-- Module: Mathlib.RingTheory.GradedAlgebra.Radical
Goal:
ι : Type u_1
σ : Type u_2
A : Type u_3
inst : CommRing A
inst_1 : AddCommMonoid ι
inst_2 : LinearOrder ι
inst_3 : IsOrderedCancelAddMonoid ι
inst_4 : SetLike σ A
inst_5 : AddSubmonoidClass σ A
𝒜 : ι → σ
inst_6 : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I
x y : A
hxy : x * y ∈ I
rid₁ : x ∉ I
rid₂ : y ∉ I
The The₁ : False
The₂ :
  let set₁ : Finset ι :=
    {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
      ((proj 𝒜 i : (a : A) → A) x : A) ∉ I};
  ∀
    (set₁_eq :
      set₁ =
        {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
          ((proj 𝒜 i : (a : A) → A) x : A) ∉ I}),
    False
The₃ :
  let set₁ : Finset ι :=
    {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
      ((proj 𝒜 i : (a : A) → A) x : A) ∉ I};
  ∀
    (set₁_eq :
      set₁ =
        {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
          ((proj 𝒜 i : (a : A) → A) x : A) ∉ I}),
    let set₂ : Finset ι :=
      {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
        ((proj 𝒜 i : (a : A) → A) y : A) ∉ I};
    ∀
      (set₂_eq :
        set₂ =
          {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
            ((proj 𝒜 i : (a : A) → A) y : A) ∉ I}),
      False
The₄ The₅ The₆ :
  let set₁ : Finset ι :=
    {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
      ((proj 𝒜 i : (a : A) → A) x : A) ∉ I};
  ∀
    (set₁_eq :
      set₁ =
        {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
          ((proj 𝒜 i : (a : A) → A) x : A) ∉ I}),
    let set₂ : Finset ι :=
      {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
        ((proj 𝒜 i : (a : A) → A) y : A) ∉ I};
    ∀
      (set₂_eq :
        set₂ =
          {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
            ((proj 𝒜 i : (a : A) → A) y : A) ∉ I})
      (nonempty :
        ∀ x ∉ I,
          Finset.Nonempty.{u_1} (α := ι)
            ({i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
              ((proj 𝒜 i : (a : A) → A) x : A) ∉ I})),
      False
The₇ :
  let set₁ : Finset ι :=
    {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
      ((proj 𝒜 i : (a : A) → A) x : A) ∉ I};
  ∀
    (set₁_eq :
      set₁ =
        {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
          ((proj 𝒜 i : (a : A) → A) x : A) ∉ I}),
    let set₂ : Finset ι :=
      {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
        ((proj 𝒜 i : (a : A) → A) y : A) ∉ I};
    ∀
      (set₂_eq :
        set₂ =
          {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
            ((proj 𝒜 i : (a : A) → A) y : A) ∉ I})
      (nonempty :
        ∀ x ∉ I,
          Finset.Nonempty.{u_1} (α := ι)
            ({i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
              ((proj 𝒜 i : (a : A) → A) x : A) ∉ I})),
      let max₁ := set₁.max' ⋯;
      ∀ (mem_max₁ : max₁ ∈ set₁), False
The₈ :
  let set₁ : Finset ι :=
    {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
      ((proj 𝒜 i : (a : A) → A) x : A) ∉ I};
  ∀
    (set₁_eq :
      set₁ =
        {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
          ((proj 𝒜 i : (a : A) → A) x : A) ∉ I}),
    let set₂ : Finset ι :=
      {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
        ((proj 𝒜 i : (a : A) → A) y : A) ∉ I};
    ∀
      (set₂_eq :
        set₂ =
          {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
            ((proj 𝒜 i : (a : A) → A) y : A) ∉ I})
      (nonempty :
        ∀ x ∉ I,
          Finset.Nonempty.{u_1} (α := ι)
            ({i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
              ((proj 𝒜 i : (a : A) → A) x : A) ∉ I})),
      let max₁ := set₁.max' ⋯;
      let max₂ := set₂.max' ⋯;
      ∀ (mem_max₁ : max₁ ∈ set₁) (mem_max₂ : max₂ ∈ set₂), False
The₉ :
  let set₁ : Finset ι :=
    {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
      ((proj 𝒜 i : (a : A) → A) x : A) ∉ I};
  ∀
    (set₁_eq :
      set₁ =
        {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
          ((proj 𝒜 i : (a : A) → A) x : A) ∉ I}),
