l3lab/ntp-mathlib huggingface dataset:
/-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
    (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by
  obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs
  replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩
  

Proof state:
case intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
H : ∀ (r : { x // x ∈ s' }), ∃ n, ↑r ^ n • x ∈ M'
⊢ x ∈ M'

Next tactic:
choose n₁ n₂ using H
/-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
    (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by
  obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs
  replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩
  choose n₁ n₂ using H
  

Proof state:
case intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : { x // x ∈ s' } → ℕ
n₂ : ∀ (r : { x // x ∈ s' }), ↑r ^ n₁ r • x ∈ M'
⊢ x ∈ M'

Next tactic:
let N := s'.attach.sup n₁
/-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
    (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by
  obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs
  replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩
  choose n₁ n₂ using H
  let N := s'.attach.sup n₁
  

Proof state:
case intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : { x // x ∈ s' } → ℕ
n₂ : ∀ (r : { x // x ∈ s' }), ↑r ^ n₁ r • x ∈ M'
N : ℕ := Finset.sup (Finset.attach s') n₁
⊢ x ∈ M'

Next tactic:
have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N
/-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
    (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by
  obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs
  replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩
  choose n₁ n₂ using H
  let N := s'.attach.sup n₁
  have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N
  

Proof state:
case intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : { x // x ∈ s' } → ℕ
n₂ : ∀ (r : { x // x ∈ s' }), ↑r ^ n₁ r • x ∈ M'
N : ℕ := Finset.sup (Finset.attach s') n₁
hs' : Ideal.span ((fun x => x ^ N) '' ↑s') = ⊤
⊢ x ∈ M'

Next tactic:
apply M'.mem_of_span_top_of_smul_mem _ hs'
/-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
    (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by
  obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs
  replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩
  choose n₁ n₂ using H
  let N := s'.attach.sup n₁
  have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N
  apply M'.mem_of_span_top_of_smul_mem _ hs'
  

Proof state:
case intro.intro.H
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : { x // x ∈ s' } → ℕ
n₂ : ∀ (r : { x // x ∈ s' }), ↑r ^ n₁ r • x ∈ M'
N : ℕ := Finset.sup (Finset.attach s') n₁
hs' : Ideal.span ((fun x => x ^ N) '' ↑s') = ⊤
⊢ ∀ (r : ↑((fun x => x ^ N) '' ↑s')), ↑r • x ∈ M'

Next tactic:
rintro ⟨_, r, hr, rfl⟩
/-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
    (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by
  obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs
  replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩
  choose n₁ n₂ using H
  let N := s'.attach.sup n₁
  have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N
  apply M'.mem_of_span_top_of_smul_mem _ hs'
  rintro ⟨_, r, hr, rfl⟩
  

Proof state:
case intro.intro.H.mk.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : { x // x ∈ s' } → ℕ
n₂ : ∀ (r : { x // x ∈ s' }), ↑r ^ n₁ r • x ∈ M'
N : ℕ := Finset.sup (Finset.attach s') n₁
hs' : Ideal.span ((fun x => x ^ N) '' ↑s') = ⊤
r : R
hr : r ∈ ↑s'
⊢ ↑{ val := (fun x => x ^ N) r, property := (_ : ∃ a ∈ ↑s', (fun x => x ^ N) a = (fun x => x ^ N) r) } • x ∈ M'

Next tactic:
convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1
/-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
    (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by
  obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs
  replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩
  choose n₁ n₂ using H
  let N := s'.attach.sup n₁
  have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N
  apply M'.mem_of_span_top_of_smul_mem _ hs'
  rintro ⟨_, r, hr, rfl⟩
  convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1
  

Proof state:
case h.e'_4
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : { x // x ∈ s' } → ℕ
n₂ : ∀ (r : { x // x ∈ s' }), ↑r ^ n₁ r • x ∈ M'
N : ℕ := Finset.sup (Finset.attach s') n₁
hs' : Ideal.span ((fun x => x ^ N) '' ↑s') = ⊤
r : R
hr : r ∈ ↑s'
⊢ ↑{ val := (fun x => x ^ N) r, property := (_ : ∃ a ∈ ↑s', (fun x => x ^ N) a = (fun x => x ^ N) r) } • x =
    r ^ (N - n₁ { val := r, property := hr }) • ↑{ val := r, property := hr } ^ n₁ { val := r, property := hr } • x

Next tactic:
simp only [Subtype.coe_mk, smul_smul, ← pow_add]
/-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/
theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
    (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by
  obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs
  replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩
  choose n₁ n₂ using H
  let N := s'.attach.sup n₁
  have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N
  apply M'.mem_of_span_top_of_smul_mem _ hs'
  rintro ⟨_, r, hr, rfl⟩
  convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1
  simp only [Subtype.coe_mk, smul_smul, ← pow_add]
  

