l3lab/ntp-mathlib huggingface dataset:
/-- A direct sum of flat `R`-modules is flat. -/
instance directSum (ι : Type v) (M : ι → Type w) [(i : ι) → AddCommGroup (M i)]
    [(i : ι) → Module R (M i)] [F : (i : ι) → (Flat R (M i))] : Flat R (⨁ i, M i) := by
  classical
  rw [iff_rTensor_injective]
  intro I hI
  rw [← Equiv.comp_injective _ (TensorProduct.lid R (⨁ i, M i)).toEquiv]
  set η₁ := TensorProduct.lid R (⨁ i, M i)
  set η := (fun i ↦ TensorProduct.lid R (M i))
  set φ := (fun i ↦ rTensor (M i) I.subtype)
  set π := (fun i ↦ component R ι (fun j ↦ M j) i)
  set ψ := (TensorProduct.directSumRight R {x // x ∈ I} (fun i ↦ M i)).symm.toLinearMap with psi_def
  set ρ := rTensor (⨁ i, M i) I.subtype
  set τ := (fun i ↦ component R ι (fun j ↦ ({x // x ∈ I} ⊗[R] (M j))) i)
  rw [← Equiv.injective_comp (TensorProduct.directSumRight _ _ _).symm.toEquiv]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [← psi_def, injective_iff_map_eq_zero ((η₁.comp ρ).comp ψ)]
  have h₁ : ∀ (i : ι), (π i).comp ((η₁.comp ρ).comp ψ) = (η i).comp ((φ i).comp (τ i))
  · intro i
    apply DirectSum.linearMap_ext
    

Proof state:
case h₁.H
R : Type u
inst✝² : CommRing R
ι : Type v
M : ι → Type w
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module R (M i)
F : ∀ (i : ι), Flat R (M i)
I : Ideal R
hI : Ideal.FG I
η₁ : (R ⊗[R] ⨁ (i : ι), M i) ≃ₗ[R] ⨁ (i : ι), M i := TensorProduct.lid R (⨁ (i : ι), M i)
η : (i : ι) → R ⊗[R] M i ≃ₗ[R] M i := fun i => TensorProduct.lid R (M i)
φ : (i : ι) → ↥I ⊗[R] M i →ₗ[R] R ⊗[R] M i := fun i => rTensor (M i) (Submodule.subtype I)
π : (i : ι) → (⨁ (i : ι), M i) →ₗ[R] M i := fun i => component R ι (fun j => M j) i
ψ : (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] ⨁ (i : ι), M i := ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
psi_def : ψ = ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
ρ : (↥I ⊗[R] ⨁ (i : ι), M i) →ₗ[R] R ⊗[R] ⨁ (i : ι), M i := rTensor (⨁ (i : ι), M i) (Submodule.subtype I)
τ : (i : ι) → (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] M i := fun i => component R ι (fun j => ↥I ⊗[R] M j) i
i : ι
⊢ ∀ (i_1 : ι),
    (π i ∘ₗ (↑η₁ ∘ₗ ρ) ∘ₗ ψ) ∘ₗ lof R ι (fun i => ↥I ⊗[R] M i) i_1 =
      (↑(η i) ∘ₗ φ i ∘ₗ τ i) ∘ₗ lof R ι (fun i => ↥I ⊗[R] M i) i_1

