l3lab/ntp-mathlib huggingface dataset:
/-- **Nakayama's Lemma**. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2,
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV) -/
theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
    [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
    ∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) := by
  rw [fg_def] at hn
  rcases hn with ⟨s, hfs, hs⟩
  have : ∃ r : R, r - 1 ∈ I ∧ N ≤ (I • span R s).comap (LinearMap.lsmul R M r) ∧ s ⊆ N := by
    refine' ⟨1, _, _, _⟩
    · rw [sub_self]
      exact I.zero_mem
    · rw [hs]
      intro n hn
      rw [mem_comap]
      change (1 : R) • n ∈ I • N
      rw [one_smul]
      exact hin hn
    · rw [← span_le, hs]
  clear hin hs
  revert this
  refine' Set.Finite.dinduction_on _ hfs (fun H => _) @fun i s _ _ ih H => _
  · rcases H with ⟨r, hr1, hrn, _⟩
    refine' ⟨r, hr1, fun n hn => _⟩
    specialize hrn hn
    rwa [mem_comap, span_empty, smul_bot, mem_bot] at hrn
  apply ih
  rcases H with ⟨r, hr1, hrn, hs⟩
  rw [← Set.singleton_union, span_union, smul_sup] at hrn
  rw [Set.insert_subset_iff] at hs
  have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
    specialize hrn hs.1
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨c, hci, rfl⟩
    use r - c
    constructor
    · rw [sub_right_comm]
      exact I.sub_mem hr1 hci
    · rw [sub_smul, ← hyz, add_sub_cancel']
      exact hz
  rcases this with ⟨c, hc1, hci⟩
  refine' ⟨c * r, _, _, hs.2⟩
  · simpa only [mul_sub, mul_one, sub_add_sub_cancel] using I.add_mem (I.mul_mem_left c hr1) hc1
  · intro n hn
    specialize hrn hn
    

Proof state:
case intro.intro.refine'_2.intro.intro.intro.intro.intro.refine'_2
R✝ : Type u_1
M✝ : Type u_2
inst✝⁵ : Semiring R✝
inst✝⁴ : AddCommMonoid M✝
inst✝³ : Module R✝ M✝
R : Type u_3
inst✝² : CommRing R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
s✝ : Set M
hfs : Set.Finite s✝
i : M
s : Set M
x✝¹ : i ∉ s
x✝ : Set.Finite s
ih :
  (∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ ↑N ∧ s ⊆ ↑N
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ I • span R s
n : M
hn : n ∈ N
hrn : n ∈ comap ((LinearMap.lsmul R M) r) (I • span R {i} ⊔ I • span R s)
⊢ n ∈ comap ((LinearMap.lsmul R M) (c * r)) (I • span R s)

Next tactic:
rw [mem_comap, mem_sup] at hrn
/-- **Nakayama's Lemma**. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2,
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV) -/
theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
    [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
    ∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) := by
  rw [fg_def] at hn
  rcases hn with ⟨s, hfs, hs⟩
  have : ∃ r : R, r - 1 ∈ I ∧ N ≤ (I • span R s).comap (LinearMap.lsmul R M r) ∧ s ⊆ N := by
    refine' ⟨1, _, _, _⟩
    · rw [sub_self]
      exact I.zero_mem
    · rw [hs]
      intro n hn
      rw [mem_comap]
      change (1 : R) • n ∈ I • N
      rw [one_smul]
      exact hin hn
    · rw [← span_le, hs]
  clear hin hs
  revert this
  refine' Set.Finite.dinduction_on _ hfs (fun H => _) @fun i s _ _ ih H => _
  · rcases H with ⟨r, hr1, hrn, _⟩
    refine' ⟨r, hr1, fun n hn => _⟩
    specialize hrn hn
    rwa [mem_comap, span_empty, smul_bot, mem_bot] at hrn
  apply ih
  rcases H with ⟨r, hr1, hrn, hs⟩
  rw [← Set.singleton_union, span_union, smul_sup] at hrn
  rw [Set.insert_subset_iff] at hs
  have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
    specialize hrn hs.1
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨c, hci, rfl⟩
    use r - c
    constructor
    · rw [sub_right_comm]
      exact I.sub_mem hr1 hci
    · rw [sub_smul, ← hyz, add_sub_cancel']
      exact hz
  rcases this with ⟨c, hc1, hci⟩
  refine' ⟨c * r, _, _, hs.2⟩
  · simpa only [mul_sub, mul_one, sub_add_sub_cancel] using I.add_mem (I.mul_mem_left c hr1) hc1
  · intro n hn
    specialize hrn hn
    rw [mem_comap, mem_sup] at hrn
    