Proof state:
case h.e'_4
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I J : Ideal R
N✝ P : Submodule R M
S : Set R
T : Set M
M' : Submodule R M
s : Set R
hs : Ideal.span s = ⊤
x : M
s' : Finset R
hs₁ : ↑s' ⊆ s
hs₂ : Ideal.span ↑s' = ⊤
n₁ : { x // x ∈ s' } → ℕ
n₂ : ∀ (r : { x // x ∈ s' }), ↑r ^ n₁ r • x ∈ M'
N : ℕ := Finset.sup (Finset.attach s') n₁
hs' : Ideal.span ((fun x => x ^ N) '' ↑s') = ⊤
r : R
hr : r ∈ ↑s'
⊢ r ^ Finset.sup (Finset.attach s') n₁ • x =
    r ^ (Finset.sup (Finset.attach s') n₁ - n₁ { val := r, property := hr } + n₁ { val := r, property := hr }) • x

Next tactic:
rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)]
theorem mem_smul_span {s : Set M} {x : M} :
    x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by
  

Proof state:
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
s : Set M
x : M
⊢ x ∈ I • span R s ↔ x ∈ span R (⋃ a ∈ I, ⋃ b ∈ s, {a • b})

Next tactic:
rw [← I.span_eq, Submodule.span_smul_span, I.span_eq]
theorem mem_smul_span {s : Set M} {x : M} :
    x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by
  rw [← I.span_eq, Submodule.span_smul_span, I.span_eq]
  

Proof state:
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
s : Set M
x : M
⊢ x ∈ span R (⋃ s_1 ∈ ↑I, ⋃ t ∈ s, {s_1 • t}) ↔ x ∈ span R (⋃ a ∈ I, ⋃ b ∈ s, {a • b})

Next tactic:
rfl
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  

Proof state:
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
⊢ x ∈ I • span R (Set.range f) ↔ ∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • f i) = x

Next tactic:
constructor
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; 

Proof state:
case mp
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
⊢ x ∈ I • span R (Set.range f) → ∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • f i) = x
case mpr
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
⊢ (∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • f i) = x) → x ∈ I • span R (Set.range f)

Next tactic:
swap
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · 

Proof state:
case mpr
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
⊢ (∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • f i) = x) → x ∈ I • span R (Set.range f)

Next tactic:
rintro ⟨a, ha, rfl⟩
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    

Proof state:
case mpr.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ (Finsupp.sum a fun i c => c • f i) ∈ I • span R (Set.range f)

Next tactic:
exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  

Proof state:
case mp
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
⊢ x ∈ I • span R (Set.range f) → ∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • f i) = x

Next tactic:
refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · 

Proof state:
case mp.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
⊢ ∀ x ∈ ⋃ a ∈ I, ⋃ b ∈ Set.range f, {a • b}, ∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • f i) = x

Next tactic:
simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    

Proof state:
case mp.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
⊢ ∀ (x : M),
    (∃ i, ∃ (_ : i ∈ I), ∃ i_1, ∃ (_ : ∃ y, f y = i_1), x = i • i_1) →
      ∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • f i) = x

Next tactic:
rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    

Proof state:
case mp.refine'_1.intro.intro.intro.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
y : R
hy : y ∈ I
i : ι
⊢ ∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • f i) = y • f i

Next tactic:
refine' ⟨Finsupp.single i y, fun j => _, _⟩
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · 

Proof state:
case mp.refine'_1.intro.intro.intro.intro.intro.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
y : R
hy : y ∈ I
i j : ι
⊢ (fun₀ | i => y) j ∈ I

Next tactic:
letI := Classical.decEq ι
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      

Proof state:
case mp.refine'_1.intro.intro.intro.intro.intro.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
y : R
hy : y ∈ I
i j : ι
this : DecidableEq ι := Classical.decEq ι
⊢ (fun₀ | i => y) j ∈ I

Next tactic:
rw [Finsupp.single_apply]
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      

Proof state:
case mp.refine'_1.intro.intro.intro.intro.intro.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
y : R
hy : y ∈ I
i j : ι
this : DecidableEq ι := Classical.decEq ι
⊢ (if i = j then y else 0) ∈ I

Next tactic:
split_ifs
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · 

Proof state:
case pos
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
y : R
hy : y ∈ I
i j : ι
this : DecidableEq ι := Classical.decEq ι
h✝ : i = j
⊢ y ∈ I

Next tactic:
assumption