Theorem: KaehlerDifferential.End_equiv_aux
File: Mathlib/RingTheory/Kaehler/Basic.lean

Proof steps:
state_before: R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
⊢ (Ideal.Quotient.mkₐ R (ideal R S).cotangentIdeal).comp f =
      IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal) ↔
    (TensorProduct.lmul' R).kerSquareLift.comp f = AlgHom.id R S
state_after: R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
⊢ (∀ (x : S),
      ((Ideal.Quotient.mkₐ R (ideal R S).cotangentIdeal).comp f) x =
        (IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x) ↔
    ∀ (x : S), ((TensorProduct.lmul' R).kerSquareLift.comp f) x = (AlgHom.id R S) x
tactic: rw [AlgHom.ext_iff, AlgHom.ext_iff]
state_before: R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
⊢ (∀ (x : S),
      ((Ideal.Quotient.mkₐ R (ideal R S).cotangentIdeal).comp f) x =
        (IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x) ↔
    ∀ (x : S), ((TensorProduct.lmul' R).kerSquareLift.comp f) x = (AlgHom.id R S) x
state_after: case h
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
⊢ ∀ (a : S),
    ((Ideal.Quotient.mkₐ R (ideal R S).cotangentIdeal).comp f) a =
        (IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) a ↔
      ((TensorProduct.lmul' R).kerSquareLift.comp f) a = (AlgHom.id R S) a
tactic: apply forall_congr'
state_before: case h
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
⊢ ∀ (a : S),
    ((Ideal.Quotient.mkₐ R (ideal R S).cotangentIdeal).comp f) a =
        (IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) a ↔
      ((TensorProduct.lmul' R).kerSquareLift.comp f) a = (AlgHom.id R S) a
state_after: case h
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
x : S
⊢ ((Ideal.Quotient.mkₐ R (ideal R S).cotangentIdeal).comp f) x =
      (IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x ↔
    ((TensorProduct.lmul' R).kerSquareLift.comp f) x = (AlgHom.id R S) x
tactic: intro x
state_before: case h
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
x : S
⊢ ((Ideal.Quotient.mkₐ R (ideal R S).cotangentIdeal).comp f) x =
      (IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x ↔
    ((TensorProduct.lmul' R).kerSquareLift.comp f) x = (AlgHom.id R S) x
state_after: case h
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
x : S
e₁ :
  (TensorProduct.lmul' R).kerSquareLift (f x) =
    (quotientCotangentIdealRingEquiv R S) ((Ideal.Quotient.mk (ideal R S).cotangentIdeal) (f x))
⊢ ((Ideal.Quotient.mkₐ R (ideal R S).cotangentIdeal).comp f) x =
      (IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x ↔
    ((TensorProduct.lmul' R).kerSquareLift.comp f) x = (AlgHom.id R S) x
tactic: have e₁ : (TensorProduct.lmul' R : S ⊗[R] S →ₐ[R] S).kerSquareLift (f x) =
      KaehlerDifferential.quotientCotangentIdealRingEquiv R S
        (Ideal.Quotient.mk (KaehlerDifferential.ideal R S).cotangentIdeal <| f x) := by
    generalize f x = y; obtain ⟨y, rfl⟩ := Ideal.Quotient.mk_surjective y; rfl
state_before: case h
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
x : S
e₁ :
  (TensorProduct.lmul' R).kerSquareLift (f x) =
    (quotientCotangentIdealRingEquiv R S) ((Ideal.Quotient.mk (ideal R S).cotangentIdeal) (f x))
⊢ ((Ideal.Quotient.mkₐ R (ideal R S).cotangentIdeal).comp f) x =
      (IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x ↔
    ((TensorProduct.lmul' R).kerSquareLift.comp f) x = (AlgHom.id R S) x
state_after: case h
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
x : S
e₁ :
  (TensorProduct.lmul' R).kerSquareLift (f x) =
    (quotientCotangentIdealRingEquiv R S) ((Ideal.Quotient.mk (ideal R S).cotangentIdeal) (f x))
e₂ :
  x =
    (quotientCotangentIdealRingEquiv R S)
      ((IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x)
⊢ ((Ideal.Quotient.mkₐ R (ideal R S).cotangentIdeal).comp f) x =
      (IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x ↔
    ((TensorProduct.lmul' R).kerSquareLift.comp f) x = (AlgHom.id R S) x
tactic: have e₂ :
    x = KaehlerDifferential.quotientCotangentIdealRingEquiv R S (IsScalarTower.toAlgHom R S _ x) :=
    (mul_one x).symm
state_before: case h
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
x : S
e₁ :
  (TensorProduct.lmul' R).kerSquareLift (f x) =
    (quotientCotangentIdealRingEquiv R S) ((Ideal.Quotient.mk (ideal R S).cotangentIdeal) (f x))
e₂ :
  x =
    (quotientCotangentIdealRingEquiv R S)
      ((IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x)
⊢ ((Ideal.Quotient.mkₐ R (ideal R S).cotangentIdeal).comp f) x =
      (IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x ↔
    ((TensorProduct.lmul' R).kerSquareLift.comp f) x = (AlgHom.id R S) x
state_after: case h.mp
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
x : S
e₁ :
  (TensorProduct.lmul' R).kerSquareLift (f x) =
    (quotientCotangentIdealRingEquiv R S) ((Ideal.Quotient.mk (ideal R S).cotangentIdeal) (f x))
e₂ :
  x =
    (quotientCotangentIdealRingEquiv R S)
      ((IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x)
⊢ ((Ideal.Quotient.mkₐ R (ideal R S).cotangentIdeal).comp f) x =
      (IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x →
    ((TensorProduct.lmul' R).kerSquareLift.comp f) x = (AlgHom.id R S) x

case h.mpr
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
x : S
e₁ :
  (TensorProduct.lmul' R).kerSquareLift (f x) =
    (quotientCotangentIdealRingEquiv R S) ((Ideal.Quotient.mk (ideal R S).cotangentIdeal) (f x))
e₂ :
  x =
    (quotientCotangentIdealRingEquiv R S)
      ((IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x)
⊢ ((TensorProduct.lmul' R).kerSquareLift.comp f) x = (AlgHom.id R S) x →
    ((Ideal.Quotient.mkₐ R (ideal R S).cotangentIdeal).comp f) x =
      (IsScalarTower.toAlgHom R S ((S ⊗[R] S ⧸ ideal R S ^ 2) ⧸ (ideal R S).cotangentIdeal)) x
tactic: constructor