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

Goal state before:
pp: R : Type u_1
M : Type u_2
A : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : Ring A
inst✝³ : Algebra R A
inst✝² : FaithfullyFlat R A
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p q : Submodule R M
⊢ baseChange A p = baseChange A q ↔ p = q

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
A : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : Ring A
inst✝³ : Algebra R A
inst✝² : FaithfullyFlat R A
inst✝¹ : AddCommGroup M
inst✝ : Module R M
h : IsNoetherian A (A ⊗[R] M)
⊢ IsNoetherian R M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SurjectiveOnStalks

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
T : Type u_3
inst✝² : CommRing T
inst✝¹ : Algebra R T
inst✝ : Algebra R S
hf₂ : (algebraMap R T).SurjectiveOnStalks
x : S ⊗[R] T
J : Ideal T
hJ : J.IsPrime
⊢ ∃ t r a, r • t ∉ J ∧ 1 ⊗ₜ[R] (r • t) * x = a ⊗ₜ[R] t

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SurjectiveOnStalks

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Submonoid R
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
⊢ (algebraMap R S).SurjectiveOnStalks

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SurjectiveOnStalks

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
T : Type u_3
inst✝² : CommRing T
inst✝¹ : Algebra R T
inst✝ : Algebra R S
hf : (algebraMap R T).SurjectiveOnStalks
⊢ (algebraMap S (S ⊗[R] T)).SurjectiveOnStalks

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SurjectiveOnStalks

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
S : Type u_2
inst✝³ : CommRing S
T : Type u_3
inst✝² : CommRing T
inst✝¹ : Algebra R T
inst✝ : Algebra R S
hf : (algebraMap R S).SurjectiveOnStalks
⊢ Algebra.TensorProduct.includeRight.SurjectiveOnStalks

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SurjectiveOnStalks

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
S : Type u_2
inst✝⁵ : CommRing S
T : Type u_3
inst✝⁴ : CommRing T
S' : Type u_4
inst✝³ : CommRing S'
inst✝² : Algebra R S
inst✝¹ : Algebra R T
inst✝ : Algebra R S'
f : S →ₐ[R] S'
Hf : f.SurjectiveOnStalks
⊢ (Algebra.TensorProduct.map f (AlgHom.id R T)).SurjectiveOnStalks

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SurjectiveOnStalks

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommRing R
S : Type u_2
inst✝⁷ : CommRing S
T : Type u_3
inst✝⁶ : CommRing T
S' : Type u_4
T' : Type u_5
inst✝⁵ : CommRing S'
inst✝⁴ : CommRing T'
inst✝³ : Algebra R S
inst✝² : Algebra R T
inst✝¹ : Algebra R S'
inst✝ : Algebra R T'
f : S →ₐ[R] S'
Hf : f.SurjectiveOnStalks
g : T →ₐ[R] T'
Hg : g.SurjectiveOnStalks
⊢ (Algebra.TensorProduct.map f g).SurjectiveOnStalks

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : Flat A B
h : Function.Surjective (PrimeSpectrum.comap (algebraMap A B))
m : Ideal A
hm : m.IsMaximal
m' : PrimeSpectrum B
hm' : PrimeSpectrum.comap (algebraMap A B) m' = { asIdeal := m, isPrime := ⋯ }
⊢ m = Ideal.comap (algebraMap A B) m'.asIdeal

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.SurjectiveOnStalks

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
f : R →+* S
inst✝¹ : IsLocalRing S
inst✝ : IsLocalHom f
⊢ f.SurjectiveOnStalks ↔ Function.Surjective ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : Flat A B
h : Function.Surjective (PrimeSpectrum.comap (algebraMap A B))
⊢ FaithfullyFlat A B

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : NonAssocSemiring R
a b c : R
mul : a * b = c * a
add_ab : a + b = 1
add_ac : a + c = 1
⊢ b = (a + c) * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : NonAssocSemiring R
a b c : R
mul : a * b = c * a
add_ab : a + b = 1
add_ac : a + c = 1
⊢ (a + c) * b = c * (a + b)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : NonAssocSemiring R
a b c : R
mul : a * b = c * a
add_ab : a + b = 1
add_ac : a + c = 1
⊢ c * (a + b) = c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
eq : (↑a).1 = (↑b).1
⊢ a = b

