-- Module: Mathlib.RingTheory.Ideal.Quotient.Index
Goal:
R : Type u_1
M : Type u_2
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
I : Ideal R
N : Submodule R M
inst_3 : Finite.{u_1 + 1} (R ⧸ I)
s : Finset M
hs : span R (↑s : Set M) = N
intro :
  ∀ (val : Fintype (R ⧸ I)),
    (I • N).toAddSubgroup.index ≤ AddSubgroup.index.{u_1} (G := R) (toAddSubgroup I) ^ s.card * N.toAddSubgroup.index
intro₁ :
  ∀ (val : Fintype (R ⧸ I)),
    (I • N).toAddSubgroup.relIndex N.toAddSubgroup * N.toAddSubgroup.index ≤
      AddSubgroup.index.{u_1} (G := R) (toAddSubgroup I) ^ s.card * N.toAddSubgroup.index
intro₂ :
  ∀ (val : Fintype (R ⧸ I)),
    (I • N).toAddSubgroup.relIndex N.toAddSubgroup ≤ AddSubgroup.index.{u_1} (G := R) (toAddSubgroup I) ^ s.card
intro₃ : ∀ (val : Fintype (R ⧸ I)), Nat.card.{u_2} (↥N ⧸ comap N.subtype (I • N)) ≤ Nat.card.{u_1} (R ⧸ I) ^ s.card
intro₄ : ∀ (val : Fintype (R ⧸ I)), Nat.card.{u_2} (↥N ⧸ comap N.subtype (I • N)) ≤ Nat.card.{u_1} (R ⧸ I) ^ s.card
intro₅ :
  ∀ (val : Fintype (R ⧸ I)), Nat.card (TensorProduct.{u_1, u_1, u_2} R (R ⧸ I) ↥N) ≤ Nat.card.{u_1} (R ⧸ I) ^ s.card
intro₆ intro₇ :
  ∀ (val : Fintype (R ⧸ I))
    (H :
      LinearMap.range (M := Finsupp.{u_2, u_1} (↥s) R)
          (Finsupp.linearCombination R (Subtype.val (p := fun (x : M) ↦ x ∈ s))) =
        N),
    Nat.card (TensorProduct.{u_1, u_1, u_2} R (R ⧸ I) ↥N) ≤ Nat.card.{u_1} (R ⧸ I) ^ s.card
intro₈ :
  ∀ (val : Fintype (R ⧸ I)),
    let e : (↥N ⧸ comap N.subtype (I • N)) ≃ₗ[R] TensorProduct.{u_1, u_1, u_2} R (R ⧸ I) ↥N :=
      (comap N.subtype (I • N)).quotEquivOfEq (HSMul.hSMul (β := Submodule R ↥N) I ⊤) ⋯ ≪≫ₗ
        LinearEquiv.symm (quotTensorEquivQuotSMul (↥N) I);
    ∀
      (H :
        LinearMap.range (M := Finsupp.{u_2, u_1} (↥s) R)
            (Finsupp.linearCombination R (Subtype.val (p := fun (x : M) ↦ x ∈ s))) =
          N),
      let f : Finsupp.{u_2, u_1} (↥s) R →ₗ[R] ↥N :=
        LinearMap.codRestrict N (Finsupp.linearCombination R (Subtype.val (p := fun (x : M) ↦ x ∈ s))) ⋯;
      ∀ (hf : Function.Surjective (α := Finsupp.{u_2, u_1} (↥s) R) (β := ↥N) (⇑f : (a : ↥s →₀ R) → ↥N)),
        Nat.card (TensorProduct.{u_1, u_1, u_2} R (R ⧸ I) ↥N) ≤ Nat.card.{u_1} (R ⧸ I) ^ s.card
intro₉ :
  ∀ (val : Fintype (R ⧸ I)),
    let e : (↥N ⧸ comap N.subtype (I • N)) ≃ₗ[R] TensorProduct.{u_1, u_1, u_2} R (R ⧸ I) ↥N :=
      (comap N.subtype (I • N)).quotEquivOfEq (HSMul.hSMul (β := Submodule R ↥N) I ⊤) ⋯ ≪≫ₗ
        LinearEquiv.symm (quotTensorEquivQuotSMul (↥N) I);
    ∀
      (H :
        LinearMap.range (M := Finsupp.{u_2, u_1} (↥s) R)
            (Finsupp.linearCombination R (Subtype.val (p := fun (x : M) ↦ x ∈ s))) =
          N),
      let f : Finsupp.{u_2, u_1} (↥s) R →ₗ[R] ↥N :=
        LinearMap.codRestrict N (Finsupp.linearCombination R (Subtype.val (p := fun (x : M) ↦ x ∈ s))) ⋯;
      ∀ (hf : Function.Surjective (α := Finsupp.{u_2, u_1} (↥s) R) (β := ↥N) (⇑f : (a : ↥s →₀ R) → ↥N))
        (this :
          Function.Surjective (α := Finsupp.{u_2, u_1} (↥s) (R ⧸ I)) (β := TensorProduct.{u_1, u_1, u_2} R (R ⧸ I) ↥N)
            (⇑(LinearMap.lTensor (R ⧸ I) f ∘ₗ