Proof state:
case intro.intro.refine'_2.intro.intro.intro.intro.intro.refine'_2
R✝ : Type u_1
M✝ : Type u_2
inst✝⁵ : Semiring R✝
inst✝⁴ : AddCommMonoid M✝
inst✝³ : Module R✝ M✝
R : Type u_3
inst✝² : CommRing R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
s✝ : Set M
hfs : Set.Finite s✝
i : M
s : Set M
x✝¹ : i ∉ s
x✝ : Set.Finite s
ih :
  (∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ ↑N ∧ s ⊆ ↑N
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ I • span R s
n : M
hn : n ∈ N
hrn : ∃ y ∈ I • span R {i}, ∃ z ∈ I • span R s, y + z = ((LinearMap.lsmul R M) r) n
⊢ n ∈ comap ((LinearMap.lsmul R M) (c * r)) (I • span R s)

Next tactic:
rcases hrn with ⟨y, hy, z, hz, hyz⟩
/-- **Nakayama's Lemma**. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2,
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV) -/
theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
    [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
    ∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) := by
  rw [fg_def] at hn
  rcases hn with ⟨s, hfs, hs⟩
  have : ∃ r : R, r - 1 ∈ I ∧ N ≤ (I • span R s).comap (LinearMap.lsmul R M r) ∧ s ⊆ N := by
    refine' ⟨1, _, _, _⟩
    · rw [sub_self]
      exact I.zero_mem
    · rw [hs]
      intro n hn
      rw [mem_comap]
      change (1 : R) • n ∈ I • N
      rw [one_smul]
      exact hin hn
    · rw [← span_le, hs]
  clear hin hs
  revert this
  refine' Set.Finite.dinduction_on _ hfs (fun H => _) @fun i s _ _ ih H => _
  · rcases H with ⟨r, hr1, hrn, _⟩
    refine' ⟨r, hr1, fun n hn => _⟩
    specialize hrn hn
    rwa [mem_comap, span_empty, smul_bot, mem_bot] at hrn
  apply ih
  rcases H with ⟨r, hr1, hrn, hs⟩
  rw [← Set.singleton_union, span_union, smul_sup] at hrn
  rw [Set.insert_subset_iff] at hs
  have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
    specialize hrn hs.1
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨c, hci, rfl⟩
    use r - c
    constructor
    · rw [sub_right_comm]
      exact I.sub_mem hr1 hci
    · rw [sub_smul, ← hyz, add_sub_cancel']
      exact hz
  rcases this with ⟨c, hc1, hci⟩
  refine' ⟨c * r, _, _, hs.2⟩
  · simpa only [mul_sub, mul_one, sub_add_sub_cancel] using I.add_mem (I.mul_mem_left c hr1) hc1
  · intro n hn
    specialize hrn hn
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    

Proof state:
case intro.intro.refine'_2.intro.intro.intro.intro.intro.refine'_2.intro.intro.intro.intro
R✝ : Type u_1
M✝ : Type u_2
inst✝⁵ : Semiring R✝
inst✝⁴ : AddCommMonoid M✝
inst✝³ : Module R✝ M✝
R : Type u_3
inst✝² : CommRing R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
s✝ : Set M
hfs : Set.Finite s✝
i : M
s : Set M
x✝¹ : i ∉ s
x✝ : Set.Finite s
ih :
  (∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ ↑N ∧ s ⊆ ↑N
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ I • span R s
n : M
hn : n ∈ N
y : M
hy : y ∈ I • span R {i}
z : M
hz : z ∈ I • span R s
hyz : y + z = ((LinearMap.lsmul R M) r) n
⊢ n ∈ comap ((LinearMap.lsmul R M) (c * r)) (I • span R s)