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
I : Ideal R
inst✝¹ : I.IsPrime
J : Ideal S
inst✝ : J.IsPrime
f : R →+* S
hf : I = comap f J
r : R
⊢ (map I J f hf) ((algebraMap R I.ResidueField) r) = (algebraMap S J.ResidueField) (f r)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
f : R →+* S
H : f.SurjectiveOnStalks
I : Ideal R
inst✝¹ : I.IsPrime
J : Ideal S
inst✝ : J.IsPrime
hf : I = Ideal.comap f J
⊢ Function.Bijective ⇑(Ideal.ResidueField.map I J f hf)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
I✝ : Ideal R
inst✝² : I✝.IsPrime
I : Ideal A
inst✝¹ : I.IsPrime
J : Ideal B
inst✝ : J.IsPrime
f : A →ₐ[R] B
hf : I = comap f.toRingHom J
r : R
⊢ (↑↑(map I J (↑f) hf)).toFun ((algebraMap R I.ResidueField) r) = (algebraMap R J.ResidueField) r

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : I.IsPrime
x : R
⊢ (algebraMap R I.ResidueField) x = 0 ↔ x ∈ I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : I.IsPrime
⊢ Function.Injective ⇑(algebraMap (R ⧸ I) I.ResidueField)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : IsLocalRing A
inst✝² : IsLocalRing B
inst✝¹ : Flat A B
inst✝ : IsLocalHom (algebraMap A B)
⊢ FaithfullyFlat A B

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : IsDomain B
inst✝² : Flat A B
inst✝¹ : Algebra.IsIntegral A B
inst✝ : FaithfulSMul A B
P : PrimeSpectrum A
⊢ RingHom.ker (algebraMap A B) ≤ P.asIdeal

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
eq : (↑a).2 = (↑b).2
⊢ a = b

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : IsDomain B
inst✝² : Flat A B
inst✝¹ : Algebra.IsIntegral A B
inst✝ : FaithfulSMul A B
⊢ FaithfullyFlat A B

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
a✝ b✝ a b c : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
hab : (↑a).1 * (↑b).1 = (↑a).1
hbc : (↑b).1 * (↑c).1 = (↑b).1
⊢ (↑a).1 * (↑c).1 = (↑a).1

Tactic:
rw [← hab, mul_assoc, hbc]

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : FaithfullyFlat A B
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module A M
⊢ lTensor B ((TensorProduct.mk A B M) 1) = ↑(leftComm A B B M).symm ∘ₗ (TensorProduct.mk A B (B ⊗[A] M)) 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
a✝ b✝ a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
hab : (↑a).1 * (↑b).1 = (↑a).1
hba : (↑b).1 * (↑a).1 = (↑b).1
⊢ (↑a).1 = (↑b).1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra R A
inst✝¹ : Algebra R B
I : Ideal R
inst✝ : I.IsPrime
x : I.ResidueField
⊢ ∃ x_1, x * (algebraMap (R ⧸ I) I.ResidueField) ↑x_1.2 = (algebraMap (R ⧸ I) I.ResidueField) x_1.1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : FaithfullyFlat A B
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module A M
⊢ Function.Injective ⇑((TensorProduct.mk A B M) 1)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : FaithfullyFlat A B
⊢ FaithfulSMul A B

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.RingHom

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
s : Set R
⊢ comap f ⁻¹' zeroLocus s = zeroLocus (⇑f '' s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
a✝ b✝ a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
⊢ ((↑a).1 + (↑a).2 * (↑b).1, (↑a).2 * (↑b).2).1 * ((↑a).1 + (↑a).2 * (↑b).1, (↑a).2 * (↑b).2).2 = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : Module.FaithfullyFlat A B
I : Ideal A
⊢ Function.Injective ⇑(↑A (quotIdealMapEquivTensorQuot B I).symm.toLinearMap ∘ₗ (TensorProduct.mk A B (A ⧸ I)) 1)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
a✝ b✝ a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
⊢ ((↑a).1 + (↑a).2 * (↑b).1, (↑a).2 * (↑b).2).1 + ((↑a).1 + (↑a).2 * (↑b).1, (↑a).2 * (↑b).2).2 = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.RingHom