Next tactic:
intro j
/-- A direct sum of flat `R`-modules is flat. -/
instance directSum (ι : Type v) (M : ι → Type w) [(i : ι) → AddCommGroup (M i)]
    [(i : ι) → Module R (M i)] [F : (i : ι) → (Flat R (M i))] : Flat R (⨁ i, M i) := by
  classical
  rw [iff_rTensor_injective]
  intro I hI
  rw [← Equiv.comp_injective _ (TensorProduct.lid R (⨁ i, M i)).toEquiv]
  set η₁ := TensorProduct.lid R (⨁ i, M i)
  set η := (fun i ↦ TensorProduct.lid R (M i))
  set φ := (fun i ↦ rTensor (M i) I.subtype)
  set π := (fun i ↦ component R ι (fun j ↦ M j) i)
  set ψ := (TensorProduct.directSumRight R {x // x ∈ I} (fun i ↦ M i)).symm.toLinearMap with psi_def
  set ρ := rTensor (⨁ i, M i) I.subtype
  set τ := (fun i ↦ component R ι (fun j ↦ ({x // x ∈ I} ⊗[R] (M j))) i)
  rw [← Equiv.injective_comp (TensorProduct.directSumRight _ _ _).symm.toEquiv]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [← psi_def, injective_iff_map_eq_zero ((η₁.comp ρ).comp ψ)]
  have h₁ : ∀ (i : ι), (π i).comp ((η₁.comp ρ).comp ψ) = (η i).comp ((φ i).comp (τ i))
  · intro i
    apply DirectSum.linearMap_ext
    intro j
    

Proof state:
case h₁.H
R : Type u
inst✝² : CommRing R
ι : Type v
M : ι → Type w
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module R (M i)
F : ∀ (i : ι), Flat R (M i)
I : Ideal R
hI : Ideal.FG I
η₁ : (R ⊗[R] ⨁ (i : ι), M i) ≃ₗ[R] ⨁ (i : ι), M i := TensorProduct.lid R (⨁ (i : ι), M i)
η : (i : ι) → R ⊗[R] M i ≃ₗ[R] M i := fun i => TensorProduct.lid R (M i)
φ : (i : ι) → ↥I ⊗[R] M i →ₗ[R] R ⊗[R] M i := fun i => rTensor (M i) (Submodule.subtype I)
π : (i : ι) → (⨁ (i : ι), M i) →ₗ[R] M i := fun i => component R ι (fun j => M j) i
ψ : (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] ⨁ (i : ι), M i := ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
psi_def : ψ = ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
ρ : (↥I ⊗[R] ⨁ (i : ι), M i) →ₗ[R] R ⊗[R] ⨁ (i : ι), M i := rTensor (⨁ (i : ι), M i) (Submodule.subtype I)
τ : (i : ι) → (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] M i := fun i => component R ι (fun j => ↥I ⊗[R] M j) i
i j : ι
⊢ (π i ∘ₗ (↑η₁ ∘ₗ ρ) ∘ₗ ψ) ∘ₗ lof R ι (fun i => ↥I ⊗[R] M i) j =
    (↑(η i) ∘ₗ φ i ∘ₗ τ i) ∘ₗ lof R ι (fun i => ↥I ⊗[R] M i) j

Next tactic:
apply TensorProduct.ext'
/-- A direct sum of flat `R`-modules is flat. -/
instance directSum (ι : Type v) (M : ι → Type w) [(i : ι) → AddCommGroup (M i)]
    [(i : ι) → Module R (M i)] [F : (i : ι) → (Flat R (M i))] : Flat R (⨁ i, M i) := by
  classical
  rw [iff_rTensor_injective]
  intro I hI
  rw [← Equiv.comp_injective _ (TensorProduct.lid R (⨁ i, M i)).toEquiv]
  set η₁ := TensorProduct.lid R (⨁ i, M i)
  set η := (fun i ↦ TensorProduct.lid R (M i))
  set φ := (fun i ↦ rTensor (M i) I.subtype)
  set π := (fun i ↦ component R ι (fun j ↦ M j) i)
  set ψ := (TensorProduct.directSumRight R {x // x ∈ I} (fun i ↦ M i)).symm.toLinearMap with psi_def
  set ρ := rTensor (⨁ i, M i) I.subtype
  set τ := (fun i ↦ component R ι (fun j ↦ ({x // x ∈ I} ⊗[R] (M j))) i)
  rw [← Equiv.injective_comp (TensorProduct.directSumRight _ _ _).symm.toEquiv]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [← psi_def, injective_iff_map_eq_zero ((η₁.comp ρ).comp ψ)]
  have h₁ : ∀ (i : ι), (π i).comp ((η₁.comp ρ).comp ψ) = (η i).comp ((φ i).comp (τ i))
  · intro i
    apply DirectSum.linearMap_ext
    intro j
    apply TensorProduct.ext'
    