Next tactic:
dsimp at hyz
/-- **Nakayama's Lemma**. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2,
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV) -/
theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
    [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
    ∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) := by
  rw [fg_def] at hn
  rcases hn with ⟨s, hfs, hs⟩
  have : ∃ r : R, r - 1 ∈ I ∧ N ≤ (I • span R s).comap (LinearMap.lsmul R M r) ∧ s ⊆ N := by
    refine' ⟨1, _, _, _⟩
    · rw [sub_self]
      exact I.zero_mem
    · rw [hs]
      intro n hn
      rw [mem_comap]
      change (1 : R) • n ∈ I • N
      rw [one_smul]
      exact hin hn
    · rw [← span_le, hs]
  clear hin hs
  revert this
  refine' Set.Finite.dinduction_on _ hfs (fun H => _) @fun i s _ _ ih H => _
  · rcases H with ⟨r, hr1, hrn, _⟩
    refine' ⟨r, hr1, fun n hn => _⟩
    specialize hrn hn
    rwa [mem_comap, span_empty, smul_bot, mem_bot] at hrn
  apply ih
  rcases H with ⟨r, hr1, hrn, hs⟩
  rw [← Set.singleton_union, span_union, smul_sup] at hrn
  rw [Set.insert_subset_iff] at hs
  have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
    specialize hrn hs.1
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨c, hci, rfl⟩
    use r - c
    constructor
    · rw [sub_right_comm]
      exact I.sub_mem hr1 hci
    · rw [sub_smul, ← hyz, add_sub_cancel']
      exact hz
  rcases this with ⟨c, hc1, hci⟩
  refine' ⟨c * r, _, _, hs.2⟩
  · simpa only [mul_sub, mul_one, sub_add_sub_cancel] using I.add_mem (I.mul_mem_left c hr1) hc1
  · intro n hn
    specialize hrn hn
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    

Proof state:
case intro.intro.refine'_2.intro.intro.intro.intro.intro.refine'_2.intro.intro.intro.intro
R✝ : Type u_1
M✝ : Type u_2
inst✝⁵ : Semiring R✝
inst✝⁴ : AddCommMonoid M✝
inst✝³ : Module R✝ M✝
R : Type u_3
inst✝² : CommRing R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
s✝ : Set M
hfs : Set.Finite s✝
i : M
s : Set M
x✝¹ : i ∉ s
x✝ : Set.Finite s
ih :
  (∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ ↑N ∧ s ⊆ ↑N
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ I • span R s
n : M
hn : n ∈ N
y : M
hy : y ∈ I • span R {i}
z : M
hz : z ∈ I • span R s
hyz : y + z = r • n
⊢ n ∈ comap ((LinearMap.lsmul R M) (c * r)) (I • span R s)

Next tactic:
rw [mem_smul_span_singleton] at hy
/-- **Nakayama's Lemma**. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2,
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV) -/
theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
    [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
    ∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) := by
  rw [fg_def] at hn
  rcases hn with ⟨s, hfs, hs⟩
  have : ∃ r : R, r - 1 ∈ I ∧ N ≤ (I • span R s).comap (LinearMap.lsmul R M r) ∧ s ⊆ N := by
    refine' ⟨1, _, _, _⟩
    · rw [sub_self]
      exact I.zero_mem
    · rw [hs]
      intro n hn
      rw [mem_comap]
      change (1 : R) • n ∈ I • N
      rw [one_smul]
      exact hin hn
    · rw [← span_le, hs]
  clear hin hs
  revert this
  refine' Set.Finite.dinduction_on _ hfs (fun H => _) @fun i s _ _ ih H => _
  · rcases H with ⟨r, hr1, hrn, _⟩
    refine' ⟨r, hr1, fun n hn => _⟩
    specialize hrn hn
    rwa [mem_comap, span_empty, smul_bot, mem_bot] at hrn
  apply ih
  rcases H with ⟨r, hr1, hrn, hs⟩
  rw [← Set.singleton_union, span_union, smul_sup] at hrn
  rw [Set.insert_subset_iff] at hs
  have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
    specialize hrn hs.1
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨c, hci, rfl⟩
    use r - c
    constructor
    · rw [sub_right_comm]
      exact I.sub_mem hr1 hci
    · rw [sub_smul, ← hyz, add_sub_cancel']
      exact hz
  rcases this with ⟨c, hc1, hci⟩
  refine' ⟨c * r, _, _, hs.2⟩
  · simpa only [mul_sub, mul_one, sub_add_sub_cancel] using I.add_mem (I.mul_mem_left c hr1) hc1
  · intro n hn
    specialize hrn hn
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    