                  (↑(LinearEquiv.symm (finsuppScalarRight R R (R ⧸ I) ↥s)) :
                    (↥s →₀ R ⧸ I) →ₗ[R] (R ⧸ I) ⊗[R] (↥s →₀ R))) :
              (a : ↥s →₀ R ⧸ I) → (R ⧸ I) ⊗[R] ↥N)),
        Nat.card (TensorProduct.{u_1, u_1, u_2} R (R ⧸ I) ↥N) ≤ Nat.card.{u_1} (R ⧸ I) ^ s.card
intro₁₀ :
  ∀ (val : Fintype (R ⧸ I)),
    let e : (↥N ⧸ comap N.subtype (I • N)) ≃ₗ[R] TensorProduct.{u_1, u_1, u_2} R (R ⧸ I) ↥N :=
      (comap N.subtype (I • N)).quotEquivOfEq (HSMul.hSMul (β := Submodule R ↥N) I ⊤) ⋯ ≪≫ₗ
        LinearEquiv.symm (quotTensorEquivQuotSMul (↥N) I);
    ∀
      (H :
        LinearMap.range (M := Finsupp.{u_2, u_1} (↥s) R)
            (Finsupp.linearCombination R (Subtype.val (p := fun (x : M) ↦ x ∈ s))) =
          N),
      let f : Finsupp.{u_2, u_1} (↥s) R →ₗ[R] ↥N :=
        LinearMap.codRestrict N (Finsupp.linearCombination R (Subtype.val (p := fun (x : M) ↦ x ∈ s))) ⋯;
      ∀ (hf : Function.Surjective (α := Finsupp.{u_2, u_1} (↥s) R) (β := ↥N) (⇑f : (a : ↥s →₀ R) → ↥N))
        (this :
          Function.Surjective (α := Finsupp.{u_2, u_1} (↥s) (R ⧸ I)) (β := TensorProduct.{u_1, u_1, u_2} R (R ⧸ I) ↥N)
            (⇑(LinearMap.lTensor (R ⧸ I) f ∘ₗ
                  (↑(LinearEquiv.symm (finsuppScalarRight R R (R ⧸ I) ↥s)) :
                    (↥s →₀ R ⧸ I) →ₗ[R] (R ⧸ I) ⊗[R] (↥s →₀ R))) :
              (a : ↥s →₀ R ⧸ I) → (R ⧸ I) ⊗[R] ↥N)),
        Nat.card (Finsupp.{u_2, u_1} (↥s) (R ⧸ I)) ≤ Nat.card.{u_1} (R ⧸ I) ^ s.card
⊢ (I • N).toAddSubgroup.index ≤ AddSubgroup.index.{u_1} (G := R) (toAddSubgroup I) ^ s.card * N.toAddSubgroup.index

Tactic:
classical
  cases nonempty_fintype (R ⧸ I)
  rw [← AddSubgroup.relIndex_mul_index
    (H := (I • N).toAddSubgroup) (K := N.toAddSubgroup) Submodule.smul_le_right]
  gcongr
  change (Nat.card (N ⧸ (I • N).comap N.subtype)) ≤ Nat.card (R ⧸ I) ^ s.card
  let e : (N ⧸ (I • N).comap N.subtype) ≃ₗ[R] (R ⧸ I) ⊗[R] N :=
    Submodule.quotEquivOfEq _ (I • (⊤ : Submodule R N)) (Submodule.map_injective_of_injective
      N.injective_subtype (by simp [Submodule.smul_le_right])) ≪≫ₗ
      (quotTensorEquivQuotSMul N I).symm
  rw [Nat.card_congr e.toEquiv]
  have H : LinearMap.range (Finsupp.linearCombination R (α := s) (↑)) = N := by
    rw [Finsupp.range_linearCombination, ← hs, Subtype.range_val]
  let f : (s →₀ R) →ₗ[R] N := (Finsupp.linearCombination R (↑)).codRestrict _
    (fun c => by rw [← H, LinearMap.mem_range]; exact exists_apply_eq_apply _ _)
  have hf : Function.Surjective f := fun x ↦ by
    obtain ⟨y, hy⟩ := H.ge x.2; exact ⟨y, Subtype.ext hy⟩
  have : Function.Surjective
      (f.lTensor (R ⧸ I) ∘ₗ (finsuppScalarRight R R (R ⧸ I) s).symm.toLinearMap) :=
    (LinearMap.lTensor_surjective (R ⧸ I) hf).comp (LinearEquiv.surjective _)
  refine (Nat.card_le_card_of_surjective _ this).trans ?_
  simp only [Nat.card_eq_fintype_card, Fintype.card_finsupp, Fintype.card_coe, le_rfl]

Premises used: le_rfl, Membership.mem, Subtype, Nat.card, AddSubgroup.relIndex_mul_index, Submodule.smul_le_right, HasQuotient.Quotient, nonempty_fintype, Submodule.comap, Submodule.subtype, Finset.card, Fintype.card_finsupp, HSMul.hSMul, Fintype.card_coe, Nat.card_eq_fintype_card