-- Mathlib/RingTheory/LinearDisjoint.lean
theorem _root_.Algebra.TensorProduct.not_isField_of_transcendental
    (A : Type v) [CommRing A] (B : Type w) [CommRing B] [Algebra R A] [Algebra R B]
    [Module.Flat R A] [Module.Flat R B] [Algebra.Transcendental R A] [Algebra.Transcendental R B] :
    ¬IsField (A ⊗[R] B) := fun H ↦ by
  letI := H.toField
  obtain ⟨a, hta⟩ := ‹Algebra.Transcendental R A›
  obtain ⟨b, htb⟩ := ‹Algebra.Transcendental R B›
  have ha : Function.Injective (algebraMap R A) := Algebra.injective_of_transcendental
  have hb : Function.Injective (algebraMap R B) := Algebra.injective_of_transcendental
  let fa : A →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeLeft
  let fb : B →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeRight
  have hfa : Function.Injective fa := Algebra.TensorProduct.includeLeft_injective hb
  have hfb : Function.Injective fb := Algebra.TensorProduct.includeRight_injective ha
  haveI := hfa.isDomain fa.toRingHom
  haveI := hfb.isDomain fb.toRingHom
  haveI := ha.isDomain _
  haveI : Module.Flat R (toSubmodule fa.range) :=
    .of_linearEquiv (AlgEquiv.ofInjective fa hfa).symm.toLinearEquiv
  have key1 : Module.rank R ↥(fa.range ⊓ fb.range) ≤ 1 :=
    (include_range R A B).rank_inf_le_one_of_flat_left
  let ga : R[X] →ₐ[R] A := aeval a
  let gb : R[X] →ₐ[R] B := aeval b
  let gab := fa.comp ga
  replace hta : Function.Injective ga := transcendental_iff_injective.1 hta
  replace htb : Function.Injective gb := transcendental_iff_injective.1 htb
  have htab : Function.Injective gab := hfa.comp hta
  algebraize_only [ga.toRingHom, gb.toRingHom]
  let f := Algebra.TensorProduct.mapOfCompatibleSMul R[X] R A B
  haveI := Algebra.TensorProduct.nontrivial_of_algebraMap_injective_of_isDomain R[X] A B hta htb
  have hf : Function.Injective f := RingHom.injective _
  have key2 : gab.range ≤ fa.range ⊓ fb.range

Proof state:
case mk.intro.mk.intro
R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Module.Flat R A
inst✝² : Module.Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
this✝³ : Field (A ⊗[R] B) := H.toField
a : A
b : B
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
fa : A →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeLeft
fb : B →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeRight
hfa : Function.Injective ⇑fa
hfb : Function.Injective ⇑fb
this✝² : IsDomain A
this✝¹ : IsDomain B
this✝ : IsDomain R
this : Module.Flat R ↥(toSubmodule fa.range)
key1 : Module.rank R ↥(fa.range ⊓ fb.range) ≤ 1
ga : R[X] →ₐ[R] A := aeval a
gb : R[X] →ₐ[R] B := aeval b
gab : R[X] →ₐ[R] A ⊗[R] B := fa.comp ga
hta : Function.Injective ⇑ga
htb : Function.Injective ⇑gb
htab : Function.Injective ⇑gab
algInst✝¹ : Algebra R[X] A := ga.toAlgebra
algInst✝ : Algebra R[X] B := gb.toAlgebra
f : A ⊗[R] B →ₐ[R] A ⊗[R[X]] B := Algebra.TensorProduct.mapOfCompatibleSMul R[X] R A B
⊢ False

Tactic:
haveI := Algebra.TensorProduct.nontrivial_of_algebraMap_injective_of_isDomain R[X] A B hta htb

Result:
case mk.intro.mk.intro
R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Module.Flat R A
inst✝² : Module.Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
this✝⁴ : Field (A ⊗[R] B) := H.toField
a : A
b : B
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
fa : A →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeLeft
fb : B →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeRight
hfa : Function.Injective ⇑fa
hfb : Function.Injective ⇑fb
this✝³ : IsDomain A
this✝² : IsDomain B
this✝¹ : IsDomain R
this✝ : Module.Flat R ↥(toSubmodule fa.range)
key1 : Module.rank R ↥(fa.range ⊓ fb.range) ≤ 1
ga : R[X] →ₐ[R] A := aeval a
gb : R[X] →ₐ[R] B := aeval b
gab : R[X] →ₐ[R] A ⊗[R] B := fa.comp ga
hta : Function.Injective ⇑ga
htb : Function.Injective ⇑gb
htab : Function.Injective ⇑gab
algInst✝¹ : Algebra R[X] A := ga.toAlgebra
algInst✝ : Algebra R[X] B := gb.toAlgebra
f : A ⊗[R] B →ₐ[R] A ⊗[R[X]] B := Algebra.TensorProduct.mapOfCompatibleSMul R[X] R A B
this : Nontrivial (A ⊗[R[X]] B)
⊢ False

Proof state:
case mk.intro.mk.intro
R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Module.Flat R A
inst✝² : Module.Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
this✝⁵ : Field (A ⊗[R] B) := H.toField
a : A
b : B
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
fa : A →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeLeft
fb : B →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeRight
hfa : Function.Injective ⇑fa
hfb : Function.Injective ⇑fb
this✝⁴ : IsDomain A
this✝³ : IsDomain B
this✝² : IsDomain R
this✝¹ : Module.Flat R ↥(toSubmodule fa.range)
key1 : Module.rank R ↥(fa.range ⊓ fb.range) ≤ 1
ga : R[X] →ₐ[R] A := aeval a
gb : R[X] →ₐ[R] B := aeval b
gab : R[X] →ₐ[R] A ⊗[R] B := fa.comp ga
hta : Function.Injective ⇑ga
htb : Function.Injective ⇑gb
htab : Function.Injective ⇑gab
algInst✝¹ : Algebra R[X] A := ga.toAlgebra
algInst✝ : Algebra R[X] B := gb.toAlgebra
f : A ⊗[R] B →ₐ[R] A ⊗[R[X]] B := Algebra.TensorProduct.mapOfCompatibleSMul R[X] R A B
this✝ : Nontrivial (A ⊗[R[X]] B)
hf : Function.Injective ⇑f
key2 : gab.range ≤ fa.range ⊓ fb.range
key3 : lift.{max v w, u} (Module.rank R R[X]) ≤ lift.{u, max v w} (Module.rank R ↥(fa.range ⊓ fb.range))
this : Module.rank R R[X] = lift.{u, 0} #ℕ
⊢ False

Tactic:
simp only [this, mk_eq_aleph0, lift_aleph0, aleph0_le_lift] at key3

Result:
case mk.intro.mk.intro
R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Module.Flat R A
inst✝² : Module.Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
this✝⁵ : Field (A ⊗[R] B) := H.toField
a : A
b : B
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
fa : A →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeLeft
fb : B →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeRight
hfa : Function.Injective ⇑fa
hfb : Function.Injective ⇑fb
this✝⁴ : IsDomain A
this✝³ : IsDomain B
this✝² : IsDomain R
this✝¹ : Module.Flat R ↥(toSubmodule fa.range)
key1 : Module.rank R ↥(fa.range ⊓ fb.range) ≤ 1
ga : R[X] →ₐ[R] A := aeval a
gb : R[X] →ₐ[R] B := aeval b
gab : R[X] →ₐ[R] A ⊗[R] B := fa.comp ga
hta : Function.Injective ⇑ga
htb : Function.Injective ⇑gb
htab : Function.Injective ⇑gab
algInst✝¹ : Algebra R[X] A := ga.toAlgebra
algInst✝ : Algebra R[X] B := gb.toAlgebra
f : A ⊗[R] B →ₐ[R] A ⊗[R[X]] B := Algebra.TensorProduct.mapOfCompatibleSMul R[X] R A B
this✝ : Nontrivial (A ⊗[R[X]] B)
hf : Function.Injective ⇑f
key2 : gab.range ≤ fa.range ⊓ fb.range
this : Module.rank R R[X] = lift.{u, 0} #ℕ
key3 : ℵ₀ ≤ Module.rank R ↥(fa.range ⊓ fb.range)
⊢ False