Proof state:
case intro.intro.refine'_2.intro.intro.intro.intro.intro.refine'_2.intro.intro.intro.intro
R✝ : Type u_1
M✝ : Type u_2
inst✝⁵ : Semiring R✝
inst✝⁴ : AddCommMonoid M✝
inst✝³ : Module R✝ M✝
R : Type u_3
inst✝² : CommRing R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
s✝ : Set M
hfs : Set.Finite s✝
i : M
s : Set M
x✝¹ : i ∉ s
x✝ : Set.Finite s
ih :
  (∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ ↑N ∧ s ⊆ ↑N
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ I • span R s
n : M
hn : n ∈ N
y : M
hy : ∃ y_1 ∈ I, y_1 • i = y
z : M
hz : z ∈ I • span R s
hyz : y + z = r • n
⊢ n ∈ comap ((LinearMap.lsmul R M) (c * r)) (I • span R s)

Next tactic:
rcases hy with ⟨d, _, rfl⟩
/-- **Nakayama's Lemma**. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2,
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV) -/
theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
    [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
    ∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) := by
  rw [fg_def] at hn
  rcases hn with ⟨s, hfs, hs⟩
  have : ∃ r : R, r - 1 ∈ I ∧ N ≤ (I • span R s).comap (LinearMap.lsmul R M r) ∧ s ⊆ N := by
    refine' ⟨1, _, _, _⟩
    · rw [sub_self]
      exact I.zero_mem
    · rw [hs]
      intro n hn
      rw [mem_comap]
      change (1 : R) • n ∈ I • N
      rw [one_smul]
      exact hin hn
    · rw [← span_le, hs]
  clear hin hs
  revert this
  refine' Set.Finite.dinduction_on _ hfs (fun H => _) @fun i s _ _ ih H => _
  · rcases H with ⟨r, hr1, hrn, _⟩
    refine' ⟨r, hr1, fun n hn => _⟩
    specialize hrn hn
    rwa [mem_comap, span_empty, smul_bot, mem_bot] at hrn
  apply ih
  rcases H with ⟨r, hr1, hrn, hs⟩
  rw [← Set.singleton_union, span_union, smul_sup] at hrn
  rw [Set.insert_subset_iff] at hs
  have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
    specialize hrn hs.1
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨c, hci, rfl⟩
    use r - c
    constructor
    · rw [sub_right_comm]
      exact I.sub_mem hr1 hci
    · rw [sub_smul, ← hyz, add_sub_cancel']
      exact hz
  rcases this with ⟨c, hc1, hci⟩
  refine' ⟨c * r, _, _, hs.2⟩
  · simpa only [mul_sub, mul_one, sub_add_sub_cancel] using I.add_mem (I.mul_mem_left c hr1) hc1
  · intro n hn
    specialize hrn hn
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨d, _, rfl⟩
    

Proof state:
case intro.intro.refine'_2.intro.intro.intro.intro.intro.refine'_2.intro.intro.intro.intro.intro.intro
R✝ : Type u_1
M✝ : Type u_2
inst✝⁵ : Semiring R✝
inst✝⁴ : AddCommMonoid M✝
inst✝³ : Module R✝ M✝
R : Type u_3
inst✝² : CommRing R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
s✝ : Set M
hfs : Set.Finite s✝
i : M
s : Set M
x✝¹ : i ∉ s
x✝ : Set.Finite s
ih :
  (∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ ↑N ∧ s ⊆ ↑N
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ I • span R s
n : M
hn : n ∈ N
z : M
hz : z ∈ I • span R s
d : R
left✝ : d ∈ I
hyz : d • i + z = r • n
⊢ n ∈ comap ((LinearMap.lsmul R M) (c * r)) (I • span R s)