Proof state:
case h₁.H.H
R : Type u
inst✝² : CommRing R
ι : Type v
M : ι → Type w
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module R (M i)
F : ∀ (i : ι), Flat R (M i)
I : Ideal R
hI : Ideal.FG I
η₁ : (R ⊗[R] ⨁ (i : ι), M i) ≃ₗ[R] ⨁ (i : ι), M i := TensorProduct.lid R (⨁ (i : ι), M i)
η : (i : ι) → R ⊗[R] M i ≃ₗ[R] M i := fun i => TensorProduct.lid R (M i)
φ : (i : ι) → ↥I ⊗[R] M i →ₗ[R] R ⊗[R] M i := fun i => rTensor (M i) (Submodule.subtype I)
π : (i : ι) → (⨁ (i : ι), M i) →ₗ[R] M i := fun i => component R ι (fun j => M j) i
ψ : (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] ⨁ (i : ι), M i := ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
psi_def : ψ = ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
ρ : (↥I ⊗[R] ⨁ (i : ι), M i) →ₗ[R] R ⊗[R] ⨁ (i : ι), M i := rTensor (⨁ (i : ι), M i) (Submodule.subtype I)
τ : (i : ι) → (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] M i := fun i => component R ι (fun j => ↥I ⊗[R] M j) i
i j : ι
⊢ ∀ (x : ↥I) (y : M j),
    ((π i ∘ₗ (↑η₁ ∘ₗ ρ) ∘ₗ ψ) ∘ₗ lof R ι (fun i => ↥I ⊗[R] M i) j) (x ⊗ₜ[R] y) =
      ((↑(η i) ∘ₗ φ i ∘ₗ τ i) ∘ₗ lof R ι (fun i => ↥I ⊗[R] M i) j) (x ⊗ₜ[R] y)

Next tactic:
intro a m
/-- A direct sum of flat `R`-modules is flat. -/
instance directSum (ι : Type v) (M : ι → Type w) [(i : ι) → AddCommGroup (M i)]
    [(i : ι) → Module R (M i)] [F : (i : ι) → (Flat R (M i))] : Flat R (⨁ i, M i) := by
  classical
  rw [iff_rTensor_injective]
  intro I hI
  rw [← Equiv.comp_injective _ (TensorProduct.lid R (⨁ i, M i)).toEquiv]
  set η₁ := TensorProduct.lid R (⨁ i, M i)
  set η := (fun i ↦ TensorProduct.lid R (M i))
  set φ := (fun i ↦ rTensor (M i) I.subtype)
  set π := (fun i ↦ component R ι (fun j ↦ M j) i)
  set ψ := (TensorProduct.directSumRight R {x // x ∈ I} (fun i ↦ M i)).symm.toLinearMap with psi_def
  set ρ := rTensor (⨁ i, M i) I.subtype
  set τ := (fun i ↦ component R ι (fun j ↦ ({x // x ∈ I} ⊗[R] (M j))) i)
  rw [← Equiv.injective_comp (TensorProduct.directSumRight _ _ _).symm.toEquiv]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [← psi_def, injective_iff_map_eq_zero ((η₁.comp ρ).comp ψ)]
  have h₁ : ∀ (i : ι), (π i).comp ((η₁.comp ρ).comp ψ) = (η i).comp ((φ i).comp (τ i))
  · intro i
    apply DirectSum.linearMap_ext
    intro j
    apply TensorProduct.ext'
    intro a m
    

