst✝ : Algebra R S := f.toAlgebra
⊢ ∀ (q : PrimeSpectrum S), Algebra.IsUnramifiedAt R q.asIdeal
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
f : A
⊢ Subsingleton (H1Cotangent R (Localization.Away f)) ∧
      Module.Projective (Localization.Away f) Ω[Localization.Away f⁄R] ↔
    Subsingleton (LocalizedModule.Away f (H1Cotangent R A)) ∧ (PrimeSpectrum.zeroLocus {f})ᶜ ⊆ Module.freeLocus A Ω[A⁄R]

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
f : A
⊢ Subsingleton (H1Cotangent R (Localization.Away f)) ∧
      Module.Projective (Localization.Away f) Ω[Localization.Away f⁄R] ↔
    Subsingleton (LocalizedModule.Away f (H1Cotangent R A)) ∧ (PrimeSpectrum.zeroLocus {f})ᶜ ⊆ Module.freeLocus A Ω[A⁄R]
Module: Mathlib.RingTheory.RingHom.Unramified

Goal state before:
pp: R S : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set S
hs : Ideal.span s = ⊤
H :
  ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] ↦ FormallyUnramified) ((algebraMap S (Localization.Away ↑r)).comp f)
algInst✝ : Algebra R S := f.toAlgebra
⊢ ∀ (q : PrimeSpectrum S), Algebra.IsUnramifiedAt R q.asIdeal

Tactic:
]

Goal state after:
R S : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set S
hs : Ideal.span s = ⊤
H :
  ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] ↦ FormallyUnramified) ((algebraMap S (Localization.Away ↑r)).comp f)
algInst✝ : Algebra R S := f.toAlgebra
⊢ ∀ (q : PrimeSpectrum S), Algebra.IsUnramifiedAt R q.asIdeal
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: case e'_1
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
f : A
this : LocalizedModule (Submonoid.powers f) (H1Cotangent R A) ≃ₗ[A] H1Cotangent R (Localization.Away f)
⊢ Subsingleton (H1Cotangent R (Localization.Away f)) ↔ Subsingleton (H1Cotangent R (Localization.Away f))

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: case e'_2
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
f : A
⊢ Module.Projective (Localization.Away f) Ω[Localization.Away f⁄R] ↔
    Module.Projective (Localization.Away f) (LocalizedModule.Away f Ω[A⁄R])

Tactic:
]

Goal state after:
case e'_2
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
f : A
⊢ Module.Projective (Localization.Away f) Ω[Localization.Away f⁄R] ↔
    Module.Projective (Localization.Away f) (LocalizedModule.Away f Ω[A⁄R])
Module: Mathlib.RingTheory.RingHom.Unramified

Goal state before:
pp: R S : Type u_3
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set S
hs : Ideal.span s = ⊤
H :
  ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] ↦ FormallyUnramified) ((algebraMap S (Localization.Away ↑r)).comp f)
algInst✝ : Algebra R S := f.toAlgebra
x : PrimeSpectrum S
r : S
hr : r ∈ s
hrx : x ∈ PrimeSpectrum.basicOpen r
⊢ Algebra.FormallyUnramified R (Localization.Away r) ↔ Algebra.FormallyUnramified R (Localization.Away r)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
f : A
this : FinitePresentation A (Localization.Away f)
⊢ FormallySmooth R (Localization.Away f) ↔ Smooth R (Localization.Away f)

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
f : A
this : FinitePresentation A (Localization.Away f)
⊢ FormallySmooth R (Localization.Away f) ↔ Smooth R (Localization.Away f)
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
this : A ≃ₐ[A] Localization.Away 1
⊢ smoothLocus R A = Set.univ ↔ ↑(PrimeSpectrum.basicOpen 1) ⊆ smoothLocus R A

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
this : A ≃ₐ[A] Localization.Away 1
⊢ smoothLocus R A = Set.univ ↔ ↑(PrimeSpectrum.basicOpen 1) ⊆ smoothLocus R A
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Smooth R A
⊢ FormallySmooth R A

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Smooth R A
⊢ FormallySmooth R A
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
⊢ ∀ x ∈ smoothLocus R A, ∃ t ⊆ smoothLocus R A, IsOpen t ∧ x ∈ t

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
⊢ ∀ x ∈ smoothLocus R A, ∃ t ⊆ smoothLocus R A, IsOpen t ∧ x ∈ t
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
hx : x ∈ smoothLocus R A
f : A
hxf : x ∈ ↑(basicOpen f)
hf : Module.Projective (Localization.Away f) (LocalizedModule.Away f Ω[A⁄R])
⊢ ∃ t ⊆ smoothLocus R A, IsOpen t ∧ x ∈ t

