-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
refine_1 :
  Finset.inf
      (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
        (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}) s))
      id =
    N
refine_2 :
  ∀ ⦃J : Submodule R M⦄,
    Membership.mem (γ := Finset (Submodule R M))
        (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s))
        J →
      J.IsPrimary
refine_3 :
  (↑(Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
            (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
              s)) :
        Set (Submodule R M)).Pairwise
    ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical)
refine_2₁ :
  ∀ ⦃J : Submodule R M⦄,
    Membership.mem (γ := Finset (Submodule R M))
        (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s))
        J →
      J.IsPrimary
refine_3₁ refine_3₂ :
  (↑(Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
            (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
              s)) :
        Set (Submodule R M)).Pairwise
    ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical)
⊢ ∃ (t : Finset (Submodule R M)),
    t.inf id = N ∧
      (∀ ⦃J : Submodule R M⦄, J ∈ t → J.IsPrimary) ∧
        (↑t : Set (Submodule R M)).Pairwise
          ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical)

Tactic:
classical
  refine ⟨(s.image fun J ↦ {I ∈ s | (I.colon .univ).radical = (J.colon .univ).radical}).image
    fun t ↦ t.inf id, ?_, ?_, ?_⟩
  · ext
    grind [Finset.inf_image, Submodule.mem_finsetInf]
  · simp only [Finset.mem_image, exists_exists_and_eq_and, forall_exists_index, and_imp,
    forall_apply_eq_imp_iff₂]
    intro J hJ
    refine isPrimary_finsetInf (i := J) ?_ ?_ (by simp)
    · simp [hJ]
    · simp only [Finset.mem_filter, id_eq, and_imp]
      intro y hy
      simp [hs' hy]
  · intro I hI J hJ hIJ
    simp only [Finset.coe_image, Set.mem_image, Finset.mem_coe, exists_exists_and_eq_and] at hI hJ
    obtain ⟨I', hI', hI⟩ := hI
    obtain ⟨J', hJ', hJ⟩ := hJ
    simp only [Function.onFun, ne_eq]
    contrapose hIJ
    suffices (I'.colon Set.univ).radical = (J'.colon Set.univ).radical by
      rw [← hI, ← hJ, this]
    · rw [← hI, colon_finsetInf,
        radical_finset_inf (i := I') (by simp [hI']) (by simp), id_eq] at hIJ
      rw [hIJ, ← hJ, colon_finsetInf,
        radical_finset_inf (i := J') (by simp [hJ']) (by simp), id_eq]

Premises used: Submodule.colon_finsetInf, Function.onFun, Finset.mem_image, and_imp, Set.mem_image, Finset.inf_image, forall_exists_index, Finset.coe_image, ne_eq, exists_exists_and_eq_and, Finset.mem_filter, Submodule.mem_finsetInf, Finset.mem_coe, forall_apply_eq_imp_iff₂, id_eq