Proof state:
case h₁.H.H
R : Type u
inst✝² : CommRing R
ι : Type v
M : ι → Type w
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module R (M i)
F : ∀ (i : ι), Flat R (M i)
I : Ideal R
hI : Ideal.FG I
η₁ : (R ⊗[R] ⨁ (i : ι), M i) ≃ₗ[R] ⨁ (i : ι), M i := TensorProduct.lid R (⨁ (i : ι), M i)
η : (i : ι) → R ⊗[R] M i ≃ₗ[R] M i := fun i => TensorProduct.lid R (M i)
φ : (i : ι) → ↥I ⊗[R] M i →ₗ[R] R ⊗[R] M i := fun i => rTensor (M i) (Submodule.subtype I)
π : (i : ι) → (⨁ (i : ι), M i) →ₗ[R] M i := fun i => component R ι (fun j => M j) i
ψ : (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] ⨁ (i : ι), M i := ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
psi_def : ψ = ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
ρ : (↥I ⊗[R] ⨁ (i : ι), M i) →ₗ[R] R ⊗[R] ⨁ (i : ι), M i := rTensor (⨁ (i : ι), M i) (Submodule.subtype I)
τ : (i : ι) → (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] M i := fun i => component R ι (fun j => ↥I ⊗[R] M j) i
i j : ι
a : ↥I
m : M j
⊢ ((π i ∘ₗ (↑η₁ ∘ₗ ρ) ∘ₗ ψ) ∘ₗ lof R ι (fun i => ↥I ⊗[R] M i) j) (a ⊗ₜ[R] m) =
    ((↑(η i) ∘ₗ φ i ∘ₗ τ i) ∘ₗ lof R ι (fun i => ↥I ⊗[R] M i) j) (a ⊗ₜ[R] m)

Next tactic:
simp only [coe_comp, LinearEquiv.coe_coe, Function.comp_apply, directSumRight_symm_lof_tmul,
      rTensor_tmul, coeSubtype, lid_tmul, map_smul]
/-- A direct sum of flat `R`-modules is flat. -/
instance directSum (ι : Type v) (M : ι → Type w) [(i : ι) → AddCommGroup (M i)]
    [(i : ι) → Module R (M i)] [F : (i : ι) → (Flat R (M i))] : Flat R (⨁ i, M i) := by
  classical
  rw [iff_rTensor_injective]
  intro I hI
  rw [← Equiv.comp_injective _ (TensorProduct.lid R (⨁ i, M i)).toEquiv]
  set η₁ := TensorProduct.lid R (⨁ i, M i)
  set η := (fun i ↦ TensorProduct.lid R (M i))
  set φ := (fun i ↦ rTensor (M i) I.subtype)
  set π := (fun i ↦ component R ι (fun j ↦ M j) i)
  set ψ := (TensorProduct.directSumRight R {x // x ∈ I} (fun i ↦ M i)).symm.toLinearMap with psi_def
  set ρ := rTensor (⨁ i, M i) I.subtype
  set τ := (fun i ↦ component R ι (fun j ↦ ({x // x ∈ I} ⊗[R] (M j))) i)
  rw [← Equiv.injective_comp (TensorProduct.directSumRight _ _ _).symm.toEquiv]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [← psi_def, injective_iff_map_eq_zero ((η₁.comp ρ).comp ψ)]
  have h₁ : ∀ (i : ι), (π i).comp ((η₁.comp ρ).comp ψ) = (η i).comp ((φ i).comp (τ i))
  · intro i
    apply DirectSum.linearMap_ext
    intro j
    apply TensorProduct.ext'
    intro a m
    simp only [coe_comp, LinearEquiv.coe_coe, Function.comp_apply, directSumRight_symm_lof_tmul,
      rTensor_tmul, coeSubtype, lid_tmul, map_smul]
    