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
hx : x ∈ smoothLocus R A
f : A
hxf : x ∈ ↑(basicOpen f)
hf : Module.Projective (Localization.Away f) (LocalizedModule.Away f Ω[A⁄R])
⊢ ∃ t ⊆ smoothLocus R A, IsOpen t ∧ x ∈ t
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
hx : x ∈ smoothLocus R A
f : A
hxf : x ∈ ↑(basicOpen f)
hf : Module.Projective (Localization.Away f) (LocalizedModule.Away f Ω[A⁄R])
Af : Type u_2 := Localization.Away f
this✝³ : FinitePresentation A (Localization.Away f)
this✝² : FinitePresentation R (Localization.Away f)
this✝¹ : LocalizedModule (Submonoid.powers f) Ω[A⁄R] ≃ₗ[A] Ω[Localization.Away f⁄R]
this✝ : LocalizedModule (Submonoid.powers f) Ω[A⁄R] ≃ₗ[Localization.Away f] Ω[Localization.Away f⁄R]
this : Module.Projective (Localization.Away f) Ω[Localization.Away f⁄R]
⊢ IsOpen ((zeroLocus ↑(Module.annihilator Af (H1Cotangent R Af)))ᶜ ∩ Module.freeLocus Af Ω[Af⁄R])

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
hx : x ∈ smoothLocus R A
f : A
hxf : x ∈ ↑(basicOpen f)
hf : Module.Projective (Localization.Away f) (LocalizedModule.Away f Ω[A⁄R])
Af : Type u_2 := Localization.Away f
this✝³ : FinitePresentation A (Localization.Away f)
this✝² : FinitePresentation R (Localization.Away f)
this✝¹ : LocalizedModule (Submonoid.powers f) Ω[A⁄R] ≃ₗ[A] Ω[Localization.Away f⁄R]
this✝ : LocalizedModule (Submonoid.powers f) Ω[A⁄R] ≃ₗ[Localization.Away f] Ω[Localization.Away f⁄R]
this : Module.Projective (Localization.Away f) Ω[Localization.Away f⁄R]
⊢ IsOpen ((zeroLocus ↑(Module.annihilator Af (H1Cotangent R Af)))ᶜ ∩ Module.freeLocus Af Ω[Af⁄R])
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
hx : x ∈ smoothLocus R A
f : A
hxf : x ∈ ↑(basicOpen f)
hf : Module.Projective (Localization.Away f) (LocalizedModule.Away f Ω[A⁄R])
Af : Type u_2 := Localization.Away f
this✝¹ : FinitePresentation A (Localization.Away f)
this✝ : FinitePresentation R (Localization.Away f)
this : IsOpen (comap (algebraMap A Af) ⁻¹' smoothLocus R A)
⊢ ∃ t ⊆ smoothLocus R A, IsOpen t ∧ x ∈ t

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
hx : x ∈ smoothLocus R A
f : A
hxf : x ∈ ↑(basicOpen f)
hf : Module.Projective (Localization.Away f) (LocalizedModule.Away f Ω[A⁄R])
Af : Type u_2 := Localization.Away f
this✝¹ : FinitePresentation A (Localization.Away f)
this✝ : FinitePresentation R (Localization.Away f)
this : IsOpen (comap (algebraMap A Af) ⁻¹' smoothLocus R A)
⊢ ∃ t ⊆ smoothLocus R A, IsOpen t ∧ x ∈ t
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
hx : x ∈ smoothLocus R A
f : A
hxf : x ∈ ↑(basicOpen f)
hf : Module.Projective (Localization.Away f) (LocalizedModule.Away f Ω[A⁄R])
Af : Type u_2 := Localization.Away f
this✝¹ : FinitePresentation A (Localization.Away f)
this✝ : FinitePresentation R (Localization.Away f)
this : IsOpen (smoothLocus R A ∩ ↑(basicOpen f))
⊢ ∃ t ⊆ smoothLocus R A, IsOpen t ∧ x ∈ t

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
hx : x ∈ smoothLocus R A
f : A
hxf : x ∈ ↑(basicOpen f)
hf : Module.Projective (Localization.Away f) (LocalizedModule.Away f Ω[A⁄R])
Af : Type u_2 := Localization.Away f
this✝¹ : FinitePresentation A (Localization.Away f)
this✝ : FinitePresentation R (Localization.Away f)
this : IsOpen (smoothLocus R A ∩ ↑(basicOpen f))
⊢ ∃ t ⊆ smoothLocus R A, IsOpen t ∧ x ∈ t
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : FinitePresentation R A
p : Ideal A
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
f : A
hxf : { asIdeal := p, isPrime := inst✝¹ } ∈ ↑(basicOpen f)
hf : FormallySmooth R (Localization.Away f)
⊢ FormallySmooth R (Localization.Away f)

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : FinitePresentation R A
p : Ideal A
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
f : A
hxf : { asIdeal := p, isPrime := inst✝¹ } ∈ ↑(basicOpen f)
hf : FormallySmooth R (Localization.Away f)
⊢ FormallySmooth R (Localization.Away f)
Module: Mathlib.RingTheory.Etale.Pi

