/-- 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
    · simp [h₂]
  intro a ha; rw [DirectSum.ext_iff R]; intro i
  have f := LinearMap.congr_arg (f := (π i)) ha
  erw [LinearMap.congr_fun (h₁ i) a] at f
  rw [(map_zero (π i) : (π i) 0 = (0 : M i))] at f
  have h₂ := (F i)
  rw [iff_rTensor_injective] at h₂
  have h₃ := h₂ hI
  

Proof state:
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
h₁ : ∀ (i : ι), π i ∘ₗ (↑η₁ ∘ₗ ρ) ∘ₗ ψ = ↑(η i) ∘ₗ φ i ∘ₗ τ i
a : ⨁ (i : ι), ↥I ⊗[R] M i
ha : ((↑η₁ ∘ₗ ρ) ∘ₗ ψ) a = 0
i : ι
f : (↑(η i) ∘ₗ φ i ∘ₗ τ i) a = 0
h₂ : ∀ ⦃I : Ideal R⦄, Ideal.FG I → Injective ⇑(rTensor (M i) (Submodule.subtype I))
h₃ : Injective ⇑(rTensor (M i) (Submodule.subtype I))
⊢ (component R ι (fun i => ↥I ⊗[R] M i) i) a = (component R ι (fun i => ↥I ⊗[R] M i) i) 0

Next tactic:
simp only [coe_comp, LinearEquiv.coe_coe, Function.comp_apply, AddEquivClass.map_eq_zero_iff,
    h₃, LinearMap.map_eq_zero_iff] at f