Proof state:
case h₁.H.H
R : Type u
inst✝² : CommRing R
ι : Type v
M : ι → Type w
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module R (M i)
F : ∀ (i : ι), Flat R (M i)
I : Ideal R
hI : Ideal.FG I
η₁ : (R ⊗[R] ⨁ (i : ι), M i) ≃ₗ[R] ⨁ (i : ι), M i := TensorProduct.lid R (⨁ (i : ι), M i)
η : (i : ι) → R ⊗[R] M i ≃ₗ[R] M i := fun i => TensorProduct.lid R (M i)
φ : (i : ι) → ↥I ⊗[R] M i →ₗ[R] R ⊗[R] M i := fun i => rTensor (M i) (Submodule.subtype I)
π : (i : ι) → (⨁ (i : ι), M i) →ₗ[R] M i := fun i => component R ι (fun j => M j) i
ψ : (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] ⨁ (i : ι), M i := ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
psi_def : ψ = ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
ρ : (↥I ⊗[R] ⨁ (i : ι), M i) →ₗ[R] R ⊗[R] ⨁ (i : ι), M i := rTensor (⨁ (i : ι), M i) (Submodule.subtype I)
τ : (i : ι) → (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] M i := fun i => component R ι (fun j => ↥I ⊗[R] M j) i
i j : ι
a : ↥I
m : M j
⊢ ↑a • (component R ι (fun j => M j) i) ((lof R ι (fun i => M i) j) m) =
    (TensorProduct.lid R (M i))
      ((rTensor (M i) (Submodule.subtype I))
        ((component R ι (fun j => ↥I ⊗[R] M j) i) ((lof R ι (fun i => ↥I ⊗[R] M i) j) (a ⊗ₜ[R] m))))

Next tactic:
rw [DirectSum.component.of, DirectSum.component.of]
/-- A direct sum of flat `R`-modules is flat. -/
instance directSum (ι : Type v) (M : ι → Type w) [(i : ι) → AddCommGroup (M i)]
    [(i : ι) → Module R (M i)] [F : (i : ι) → (Flat R (M i))] : Flat R (⨁ i, M i) := by
  classical
  rw [iff_rTensor_injective]
  intro I hI
  rw [← Equiv.comp_injective _ (TensorProduct.lid R (⨁ i, M i)).toEquiv]
  set η₁ := TensorProduct.lid R (⨁ i, M i)
  set η := (fun i ↦ TensorProduct.lid R (M i))
  set φ := (fun i ↦ rTensor (M i) I.subtype)
  set π := (fun i ↦ component R ι (fun j ↦ M j) i)
  set ψ := (TensorProduct.directSumRight R {x // x ∈ I} (fun i ↦ M i)).symm.toLinearMap with psi_def
  set ρ := rTensor (⨁ i, M i) I.subtype
  set τ := (fun i ↦ component R ι (fun j ↦ ({x // x ∈ I} ⊗[R] (M j))) i)
  rw [← Equiv.injective_comp (TensorProduct.directSumRight _ _ _).symm.toEquiv]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [← psi_def, injective_iff_map_eq_zero ((η₁.comp ρ).comp ψ)]
  have h₁ : ∀ (i : ι), (π i).comp ((η₁.comp ρ).comp ψ) = (η i).comp ((φ i).comp (τ i))
  · intro i
    apply DirectSum.linearMap_ext
    intro j
    apply TensorProduct.ext'
    intro a m
    simp only [coe_comp, LinearEquiv.coe_coe, Function.comp_apply, directSumRight_symm_lof_tmul,
      rTensor_tmul, coeSubtype, lid_tmul, map_smul]
    rw [DirectSum.component.of, DirectSum.component.of]
    