    let set₂ : Finset ι :=
      {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
        ((proj 𝒜 i : (a : A) → A) y : A) ∉ I};
    ∀
      (set₂_eq :
        set₂ =
          {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
            ((proj 𝒜 i : (a : A) → A) y : A) ∉ I})
      (nonempty :
        ∀ x ∉ I,
          Finset.Nonempty.{u_1} (α := ι)
            ({i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
              ((proj 𝒜 i : (a : A) → A) x : A) ∉ I})),
      let max₁ := set₁.max' ⋯;
      let max₂ := set₂.max' ⋯;
      ∀ (mem_max₁ : max₁ ∈ set₁) (mem_max₂ : max₂ ∈ set₂)
        (hxy : Membership.mem (α := A) I ((proj 𝒜 (max₁ + max₂) : (a : A) → A) (x * y) : A)), False
The₁₀ :
  let set₁ : Finset ι :=
    {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
      ((proj 𝒜 i : (a : A) → A) x : A) ∉ I};
  ∀
    (set₁_eq :
      set₁ =
        {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
          ((proj 𝒜 i : (a : A) → A) x : A) ∉ I}),
    let set₂ : Finset ι :=
      {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
        ((proj 𝒜 i : (a : A) → A) y : A) ∉ I};
    ∀
      (set₂_eq :
        set₂ =
          {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
            ((proj 𝒜 i : (a : A) → A) y : A) ∉ I})
      (nonempty :
        ∀ x ∉ I,
          Finset.Nonempty.{u_1} (α := ι)
            ({i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
              ((proj 𝒜 i : (a : A) → A) x : A) ∉ I})),
      let max₁ := set₁.max' ⋯;
      let max₂ := set₂.max' ⋯;
      ∀ (mem_max₁ : max₁ ∈ set₁) (mem_max₂ : max₂ ∈ set₂)
        (hxy : Membership.mem (α := A) I ((proj 𝒜 (max₁ + max₂) : (a : A) → A) (x * y) : A))
        (mem_I :
          HMul.hMul (α := A) (β := A) ((proj 𝒜 max₁ : (a : A) → A) x : A) ((proj 𝒜 max₂ : (a : A) → A) y : A) ∈ I),
        False
The₁₁ :
  let set₁ : Finset ι :=
    {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
      ((proj 𝒜 i : (a : A) → A) x : A) ∉ I};
  ∀
    (set₁_eq :
      set₁ =
        {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
          ((proj 𝒜 i : (a : A) → A) x : A) ∉ I}),
    let set₂ : Finset ι :=
      {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
        ((proj 𝒜 i : (a : A) → A) y : A) ∉ I};
    ∀
      (set₂_eq :
        set₂ =
          {i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) y) |
            ((proj 𝒜 i : (a : A) → A) y : A) ∉ I})
      (nonempty :
        ∀ x ∉ I,
          Finset.Nonempty.{u_1} (α := ι)
            ({i ∈ DFinsupp.support (β := fun (i : ι) ↦ ↥(𝒜 i)) ((decompose 𝒜 : (a : A) → ⨁ (i : ι), ↥(𝒜 i)) x) |
              ((proj 𝒜 i : (a : A) → A) x : A) ∉ I})),
      let max₁ := set₁.max' ⋯;
      let max₂ := set₂.max' ⋯;
      ∀ (mem_max₁ : max₁ ∈ set₁) (mem_max₂ : max₂ ∈ set₂)
        (hxy : Membership.mem (α := A) I ((proj 𝒜 (max₁ + max₂) : (a : A) → A) (x * y) : A))
        (mem_I :
          HMul.hMul (α := A) (β := A) ((proj 𝒜 max₁ : (a : A) → A) x : A) ((proj 𝒜 max₂ : (a : A) → A) y : A) ∈ I)
        (notMem_I :
          HMul.hMul (α := A) (β := A) ((proj 𝒜 max₁ : (a : A) → A) x : A) ((proj 𝒜 max₂ : (a : A) → A) y : A) ∉ I),
        False
⊢ False

Tactic:
classical
      /-
        The idea of the proof is the following :
        since `x * y ∈ I` and `I` homogeneous, then `proj i (x * y) ∈ I` for any `i : ι`.
        Then consider two sets `{i ∈ x.support | xᵢ ∉ I}` and `{j ∈ y.support | yⱼ ∉ J}`;
        let `max₁, max₂` be the maximum of the two sets, then `proj (max₁ + max₂) (x * y) ∈ I`.
        Then, `proj max₁ x ∉ I` and `proj max₂ j ∉ I`
        but `proj i x ∈ I` for all `max₁ < i` and `proj j y ∈ I` for all `max₂ < j`.
        `  proj (max₁ + max₂) (x * y)`
        `= ∑ {(i, j) ∈ supports | i + j = max₁ + max₂}, xᵢ * yⱼ`
        `= proj max₁ x * proj max₂ y`
        `  + ∑ {(i, j) ∈ supports \ {(max₁, max₂)} | i + j = max₁ + max₂}, xᵢ * yⱼ`.
        This is a contradiction, because both `proj (max₁ + max₂) (x * y) ∈ I` and the sum on the
        right-hand side is in `I` however `proj max₁ x * proj max₂ y` is not in `I`.