Goal state before:
pp: R : Type u_1
I : Type u_2
A : I → Type u_3
inst✝³ : CommRing R
inst✝² : (i : I) → CommRing (A i)
inst✝¹ : (i : I) → Algebra R (A i)
inst✝ : Finite I
⊢ ((∀ (i : I), FormallyUnramified R (A i)) ∧ ∀ (i : I), FormallySmooth R (A i)) ↔
    (∀ (x : I), FormallyUnramified R (A x)) ∧ ∀ (x : I), FormallySmooth R (A x)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Smooth.Pi

Goal state before:
pp: case g.map_one
R : Type u_1
I : Type u_2
A : I → Type u_3
inst✝³ : CommRing R
inst✝² : (i : I) → CommRing (A i)
inst✝¹ : (i : I) → Algebra R (A i)
inst✝ : FormallySmooth R ((i : I) → A i)
i : I
⊢ Pi.single i 1 - 1 ∈ RingHom.ker (Pi.evalAlgHom R A i).toRingHom ^ 2

Tactic:
]

Goal state after:
case g.map_one
R : Type u_1
I : Type u_2
A : I → Type u_3
inst✝³ : CommRing R
inst✝² : (i : I) → CommRing (A i)
inst✝¹ : (i : I) → Algebra R (A i)
inst✝ : FormallySmooth R ((i : I) → A i)
i : I
⊢ Pi.single i 1 - 1 ∈ RingHom.ker (Pi.evalAlgHom R A i).toRingHom ^ 2
Module: Mathlib.RingTheory.Smooth.Pi

Goal state before:
pp: R : Type u_1
I : Type u_2
A : I → Type u_3
inst✝³ : CommRing R
inst✝² : (i : I) → CommRing (A i)
inst✝¹ : (i : I) → Algebra R (A i)
inst✝ : Finite I
val✝ : Fintype I
H : ∀ (i : I), FormallySmooth R (A i)
B : Type (max u_1 u_2 u_3)
x✝¹ : CommRing B
x✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
g : ((i : I) → A i) →ₐ[R] B ⧸ J
x : B
hx : x ∈ RingHom.ker (Ideal.Quotient.mk J)
⊢ x ^ 2 ∈ J ^ 2

Tactic:
]

Goal state after:
R : Type u_1
I : Type u_2
A : I → Type u_3
inst✝³ : CommRing R
inst✝² : (i : I) → CommRing (A i)
inst✝¹ : (i : I) → Algebra R (A i)
inst✝ : Finite I
val✝ : Fintype I
H : ∀ (i : I), FormallySmooth R (A i)
B : Type (max u_1 u_2 u_3)
x✝¹ : CommRing B
x✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
g : ((i : I) → A i) →ₐ[R] B ⧸ J
x : B
hx : x ∈ RingHom.ker (Ideal.Quotient.mk J)
⊢ x ^ 2 ∈ J ^ 2
Module: Mathlib.RingTheory.Unramified.Pi

Goal state before:
pp: case intro.mpr
R : Type u_1
I : Type u_2
inst✝³ : Finite I
f : I → Type u_3
inst✝² : CommRing R
inst✝¹ : (i : I) → CommRing (f i)
inst✝ : (i : I) → Algebra R (f i)
val✝ : Fintype I
H : ∀ (i : I), FormallyUnramified R (f i)
⊢ ∀ ⦃B : Type (max u_2 u_3)⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I_1 : Ideal B),
    I_1 ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I_1).comp

Tactic:
]

Goal state after:
case intro.mpr
R : Type u_1
I : Type u_2
inst✝³ : Finite I
f : I → Type u_3
inst✝² : CommRing R
inst✝¹ : (i : I) → CommRing (f i)
inst✝ : (i : I) → Algebra R (f i)
val✝ : Fintype I
H : ∀ (i : I), FormallyUnramified R (f i)
⊢ ∀ ⦃B : Type (max u_2 u_3)⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I_1 : Ideal B),
    I_1 ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I_1).comp
Module: Mathlib.RingTheory.Unramified.Pi

Goal state before:
pp: case intro.mpr
R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g : (i : I) → f i
⊢ ∑ x, f₁ (Pi.single x (g x)) = ∑ x, f₂ (Pi.single x (g x))

Tactic:
]

Goal state after:
case intro.mpr
R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g : (i : I) → f i
⊢ ∑ x, f₁ (Pi.single x (g x)) = ∑ x, f₂ (Pi.single x (g x))
Module: Mathlib.RingTheory.Unramified.Pi

Goal state before:
pp: R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g✝ : (i : I) → f i
x : I
g : (i : I) → f i
⊢ (Ideal.Quotient.mk J) (f₁ g) = (Ideal.Quotient.mk J) (f₂ g)

Tactic:
]

Goal state after:
R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g✝ : (i : I) → f i
x : I
g : (i : I) → f i
⊢ (Ideal.Quotient.mk J) (f₁ g) = (Ideal.Quotient.mk J) (f₂ g)