Next tactic:
simp only [mem_comap, LinearMap.lsmul_apply]
/-- **Nakayama's Lemma**. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2,
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV) -/
theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
    [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
    ∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) := by
  rw [fg_def] at hn
  rcases hn with ⟨s, hfs, hs⟩
  have : ∃ r : R, r - 1 ∈ I ∧ N ≤ (I • span R s).comap (LinearMap.lsmul R M r) ∧ s ⊆ N := by
    refine' ⟨1, _, _, _⟩
    · rw [sub_self]
      exact I.zero_mem
    · rw [hs]
      intro n hn
      rw [mem_comap]
      change (1 : R) • n ∈ I • N
      rw [one_smul]
      exact hin hn
    · rw [← span_le, hs]
  clear hin hs
  revert this
  refine' Set.Finite.dinduction_on _ hfs (fun H => _) @fun i s _ _ ih H => _
  · rcases H with ⟨r, hr1, hrn, _⟩
    refine' ⟨r, hr1, fun n hn => _⟩
    specialize hrn hn
    rwa [mem_comap, span_empty, smul_bot, mem_bot] at hrn
  apply ih
  rcases H with ⟨r, hr1, hrn, hs⟩
  rw [← Set.singleton_union, span_union, smul_sup] at hrn
  rw [Set.insert_subset_iff] at hs
  have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
    specialize hrn hs.1
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨c, hci, rfl⟩
    use r - c
    constructor
    · rw [sub_right_comm]
      exact I.sub_mem hr1 hci
    · rw [sub_smul, ← hyz, add_sub_cancel']
      exact hz
  rcases this with ⟨c, hc1, hci⟩
  refine' ⟨c * r, _, _, hs.2⟩
  · simpa only [mul_sub, mul_one, sub_add_sub_cancel] using I.add_mem (I.mul_mem_left c hr1) hc1
  · intro n hn
    specialize hrn hn
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨d, _, rfl⟩
    simp only [mem_comap, LinearMap.lsmul_apply]
    

Proof state:
case intro.intro.refine'_2.intro.intro.intro.intro.intro.refine'_2.intro.intro.intro.intro.intro.intro
R✝ : Type u_1
M✝ : Type u_2
inst✝⁵ : Semiring R✝
inst✝⁴ : AddCommMonoid M✝
inst✝³ : Module R✝ M✝
R : Type u_3
inst✝² : CommRing R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
s✝ : Set M
hfs : Set.Finite s✝
i : M
s : Set M
x✝¹ : i ∉ s
x✝ : Set.Finite s
ih :
  (∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ ↑N ∧ s ⊆ ↑N
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ I • span R s
n : M
hn : n ∈ N
z : M
hz : z ∈ I • span R s
d : R
left✝ : d ∈ I
hyz : d • i + z = r • n
⊢ (c * r) • n ∈ I • span R s

Next tactic:
rw [mul_smul, ← hyz, smul_add, smul_smul, mul_comm, mul_smul]
/-- **Nakayama's Lemma**. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2,
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV) -/
theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
    [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
    ∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) := by
  rw [fg_def] at hn
  rcases hn with ⟨s, hfs, hs⟩
  have : ∃ r : R, r - 1 ∈ I ∧ N ≤ (I • span R s).comap (LinearMap.lsmul R M r) ∧ s ⊆ N := by
    refine' ⟨1, _, _, _⟩
    · rw [sub_self]
      exact I.zero_mem
    · rw [hs]
      intro n hn
      rw [mem_comap]
      change (1 : R) • n ∈ I • N
      rw [one_smul]
      exact hin hn
    · rw [← span_le, hs]
  clear hin hs
  revert this
  refine' Set.Finite.dinduction_on _ hfs (fun H => _) @fun i s _ _ ih H => _
  · rcases H with ⟨r, hr1, hrn, _⟩
    refine' ⟨r, hr1, fun n hn => _⟩
    specialize hrn hn
    rwa [mem_comap, span_empty, smul_bot, mem_bot] at hrn
  apply ih
  rcases H with ⟨r, hr1, hrn, hs⟩
  rw [← Set.singleton_union, span_union, smul_sup] at hrn
  rw [Set.insert_subset_iff] at hs
  have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
    specialize hrn hs.1
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨c, hci, rfl⟩
    use r - c
    constructor
    · rw [sub_right_comm]
      exact I.sub_mem hr1 hci
    · rw [sub_smul, ← hyz, add_sub_cancel']
      exact hz
  rcases this with ⟨c, hc1, hci⟩
  refine' ⟨c * r, _, _, hs.2⟩
  · simpa only [mul_sub, mul_one, sub_add_sub_cancel] using I.add_mem (I.mul_mem_left c hr1) hc1
  · intro n hn
    specialize hrn hn
    rw [mem_comap, mem_sup] at hrn
    rcases hrn with ⟨y, hy, z, hz, hyz⟩
    dsimp at hyz
    rw [mem_smul_span_singleton] at hy
    rcases hy with ⟨d, _, rfl⟩
    simp only [mem_comap, LinearMap.lsmul_apply]
    rw [mul_smul, ← hyz, smul_add, smul_smul, mul_comm, mul_smul]
    