        -/
      set set₁ := {i ∈ (decompose 𝒜 x).support | proj 𝒜 i x ∉ I} with set₁_eq
      set set₂ := {i ∈ (decompose 𝒜 y).support | proj 𝒜 i y ∉ I} with set₂_eq
      have nonempty :
        ∀ x : A, x ∉ I → {i ∈ (decompose 𝒜 x).support | proj 𝒜 i x ∉ I}.Nonempty := by
        intro x hx
        rw [filter_nonempty_iff]
        contrapose! hx
        simp_rw [proj_apply] at hx
        rw [← sum_support_decompose 𝒜 x]
        exact Ideal.sum_mem _ hx
      set max₁ := set₁.max' (nonempty x rid₁)
      set max₂ := set₂.max' (nonempty y rid₂)
      have mem_max₁ : max₁ ∈ set₁ := max'_mem set₁ (nonempty x rid₁)
      have mem_max₂ : max₂ ∈ set₂ := max'_mem set₂ (nonempty y rid₂)
      replace hxy : proj 𝒜 (max₁ + max₂) (x * y) ∈ I := hI _ hxy
      have mem_I : proj 𝒜 max₁ x * proj 𝒜 max₂ y ∈ I := by
        set antidiag :=
          {z ∈ (decompose 𝒜 x).support ×ˢ (decompose 𝒜 y).support | z.1 + z.2 = max₁ + max₂}
           with ha
        have mem_antidiag : (max₁, max₂) ∈ antidiag := by
          simp only [antidiag, mem_filter, mem_product]
          exact ⟨⟨mem_of_mem_filter _ mem_max₁, mem_of_mem_filter _ mem_max₂⟩, trivial⟩
        have eq_add_sum :=
          calc
            proj 𝒜 (max₁ + max₂) (x * y) = ∑ ij ∈ antidiag, proj 𝒜 ij.1 x * proj 𝒜 ij.2 y := by
              simp_rw [ha, proj_apply, DirectSum.decompose_mul, DirectSum.coe_mul_apply 𝒜]
            _ =
                proj 𝒜 max₁ x * proj 𝒜 max₂ y +
                  ∑ ij ∈ antidiag.erase (max₁, max₂), proj 𝒜 ij.1 x * proj 𝒜 ij.2 y :=
              (add_sum_erase _ _ mem_antidiag).symm
        rw [eq_sub_of_add_eq eq_add_sum.symm]
        refine Ideal.sub_mem _ hxy (Ideal.sum_mem _ fun z H => ?_)
        rcases z with ⟨i, j⟩
        simp only [antidiag, mem_erase, Prod.mk_inj, Ne, mem_filter, mem_product] at H
        rcases H with ⟨H₁, ⟨H₂, H₃⟩, H₄⟩
        have max_lt : max₁ < i ∨ max₂ < j := by
          convert! le_or_lt_of_add_le_add H₄.ge using 1
          rw [Ne.le_iff_lt]
          rintro rfl
          cases H₁ ⟨rfl, add_left_cancel H₄⟩
        rcases max_lt with max_lt | max_lt
        · -- in this case `max₁ < i`, then `xᵢ ∈ I`; for otherwise `i ∈ set₁` then `i ≤ max₁`.
          have notMem : i ∉ set₁ := fun h =>
            lt_irrefl _ ((max'_lt_iff set₁ (nonempty x rid₁)).mp max_lt i h)
          rw [set₁_eq] at notMem
          simp only [not_and, Classical.not_not, mem_filter] at notMem
          exact Ideal.mul_mem_right _ I (notMem H₂)
        · -- in this case `max₂ < j`, then `yⱼ ∈ I`; for otherwise `j ∈ set₂`, then `j ≤ max₂`.
          have notMem : j ∉ set₂ := fun h =>
            lt_irrefl _ ((max'_lt_iff set₂ (nonempty y rid₂)).mp max_lt j h)
          rw [set₂_eq] at notMem
          simp only [not_and, Classical.not_not, mem_filter] at notMem
          exact Ideal.mul_mem_left I _ (notMem H₃)
      have notMem_I : proj 𝒜 max₁ x * proj 𝒜 max₂ y ∉ I := by
        have neither_mem : proj 𝒜 max₁ x ∉ I ∧ proj 𝒜 max₂ y ∉ I := by
          rw [mem_filter] at mem_max₁ mem_max₂
          exact ⟨mem_max₁.2, mem_max₂.2⟩
        intro _rid
        rcases homogeneous_mem_or_mem ⟨max₁, SetLike.coe_mem _⟩ ⟨max₂, SetLike.coe_mem _⟩ mem_I
          with h | h
        · apply neither_mem.1 h
        · apply neither_mem.2 h
      exact notMem_I mem_I