Goal state before:
pp: R : Type u
S : Type v
S' : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : CommSemiring S'
f : R →+* S
e : R ≃+* S
x : PrimeSpectrum R
⊢ comap e.toRingHom (comap e.symm.toRingHom x) = x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
a✝ b✝ a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
⊢ a ≤ ⟨((↑a).1 + (↑a).2 * (↑b).1, (↑a).2 * (↑b).2), ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
a✝ b✝ a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
⊢ b ≤ ⟨((↑a).1 + (↑a).2 * (↑b).1, (↑a).2 * (↑b).2), ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra R A
inst✝¹ : Algebra R B
I : Ideal R
inst✝ : I.IsPrime
x y : R ⧸ I
e : (algebraMap (R ⧸ I) I.ResidueField) x = (algebraMap (R ⧸ I) I.ResidueField) y
⊢ ∃ c, ↑c * x = ↑c * y

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.RingHom

Goal state before:
pp: R : Type u
S : Type v
S' : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : CommSemiring S'
f : R →+* S
e : R ≃+* S
x : PrimeSpectrum S
⊢ comap e.symm.toRingHom (comap e.toRingHom x) = x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
a✝ b✝ a b c : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
hac : a ≤ c
hbc : b ≤ c
⊢ ⟨((↑a).1 + (↑a).2 * (↑b).1, (↑a).2 * (↑b).2), ⋯⟩ ≤ c

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra R B
I : Ideal R
inst✝¹ : I.IsPrime
inst✝ : IsDomain R
⊢ ∀ (x : R), (algebraMap R ⊥.ResidueField) x = (algebraMap (R ⧸ ⊥) ⊥.ResidueField) ((RingEquiv.quotientBot R).symm x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : Module.FaithfullyFlat A B
I : Ideal A
inj : Function.Injective ⇑(↑A (quotIdealMapEquivTensorQuot B I).symm.toLinearMap ∘ₗ (TensorProduct.mk A B (A ⧸ I)) 1)
x : A
hx : (Quotient.mk (map (algebraMap A B) I)) ((algebraMap A B) x) = 0
⊢ (↑A ↑↑(quotIdealMapEquivTensorQuot B I).symm ∘ₗ (TensorProduct.mk A B (A ⧸ I)) 1) ((Quotient.mk I) x) = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : I.IsMaximal
⊢ Function.Surjective ⇑(algebraMap R I.ResidueField)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.RingHom

Goal state before:
pp: ι : Type u_3
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
x✝¹ x✝ : (i : ι) × PrimeSpectrum (R i)
i : ι
p : PrimeSpectrum (R i)
j : ι
q : PrimeSpectrum (R j)
eq : sigmaToPi R ⟨i, p⟩ = sigmaToPi R ⟨j, q⟩
ne : i ≠ j
⊢ Function.update 1 j 0 ∈ (sigmaToPi R ⟨j, q⟩).asIdeal

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
a✝ b✝ a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
⊢ (aᶜ ⊔ bᶜ)ᶜ ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
I : Ideal R
inst✝² : I.IsPrime
k : Type u_5
inst✝¹ : Field k
p : Ideal k
inst✝ : p.IsPrime
⊢ p.IsMaximal

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.RingHom

Goal state before:
pp: ι : Type u_3
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
x✝¹ x✝ : (i : ι) × PrimeSpectrum (R i)
i : ι
p : PrimeSpectrum (R i)
j : ι
q : PrimeSpectrum (R j)
eq : sigmaToPi R ⟨i, p⟩ = sigmaToPi R ⟨j, q⟩
⊢ ⟨i, p⟩ = ⟨j, q⟩

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : CommRing A
inst✝¹¹ : CommRing B
inst✝¹⁰ : Algebra R A
inst✝⁹ : Algebra R B
I : Ideal R
inst✝⁸ : I.IsPrime
J : Ideal A
K : Ideal B
inst✝⁷ : J.IsPrime
inst✝⁶ : K.IsPrime
inst✝⁵ : J.LiesOver I
inst✝⁴ : Algebra (Localization.AtPrime I) (Localization.AtPrime J)
inst✝³ : IsLiesOverAlgebra I J
inst✝² : K.LiesOver I
inst✝¹ : Algebra (Localization.AtPrime I) (Localization.AtPrime K)
inst✝ : IsLiesOverAlgebra I K
⊢ IsLocalHom (algebraMap (Localization.AtPrime I) (Localization.AtPrime J))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
a✝ b✝ a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
⊢ (aᶜ ⊔ bᶜ)ᶜ ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : Module.FaithfullyFlat A B
I : Ideal A
⊢ comap (algebraMap A B) (map (algebraMap A B) I) = I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : Module.FaithfullyFlat A B
x✝¹ x✝ : Ideal A
h : map (algebraMap A B) x✝¹ = map (algebraMap A B) x✝
⊢ x✝¹ = x✝

