-- Mathlib/RingTheory/Spectrum/Prime/TensorProduct.lean
lemma PrimeSpectrum.isEmbedding_tensorProductTo_of_surjectiveOnStalks :
    IsEmbedding (tensorProductTo R S T)

Proof state:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
⊢ IsEmbedding (tensorProductTo R S T)

Tactic:
refine ⟨?_, fun p₁ p₂ e ↦
  (isEmbedding_tensorProductTo_of_surjectiveOnStalks_aux R S T hRT p₁ p₂ e).antisymm
    (isEmbedding_tensorProductTo_of_surjectiveOnStalks_aux R S T hRT p₂ p₁ e.symm)⟩

Result:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
⊢ IsInducing (tensorProductTo R S T)

Proof state:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
⊢ IsInducing (tensorProductTo R S T)

Tactic:
refine ⟨(continuous_tensorProductTo ..).le_induced.antisymm (isBasis_basic_opens.le_iff.mpr ?_)⟩

Result:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
⊢ ∀ U ∈ Set.range basicOpen, IsOpen ↑U

Proof state:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
f : S ⊗[R] T
J : PrimeSpectrum (S ⊗[R] T)
hJ : f ∉ J.asIdeal
t : T
r : R
a : S
ht : r • t ∉ Ideal.comap g J.asIdeal
e : 1 ⊗ₜ[R] (r • t) * f = a ⊗ₜ[R] t
⊢ a ⊗ₜ[R] 1 * 1 ⊗ₜ[R] t ∉ J.asIdeal

Tactic:
rw [Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul, ← e]

Result:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
f : S ⊗[R] T
J : PrimeSpectrum (S ⊗[R] T)
hJ : f ∉ J.asIdeal
t : T
r : R
a : S
ht : r • t ∉ Ideal.comap g J.asIdeal
e : 1 ⊗ₜ[R] (r • t) * f = a ⊗ₜ[R] t
⊢ 1 ⊗ₜ[R] (r • t) * f ∉ J.asIdeal

Proof state:
case intro
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
f : S ⊗[R] T
⊢ ∀ x ∈ ↑(basicOpen f), ∃ t ⊆ ↑(basicOpen f), IsOpen t ∧ x ∈ t

Tactic:
rintro J (hJ : f ∉ J.asIdeal)

Result:
case intro
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
f : S ⊗[R] T
J : PrimeSpectrum (S ⊗[R] T)
hJ : f ∉ J.asIdeal
⊢ ∃ t ⊆ ↑(basicOpen f), IsOpen t ∧ J ∈ t

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

Tactic:
rintro x ⟨hx₁ : a ⊗ₜ[R] (1 : T) ∉ x.asIdeal, hx₂ : (1 : S) ⊗ₜ[R] t ∉ x.asIdeal⟩
  (hx₃ : f ∈ x.asIdeal)

Result:
case intro.intro.intro.intro.intro.refine_1.intro
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
f : S ⊗[R] T
J : PrimeSpectrum (S ⊗[R] T)
hJ : f ∉ J.asIdeal
t : T
r : R
a : S
ht : r • t ∉ Ideal.comap g J.asIdeal
e : 1 ⊗ₜ[R] (r • t) * f = a ⊗ₜ[R] t
x : PrimeSpectrum (S ⊗[R] T)
hx₁ : a ⊗ₜ[R] 1 ∉ x.asIdeal
hx₂ : 1 ⊗ₜ[R] t ∉ x.asIdeal
hx₃ : f ∈ x.asIdeal
⊢ False

Proof state:
case intro.intro.intro.intro.intro
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
f : S ⊗[R] T
J : PrimeSpectrum (S ⊗[R] T)
hJ : f ∉ J.asIdeal
t : T
r : R
a : S
ht : r • t ∉ Ideal.comap g J.asIdeal
e : 1 ⊗ₜ[R] (r • t) * f = a ⊗ₜ[R] t
⊢ ∃ t ⊆ ↑(basicOpen f), IsOpen t ∧ J ∈ t

Tactic:
refine ⟨_, ?_, ⟨_, (basicOpen a).2.prod (basicOpen t).2, rfl⟩, ?_⟩

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

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

Proof state:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
⊢ ∀ U ∈ Set.range basicOpen, IsOpen ↑U

Tactic:
rintro _ ⟨f, rfl⟩

Result:
case intro
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
f : S ⊗[R] T
⊢ IsOpen ↑(basicOpen f)