Proof state:
case intro.intro.refine'_2.intro.intro.intro.intro.intro.refine'_2.intro.intro.intro.intro.intro.intro
R✝ : Type u_1
M✝ : Type u_2
inst✝⁵ : Semiring R✝
inst✝⁴ : AddCommMonoid M✝
inst✝³ : Module R✝ M✝
R : Type u_3
inst✝² : CommRing R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
s✝ : Set M
hfs : Set.Finite s✝
i : M
s : Set M
x✝¹ : i ∉ s
x✝ : Set.Finite s
ih :
  (∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ ↑N ∧ s ⊆ ↑N
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ I • span R s
n : M
hn : n ∈ N
z : M
hz : z ∈ I • span R s
d : R
left✝ : d ∈ I
hyz : d • i + z = r • n
⊢ d • c • i + c • z ∈ I • span R s

Next tactic:
exact add_mem (smul_mem _ _ hci) (smul_mem _ _ hz)
theorem exists_mem_and_smul_eq_self_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
    [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
    ∃ r ∈ I, ∀ n ∈ N, r • n = n := by
  

Proof state:
R✝ : Type u_1
M✝ : Type u_2
inst✝⁵ : Semiring R✝
inst✝⁴ : AddCommMonoid M✝
inst✝³ : Module R✝ M✝
R : Type u_3
inst✝² : CommRing R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
hn : FG N
hin : N ≤ I • N
⊢ ∃ r ∈ I, ∀ n ∈ N, r • n = n

Next tactic:
obtain ⟨r, hr, hr'⟩ := exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul I N hn hin
theorem exists_mem_and_smul_eq_self_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
    [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
    ∃ r ∈ I, ∀ n ∈ N, r • n = n := by
  obtain ⟨r, hr, hr'⟩ := exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul I N hn hin
  

Proof state:
case intro.intro
R✝ : Type u_1
M✝ : Type u_2
inst✝⁵ : Semiring R✝
inst✝⁴ : AddCommMonoid M✝
inst✝³ : Module R✝ M✝
R : Type u_3
inst✝² : CommRing R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
hn : FG N
hin : N ≤ I • N
r : R
hr : r - 1 ∈ I
hr' : ∀ n ∈ N, r • n = 0
⊢ ∃ r ∈ I, ∀ n ∈ N, r • n = n

Next tactic:
exact ⟨-(r - 1), I.neg_mem hr, fun n hn => by simpa [sub_smul] using hr' n hn⟩
theorem exists_mem_and_smul_eq_self_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
    [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
    ∃ r ∈ I, ∀ n ∈ N, r • n = n := by
  obtain ⟨r, hr, hr'⟩ := exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul I N hn hin
  exact ⟨-(r - 1), I.neg_mem hr, fun n hn => by 

Proof state:
R✝ : Type u_1
M✝ : Type u_2
inst✝⁵ : Semiring R✝
inst✝⁴ : AddCommMonoid M✝
inst✝³ : Module R✝ M✝
R : Type u_3
inst✝² : CommRing R
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
hn✝ : FG N
hin : N ≤ I • N
r : R
hr : r - 1 ∈ I
hr' : ∀ n ∈ N, r • n = 0
n : M
hn : n ∈ N
⊢ -(r - 1) • n = n

Next tactic:
simpa [sub_smul] using hr' n hn
theorem fg_bot : (⊥ : Submodule R M).FG :=
  ⟨∅, by 

Proof state:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ span R ↑∅ = ⊥

Next tactic:
rw [Finset.coe_empty, span_empty]
theorem _root_.Subalgebra.fg_bot_toSubmodule {R A : Type*} [CommSemiring R] [Semiring A]
    [Algebra R A] : (⊥ : Subalgebra R A).toSubmodule.FG :=
  ⟨{1}, by 

Proof state:
R✝ : Type u_1
M : Type u_2
inst✝⁵ : Semiring R✝
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R✝ M
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
⊢ span R ↑{1} = Subalgebra.toSubmodule ⊥

Next tactic:
simp [Algebra.toSubmodule_bot, one_eq_span]