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : CommRing A
inst✝⁹ : CommRing B
inst✝⁸ : Algebra R A
inst✝⁷ : Algebra R B
I : Ideal R
inst✝⁶ : I.IsPrime
inst✝⁵ : Algebra.EssFiniteType R A
p : Ideal R
inst✝⁴ : p.IsPrime
q : Ideal A
inst✝³ : q.IsPrime
inst✝² : q.LiesOver p
inst✝¹ : Algebra (Localization.AtPrime p) (Localization.AtPrime q)
inst✝ : Localization.AtPrime.IsLiesOverAlgebra p q
⊢ Algebra.EssFiniteType p.ResidueField q.ResidueField

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra A B
inst✝¹ : Module.FaithfullyFlat A B
p : Ideal A
inst✝ : p.IsPrime
⊢ ∃ P, P.IsPrime ∧ P.LiesOver p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra R A
inst✝¹ : Algebra R B
I : Ideal R
inst✝ : I.IsPrime
f : R →+* S
hf₁ : I ≤ RingHom.ker f
hf₂ : I.primeCompl ≤ Submonoid.comap f (IsUnit.submonoid S)
⊢ ∀ (y : ↥(R ⧸ I)⁰), IsUnit ((Quotient.lift I f hf₁) ↑y)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.RingHom

Goal state before:
pp: ι : Type u_3
R : ι → Type u_2
inst✝² : (i : ι) → CommSemiring (R i)
inst✝¹ : Infinite ι
inst✝ : ∀ (i : ι), Nontrivial (R i)
⊢ ∀ (c : (i : ι) → R i) {x : (i : ι) → R i},
    x ∈ (AddMonoidHom.mrange DFinsupp.coeFnAddMonoidHom).carrier →
      c • x ∈ (AddMonoidHom.mrange DFinsupp.coeFnAddMonoidHom).carrier

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
I : Ideal R
inst✝ : I.IsPrime
f : R →+* S
hf₁ : I ≤ RingHom.ker f
hf₂ : I.primeCompl ≤ Submonoid.comap f (IsUnit.submonoid S)
r : R
⊢ (lift I f hf₁ hf₂) ((algebraMap R I.ResidueField) r) = f r

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra R B
I✝ : Ideal R
inst✝¹ : I✝.IsPrime
I : Ideal A
inst✝ : I.IsPrime
f : A →ₐ[R] B
hf₁ : I ≤ RingHom.ker f
hf₂ : I.primeCompl ≤ Submonoid.comap f (IsUnit.submonoid B)
r : R
⊢ (↑↑(lift I f.toRingHom hf₁ hf₂)).toFun ((algebraMap R I.ResidueField) r) = (algebraMap R B) r

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Ideal

Goal state before:
pp: R : Type u_1
A : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
I : Ideal A
inst✝ : I.IsPrime
⊢ mapₐ I I (AlgHom.id R A) ⋯ = AlgHom.id R I.ResidueField

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
a✝ b✝ a b c : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
hab : a ≤ b
hac : a ≤ c
⊢ a ≤ (bᶜ ⊔ cᶜ)ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.RingHom

Goal state before:
pp: ι : Type u_3
R : ι → Type u_2
inst✝² : (i : ι) → CommSemiring (R i)
inst✝¹ : Infinite ι
inst✝ : ∀ (i : ι), Nontrivial (R i)
J : Ideal ((i : ι) → R i) :=
  let __spread.0 := AddMonoidHom.mrange DFinsupp.coeFnAddMonoidHom;
  { toAddSubmonoid := __spread.0, smul_mem' := ⋯ }
⊢ 1 ∉ J

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.MinimalPrime.Localization

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
x : R
⊢ x ∈ ⋃ p ∈ I.minimalPrimes, ↑p ↔ x ∈ {x | ∃ y ∉ I.radical, x * y ∈ I.radical}

