-- Mathlib/RingTheory/Unramified/Field.lean
theorem bijective_of_isAlgClosed_of_isLocalRing
    [IsAlgClosed K] [IsLocalRing A] :
    Function.Bijective (algebraMap K A)

Proof state:
case intro
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝² : Module.Finite K A
this✝¹ : IsArtinianRing A
this✝ : Function.Bijective ⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
  let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
  let __spread.1 := Equiv.ofBijective (⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))) this✝;
  { toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
    map_add' := ⋯, commutes' := ⋯ }
e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
  (TensorProduct.congr AlgEquiv.refl e.symm).trans (TensorProduct.rid K A A)
f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A)
hf : (ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ f = LinearMap.id
hf₁ : f 1 • 1 = 1
hf₂ : 1 - f 1 ∈ IsLocalRing.maximalIdeal A
hf₃ : IsIdempotentElem (1 - f 1)
n : ℕ
hn : IsLocalRing.maximalIdeal A ^ n = 0
this : (1 - f 1) ^ n = 0
⊢ f 1 = 1

Tactic:
rw [eq_comm, ← sub_eq_zero, ← hf₃.pow_succ_eq n, pow_succ, this, zero_mul]

Result:
no goals

Proof state:
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝¹ : Module.Finite K A
this✝ : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
this : Function.Bijective ⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
  let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
  let __spread.1 := Equiv.ofBijective (⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))) this;
  { toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
    map_add' := ⋯, commutes' := ⋯ }
e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
  (TensorProduct.congr AlgEquiv.refl e.symm).trans (TensorProduct.rid K A A)
f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A)
⊢ ((ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ e'.toLinearMap) ∘ₗ
      sec K A (A ⧸ IsLocalRing.maximalIdeal A) =
    TensorProduct.AlgebraTensorModule.lift (↑K (lsmul A A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap.flip).flip ∘ₗ
      sec K A (A ⧸ IsLocalRing.maximalIdeal A)

Tactic:
congr 1

Result:
case e_f
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝¹ : Module.Finite K A
this✝ : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
this : Function.Bijective ⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
  let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
  let __spread.1 := Equiv.ofBijective (⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))) this;
  { toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
    map_add' := ⋯, commutes' := ⋯ }
e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
  (TensorProduct.congr AlgEquiv.refl e.symm).trans (TensorProduct.rid K A A)
f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A)
⊢ (ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ e'.toLinearMap =
    TensorProduct.AlgebraTensorModule.lift (↑K (lsmul A A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap.flip).flip

Proof state:
case h
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝¹ : Module.Finite K A
this✝ : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
this : Function.Bijective ⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
  let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
  let __spread.1 := Equiv.ofBijective (⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))) this;
  { toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
    map_add' := ⋯, commutes' := ⋯ }
e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
  (TensorProduct.congr AlgEquiv.refl e.symm).trans (TensorProduct.rid K A A)
f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A)
hf : (ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ f = LinearMap.id
hf₁ : f 1 • 1 = 1
hf₂ : 1 - f 1 ∈ IsLocalRing.maximalIdeal A
hf₃ : IsIdempotentElem (1 - f 1)
hf₄ : f 1 = 1
x : A
k : K
⊢ (algebraMap K A) k = (⇑f ∘ ⇑e) k

Tactic:
conv_rhs => rw [← mul_one k, ← smul_eq_mul, Function.comp_apply, map_smul,
  LinearMap.map_smul_of_tower, map_one, hf₄, ← algebraMap_eq_smul_one]

Result:
no goals

Proof state:
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝¹ : Module.Finite K A
this✝ : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
this : Function.Bijective ⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
  let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
  let __spread.1 := Equiv.ofBijective (⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))) this;
  { toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
    map_add' := ⋯, commutes' := ⋯ }
⊢ Function.Bijective ⇑(algebraMap K A)

Tactic:
let e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
  (Algebra.TensorProduct.congr AlgEquiv.refl e.symm).trans (Algebra.TensorProduct.rid K A A)

Result:
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝¹ : Module.Finite K A
this✝ : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
this : Function.Bijective ⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
  let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
  let __spread.1 := Equiv.ofBijective (⇑(ofId K (A ⧸ IsLocalRing.maximalIdeal A))) this;
  { toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
    map_add' := ⋯, commutes' := ⋯ }
e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
  (TensorProduct.congr AlgEquiv.refl e.symm).trans (TensorProduct.rid K A A)
⊢ Function.Bijective ⇑(algebraMap K A)

Proof state:
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this : Module.Finite K A
⊢ Function.Bijective ⇑(algebraMap K A)

Tactic:
have : IsArtinianRing A := isArtinian_of_tower K inferInstance

Result:
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝ : Module.Finite K A
this : IsArtinianRing A
⊢ Function.Bijective ⇑(algebraMap K A)