Proof state:
case h₁.H.H
R : Type u
inst✝² : CommRing R
ι : Type v
M : ι → Type w
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module R (M i)
F : ∀ (i : ι), Flat R (M i)
I : Ideal R
hI : Ideal.FG I
η₁ : (R ⊗[R] ⨁ (i : ι), M i) ≃ₗ[R] ⨁ (i : ι), M i := TensorProduct.lid R (⨁ (i : ι), M i)
η : (i : ι) → R ⊗[R] M i ≃ₗ[R] M i := fun i => TensorProduct.lid R (M i)
φ : (i : ι) → ↥I ⊗[R] M i →ₗ[R] R ⊗[R] M i := fun i => rTensor (M i) (Submodule.subtype I)
π : (i : ι) → (⨁ (i : ι), M i) →ₗ[R] M i := fun i => component R ι (fun j => M j) i
ψ : (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] ⨁ (i : ι), M i := ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
psi_def : ψ = ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
ρ : (↥I ⊗[R] ⨁ (i : ι), M i) →ₗ[R] R ⊗[R] ⨁ (i : ι), M i := rTensor (⨁ (i : ι), M i) (Submodule.subtype I)
τ : (i : ι) → (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] M i := fun i => component R ι (fun j => ↥I ⊗[R] M j) i
i j : ι
a : ↥I
m : M j
⊢ (↑a • if h : j = i then Eq.recOn h m else 0) =
    (TensorProduct.lid R (M i))
      ((rTensor (M i) (Submodule.subtype I)) (if h : j = i then Eq.recOn h (a ⊗ₜ[R] m) else 0))

Next tactic:
by_cases h₂ : j = i
/-- A direct sum of flat `R`-modules is flat. -/
instance directSum (ι : Type v) (M : ι → Type w) [(i : ι) → AddCommGroup (M i)]
    [(i : ι) → Module R (M i)] [F : (i : ι) → (Flat R (M i))] : Flat R (⨁ i, M i) := by
  classical
  rw [iff_rTensor_injective]
  intro I hI
  rw [← Equiv.comp_injective _ (TensorProduct.lid R (⨁ i, M i)).toEquiv]
  set η₁ := TensorProduct.lid R (⨁ i, M i)
  set η := (fun i ↦ TensorProduct.lid R (M i))
  set φ := (fun i ↦ rTensor (M i) I.subtype)
  set π := (fun i ↦ component R ι (fun j ↦ M j) i)
  set ψ := (TensorProduct.directSumRight R {x // x ∈ I} (fun i ↦ M i)).symm.toLinearMap with psi_def
  set ρ := rTensor (⨁ i, M i) I.subtype
  set τ := (fun i ↦ component R ι (fun j ↦ ({x // x ∈ I} ⊗[R] (M j))) i)
  rw [← Equiv.injective_comp (TensorProduct.directSumRight _ _ _).symm.toEquiv]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [← psi_def, injective_iff_map_eq_zero ((η₁.comp ρ).comp ψ)]
  have h₁ : ∀ (i : ι), (π i).comp ((η₁.comp ρ).comp ψ) = (η i).comp ((φ i).comp (τ i))
  · intro i
    apply DirectSum.linearMap_ext
    intro j
    apply TensorProduct.ext'
    intro a m
    simp only [coe_comp, LinearEquiv.coe_coe, Function.comp_apply, directSumRight_symm_lof_tmul,
      rTensor_tmul, coeSubtype, lid_tmul, map_smul]
    rw [DirectSum.component.of, DirectSum.component.of]
    by_cases h₂ : j = i
    · 

Proof state:
case pos
R : Type u
inst✝² : CommRing R
ι : Type v
M : ι → Type w
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module R (M i)
F : ∀ (i : ι), Flat R (M i)
I : Ideal R
hI : Ideal.FG I
η₁ : (R ⊗[R] ⨁ (i : ι), M i) ≃ₗ[R] ⨁ (i : ι), M i := TensorProduct.lid R (⨁ (i : ι), M i)
η : (i : ι) → R ⊗[R] M i ≃ₗ[R] M i := fun i => TensorProduct.lid R (M i)
φ : (i : ι) → ↥I ⊗[R] M i →ₗ[R] R ⊗[R] M i := fun i => rTensor (M i) (Submodule.subtype I)
π : (i : ι) → (⨁ (i : ι), M i) →ₗ[R] M i := fun i => component R ι (fun j => M j) i
ψ : (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] ⨁ (i : ι), M i := ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
psi_def : ψ = ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
ρ : (↥I ⊗[R] ⨁ (i : ι), M i) →ₗ[R] R ⊗[R] ⨁ (i : ι), M i := rTensor (⨁ (i : ι), M i) (Submodule.subtype I)
τ : (i : ι) → (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] M i := fun i => component R ι (fun j => ↥I ⊗[R] M j) i
i j : ι
a : ↥I
m : M j
h₂ : j = i
⊢ (↑a • if h : j = i then Eq.recOn h m else 0) =
    (TensorProduct.lid R (M i))
      ((rTensor (M i) (Submodule.subtype I)) (if h : j = i then Eq.recOn h (a ⊗ₜ[R] m) else 0))

Next tactic:
subst j
/-- A direct sum of flat `R`-modules is flat. -/
instance directSum (ι : Type v) (M : ι → Type w) [(i : ι) → AddCommGroup (M i)]
    [(i : ι) → Module R (M i)] [F : (i : ι) → (Flat R (M i))] : Flat R (⨁ i, M i) := by
  classical
  rw [iff_rTensor_injective]
  intro I hI
  rw [← Equiv.comp_injective _ (TensorProduct.lid R (⨁ i, M i)).toEquiv]
  set η₁ := TensorProduct.lid R (⨁ i, M i)
  set η := (fun i ↦ TensorProduct.lid R (M i))
  set φ := (fun i ↦ rTensor (M i) I.subtype)
  set π := (fun i ↦ component R ι (fun j ↦ M j) i)
  set ψ := (TensorProduct.directSumRight R {x // x ∈ I} (fun i ↦ M i)).symm.toLinearMap with psi_def
  set ρ := rTensor (⨁ i, M i) I.subtype
  set τ := (fun i ↦ component R ι (fun j ↦ ({x // x ∈ I} ⊗[R] (M j))) i)
  rw [← Equiv.injective_comp (TensorProduct.directSumRight _ _ _).symm.toEquiv]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [← psi_def, injective_iff_map_eq_zero ((η₁.comp ρ).comp ψ)]
  have h₁ : ∀ (i : ι), (π i).comp ((η₁.comp ρ).comp ψ) = (η i).comp ((φ i).comp (τ i))
  · intro i
    apply DirectSum.linearMap_ext
    intro j
    apply TensorProduct.ext'
    intro a m
    simp only [coe_comp, LinearEquiv.coe_coe, Function.comp_apply, directSumRight_symm_lof_tmul,
      rTensor_tmul, coeSubtype, lid_tmul, map_smul]
    rw [DirectSum.component.of, DirectSum.component.of]
    by_cases h₂ : j = i
    · subst j; 

Proof state:
case pos
R : Type u
inst✝² : CommRing R
ι : Type v
M : ι → Type w
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module R (M i)
F : ∀ (i : ι), Flat R (M i)
I : Ideal R
hI : Ideal.FG I
η₁ : (R ⊗[R] ⨁ (i : ι), M i) ≃ₗ[R] ⨁ (i : ι), M i := TensorProduct.lid R (⨁ (i : ι), M i)
η : (i : ι) → R ⊗[R] M i ≃ₗ[R] M i := fun i => TensorProduct.lid R (M i)
φ : (i : ι) → ↥I ⊗[R] M i →ₗ[R] R ⊗[R] M i := fun i => rTensor (M i) (Submodule.subtype I)
π : (i : ι) → (⨁ (i : ι), M i) →ₗ[R] M i := fun i => component R ι (fun j => M j) i
ψ : (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] ⨁ (i : ι), M i := ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
psi_def : ψ = ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
ρ : (↥I ⊗[R] ⨁ (i : ι), M i) →ₗ[R] R ⊗[R] ⨁ (i : ι), M i := rTensor (⨁ (i : ι), M i) (Submodule.subtype I)
τ : (i : ι) → (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] M i := fun i => component R ι (fun j => ↥I ⊗[R] M j) i
i : ι
a : ↥I
m : M i
⊢ (↑a • if h : i = i then Eq.recOn h m else 0) =
    (TensorProduct.lid R (M i))
      ((rTensor (M i) (Submodule.subtype I)) (if h : i = i then Eq.recOn h (a ⊗ₜ[R] m) else 0))

Next tactic:
simp
/-- A direct sum of flat `R`-modules is flat. -/
instance directSum (ι : Type v) (M : ι → Type w) [(i : ι) → AddCommGroup (M i)]
    [(i : ι) → Module R (M i)] [F : (i : ι) → (Flat R (M i))] : Flat R (⨁ i, M i) := by
  classical
  rw [iff_rTensor_injective]
  intro I hI
  rw [← Equiv.comp_injective _ (TensorProduct.lid R (⨁ i, M i)).toEquiv]
  set η₁ := TensorProduct.lid R (⨁ i, M i)
  set η := (fun i ↦ TensorProduct.lid R (M i))
  set φ := (fun i ↦ rTensor (M i) I.subtype)
  set π := (fun i ↦ component R ι (fun j ↦ M j) i)
  set ψ := (TensorProduct.directSumRight R {x // x ∈ I} (fun i ↦ M i)).symm.toLinearMap with psi_def
  set ρ := rTensor (⨁ i, M i) I.subtype
  set τ := (fun i ↦ component R ι (fun j ↦ ({x // x ∈ I} ⊗[R] (M j))) i)
  rw [← Equiv.injective_comp (TensorProduct.directSumRight _ _ _).symm.toEquiv]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [LinearEquiv.coe_toEquiv, ← LinearEquiv.coe_coe, ← LinearMap.coe_comp]
  rw [← psi_def, injective_iff_map_eq_zero ((η₁.comp ρ).comp ψ)]
  have h₁ : ∀ (i : ι), (π i).comp ((η₁.comp ρ).comp ψ) = (η i).comp ((φ i).comp (τ i))
  · intro i
    apply DirectSum.linearMap_ext
    intro j
    apply TensorProduct.ext'
    intro a m
    simp only [coe_comp, LinearEquiv.coe_coe, Function.comp_apply, directSumRight_symm_lof_tmul,
      rTensor_tmul, coeSubtype, lid_tmul, map_smul]
    rw [DirectSum.component.of, DirectSum.component.of]
    by_cases h₂ : j = i
    · subst j; simp
    · 

Proof state:
case neg
R : Type u
inst✝² : CommRing R
ι : Type v
M : ι → Type w
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module R (M i)
F : ∀ (i : ι), Flat R (M i)
I : Ideal R
hI : Ideal.FG I
η₁ : (R ⊗[R] ⨁ (i : ι), M i) ≃ₗ[R] ⨁ (i : ι), M i := TensorProduct.lid R (⨁ (i : ι), M i)
η : (i : ι) → R ⊗[R] M i ≃ₗ[R] M i := fun i => TensorProduct.lid R (M i)
φ : (i : ι) → ↥I ⊗[R] M i →ₗ[R] R ⊗[R] M i := fun i => rTensor (M i) (Submodule.subtype I)
π : (i : ι) → (⨁ (i : ι), M i) →ₗ[R] M i := fun i => component R ι (fun j => M j) i
ψ : (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] ⨁ (i : ι), M i := ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
psi_def : ψ = ↑(LinearEquiv.symm (directSumRight R ↥I fun i => M i))
ρ : (↥I ⊗[R] ⨁ (i : ι), M i) →ₗ[R] R ⊗[R] ⨁ (i : ι), M i := rTensor (⨁ (i : ι), M i) (Submodule.subtype I)
τ : (i : ι) → (⨁ (i : ι), ↥I ⊗[R] M i) →ₗ[R] ↥I ⊗[R] M i := fun i => component R ι (fun j => ↥I ⊗[R] M j) i
i j : ι
a : ↥I
m : M j
h₂ : ¬j = i
⊢ (↑a • if h : j = i then Eq.recOn h m else 0) =
    (TensorProduct.lid R (M i))
      ((rTensor (M i) (Submodule.subtype I)) (if h : j = i then Eq.recOn h (a ⊗ₜ[R] m) else 0))

Next tactic:
simp [h₂]