Tactic:
simp only [Set.mem_iUnion, SetLike.mem_coe, exists_prop, Set.mem_ofPred_eq]

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
x : R
⊢ (∃ i, I.IsMinimalPrime i ∧ x ∈ i) ↔ ∃ y ∉ I.radical, x * y ∈ I.radical
Module: Mathlib.RingTheory.Spectrum.Prime.RingHom

Goal state before:
pp: ι : Type u_3
R : ι → Type u_2
inst✝² : (i : ι) → CommSemiring (R i)
inst✝¹ : Infinite ι
inst✝ : ∀ (i : ι), Nontrivial (R i)
J : Ideal ((i : ι) → R i) :=
  let __spread.0 := AddMonoidHom.mrange DFinsupp.coeFnAddMonoidHom;
  { toAddSubmonoid := __spread.0, smul_mem' := ⋯ }
I : Ideal ((i : ι) → R i)
max : I.IsMaximal
le : J ≤ I
x✝ : { asIdeal := I, isPrime := ⋯ } ∈ Set.range (sigmaToPi R)
i : ι
p : PrimeSpectrum (R i)
eq : sigmaToPi R ⟨i, p⟩ = { asIdeal := I, isPrime := ⋯ }
⊢ ⇑(DFinsupp.single i 1) ∉ (sigmaToPi R ⟨i, p⟩).asIdeal

Tactic:
by

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
hf : Function.Surjective ⇑f
x✝¹ x✝ : PrimeSpectrum S
h : { asIdeal := Ideal.comap f x✝¹.asIdeal, isPrime := ⋯ } = { asIdeal := Ideal.comap f x✝.asIdeal, isPrime := ⋯ }
⊢ Ideal.comap f x✝¹.asIdeal = Ideal.comap f x✝.asIdeal

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : FiniteRingKrullDim R
⊢ Nontrivial R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.RingHom

Goal state before:
pp: ι : Type u_3
R : ι → Type u_2
inst✝² : (i : ι) → CommSemiring (R i)
inst✝¹ : Infinite ι
inst✝ : ∀ (i : ι), Nontrivial (R i)
⊢ ∃ I, ∃ (x : I.IsMaximal), { asIdeal := I, isPrime := ⋯ } ∉ Set.range (sigmaToPi R)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.RingHom

Goal state before:
pp: ι : Type u_3
R : ι → Type u_4
inst✝¹ : (i : ι) → CommRing (R i)
inst✝ : _root_.Finite ι
p : PrimeSpectrum ((i : ι) → R i)
val✝ : Fintype ι
e : ι → (i : ι) → R i := fun i ↦ Function.update 1 i 0
⊢ ∏ i, e i = 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
⊢ KrullDimLE 0 R ↔ ∀ (I : Ideal R), I.IsPrime → I.IsMaximal

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.RingHom

Goal state before:
pp: ι : Type u_3
R : ι → Type u_4
inst✝¹ : (i : ι) → CommRing (R i)
inst✝ : _root_.Finite ι
p : PrimeSpectrum ((i : ι) → R i)
val✝ : Fintype ι
e : ι → (i : ι) → R i := fun i ↦ Function.update 1 i 0
H : ∏ i, e i = 0
⊢ ∃ i, e i ∈ p.asIdeal

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
⊢ KrullDimLE 0 R ↔ ∀ I ∈ minimalPrimes R, I.IsMaximal

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
H : ∀ (I : Ideal R), I.IsPrime → I.IsMaximal
⊢ KrullDimLE 0 R

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Idempotent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
a✝ b✝ a b c : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }
⊢ (↑((a ⊔ b) ⊓ (a ⊔ c))).2 = (↑(a ⊔ b ⊓ c)).2

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.MinimalPrime.Localization

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
x : R
p : Ideal R
hxp : x ∈ p
hp₃ : ∀ ⦃y : Ideal R⦄, (fun q ↦ q.IsPrime ∧ I ≤ q) y → y ≤ p → p ≤ y
hp₁ : p.IsPrime
hp₂ : I ≤ p
⊢ map (algebraMap R (Localization.AtPrime p)) p ≤ (map (algebraMap R (Localization.AtPrime p)) I).radical

Tactic:
by

Goal state after: