Theorem: DividedPowerAlgebra.T_free_and_D_to_QSplit
Proof state before:
case h.refine_1.mk.codisjoint
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : CondD A
hM✝ : DividedPowers (augIdeal A (↥I✝ →₀ A))
hM_eq✝ : ∀ (n : ℕ) (x : ↥I✝ →₀ A), hM✝.dpow n ((ι A (↥I✝ →₀ A)) x) = dp A n x
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsAugmentation (↥S₀) I
M : Type u := ↥I →₀ A
R : Type u := MvPolynomial (↥S₀) A
D : Type u := DividedPowerAlgebra A M
hM : DividedPowers (augIdeal A M)
hM_eq : ∀ (n : ℕ) (x : M), hM.dpow n ((ι A M) x) = dp A n x
hdpM_free : Module.Free A D
hR_free : Module.Free A R
T : Type u := R ⊗[A] D
idK : Ideal T := K A ⊥ (augIdeal A M)
hidK : idK = Ideal.map Algebra.TensorProduct.includeRight (augIdeal A M)
hK : DividedPowers idK := Exists.choose ⋯
this : MvPolynomial (↥S₀) A →ₐ[A] MvPolynomial (↥S₀) A ⊗[A] DividedPowerAlgebra A M
hd :
  Disjoint (Subalgebra.toSubmodule ⊥)
    (Submodule.restrictScalars (MvPolynomial (↥S₀) A) (Ideal.map Algebra.TensorProduct.includeRight (augIdeal A M)))
hc :
  Codisjoint (Subalgebra.toSubmodule ⊥)
    (Submodule.restrictScalars (MvPolynomial (↥S₀) A) (Ideal.map Algebra.TensorProduct.includeRight (augIdeal A M)))
x : T
a✝ : x ∈ ⊤
⊢ x ∈
    Subalgebra.toSubmodule ⊥ ⊔
      Submodule.restrictScalars (↥(Subalgebra.restrictScalars A ⊥))
        (Ideal.map Algebra.TensorProduct.includeRight (augIdeal A M))
Tactic:
rw [Submodule.mem_sup]
Proof state after:
case h.refine_1.mk.codisjoint
R✝ : Type u
inst✝⁶ : CommRing R✝
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : DecidableEq A
S✝ : Type u
inst✝³ : CommRing S✝
inst✝² : Algebra A S✝
I✝ : Ideal S✝
hI✝ : DividedPowers I✝
S₀✝ : Subalgebra A S✝
hIS₀✝ : IsCompl (Subalgebra.toSubmodule S₀✝) (Submodule.restrictScalars A I✝)
condTFree : CondTFree A
condD : CondD A
hM✝ : DividedPowers (augIdeal A (↥I✝ →₀ A))
hM_eq✝ : ∀ (n : ℕ) (x : ↥I✝ →₀ A), hM✝.dpow n ((ι A (↥I✝ →₀ A)) x) = dp A n x
S : Type u
inst✝¹ : CommRing S
inst✝ : Algebra A S
I : Ideal S
hI : DividedPowers I
S₀ : Subalgebra A S
hIS₀ : IsAugmentation (↥S₀) I
M : Type u := ↥I →₀ A
R : Type u := MvPolynomial (↥S₀) A
D : Type u := DividedPowerAlgebra A M
hM : DividedPowers (augIdeal A M)
hM_eq : ∀ (n : ℕ) (x : M), hM.dpow n ((ι A M) x) = dp A n x
hdpM_free : Module.Free A D
hR_free : Module.Free A R
T : Type u := R ⊗[A] D
idK : Ideal T := K A ⊥ (augIdeal A M)
hidK : idK = Ideal.map Algebra.TensorProduct.includeRight (augIdeal A M)
hK : DividedPowers idK := Exists.choose ⋯
this : MvPolynomial (↥S₀) A →ₐ[A] MvPolynomial (↥S₀) A ⊗[A] DividedPowerAlgebra A M
hd :
  Disjoint (Subalgebra.toSubmodule ⊥)
    (Submodule.restrictScalars (MvPolynomial (↥S₀) A) (Ideal.map Algebra.TensorProduct.includeRight (augIdeal A M)))
hc :
  Codisjoint (Subalgebra.toSubmodule ⊥)
    (Submodule.restrictScalars (MvPolynomial (↥S₀) A) (Ideal.map Algebra.TensorProduct.includeRight (augIdeal A M)))
x : T
a✝ : x ∈ ⊤
⊢ ∃ y ∈ Subalgebra.toSubmodule ⊥,
    ∃
      z ∈
        Submodule.restrictScalars (↥(Subalgebra.restrictScalars A ⊥))
          (Ideal.map Algebra.TensorProduct.includeRight (augIdeal A M)),
      y + z = x