l3lab/ntp-mathlib huggingface dataset:
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        

Proof state:
case h.hf
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : Π₀ (i : ι), ↥(⨆ (_ : i ∈ s), p i)
hμ : (sum μ fun i xi => ↑xi) = a
x : ι
a✝ : x ∈ s
hx : x ∉ support μ
⊢ (fun i xi => ↑xi) x (μ x) = 0

Next tactic:
rw [mem_support_iff, not_ne_iff] at hx
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        

Proof state:
case h.hf
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : Π₀ (i : ι), ↥(⨆ (_ : i ∈ s), p i)
hμ : (sum μ fun i xi => ↑xi) = a
x : ι
a✝ : x ∈ s
hx : μ x = 0
⊢ (fun i xi => ↑xi) x (μ x) = 0

Next tactic:
rw [hx]
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        

Proof state:
case h.hf
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : Π₀ (i : ι), ↥(⨆ (_ : i ∈ s), p i)
hμ : (sum μ fun i xi => ↑xi) = a
x : ι
a✝ : x ∈ s
hx : μ x = 0
⊢ (fun i xi => ↑xi) x 0 = 0

Next tactic:
rfl
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        rfl
    · 

Proof state:
case mpr.intro
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
⊢ ∃ f, (sum f fun i xi => ↑xi) = a

Next tactic:
refine' ⟨DFinsupp.mk s _, _⟩
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        rfl
    · refine' ⟨DFinsupp.mk s _, _⟩
      · 

Proof state:
case mpr.intro.refine'_1
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
⊢ (i : ↑↑s) → ↥(⨆ (_ : ↑i ∈ s), p ↑i)

Next tactic:
rintro ⟨i, hi⟩
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        rfl
    · refine' ⟨DFinsupp.mk s _, _⟩
      · rintro ⟨i, hi⟩
        

Proof state:
case mpr.intro.refine'_1.mk
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
i : ι
hi : i ∈ ↑s
⊢ ↥(⨆ (_ : ↑{ val := i, property := hi } ∈ s), p ↑{ val := i, property := hi })

Next tactic:
refine' ⟨μ i, _⟩
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        rfl
    · refine' ⟨DFinsupp.mk s _, _⟩
      · rintro ⟨i, hi⟩
        refine' ⟨μ i, _⟩
        

Proof state:
case mpr.intro.refine'_1.mk
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
i : ι
hi : i ∈ ↑s
⊢ ↑(μ i) ∈ ⨆ (_ : ↑{ val := i, property := hi } ∈ s), p ↑{ val := i, property := hi }

Next tactic:
rw [iSup_pos]
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        rfl
    · refine' ⟨DFinsupp.mk s _, _⟩
      · rintro ⟨i, hi⟩
        refine' ⟨μ i, _⟩
        rw [iSup_pos]
        · 

Proof state:
case mpr.intro.refine'_1.mk
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
i : ι
hi : i ∈ ↑s
⊢ ↑(μ i) ∈ p ↑{ val := i, property := hi }

Next tactic:
exact coe_mem _
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        rfl
    · refine' ⟨DFinsupp.mk s _, _⟩
      · rintro ⟨i, hi⟩
        refine' ⟨μ i, _⟩
        rw [iSup_pos]
        · exact coe_mem _
        · 

Proof state:
case mpr.intro.refine'_1.mk
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
i : ι
hi : i ∈ ↑s
⊢ ↑{ val := i, property := hi } ∈ s

Next tactic:
exact hi
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        rfl
    · refine' ⟨DFinsupp.mk s _, _⟩
      · rintro ⟨i, hi⟩
        refine' ⟨μ i, _⟩
        rw [iSup_pos]
        · exact coe_mem _
        · exact hi
      

Proof state:
case mpr.intro.refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
⊢ (sum
      (DFinsupp.mk s fun i =>
        Subtype.casesOn i fun i hi =>
          { val := ↑(μ i),
            property := (_ : ↑(μ i) ∈ ⨆ (_ : ↑{ val := i, property := hi } ∈ s), p ↑{ val := i, property := hi }) })
      fun i xi => ↑xi) =
    a

Next tactic:
simp only [DFinsupp.sum]
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        rfl
    · refine' ⟨DFinsupp.mk s _, _⟩
      · rintro ⟨i, hi⟩
        refine' ⟨μ i, _⟩
        rw [iSup_pos]
        · exact coe_mem _
        · exact hi
      simp only [DFinsupp.sum]
      

Proof state:
case mpr.intro.refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
⊢ ∑ x in
      support
        (DFinsupp.mk s fun i =>
          { val := ↑(μ ↑i),
            property :=
              (_ :
                ↑(μ ↑i) ∈
                  ⨆ (_ : ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) } ∈ s),
                    p ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) }) }),
      ↑((DFinsupp.mk s fun i =>
            { val := ↑(μ ↑i),
              property :=
                (_ :
                  ↑(μ ↑i) ∈
                    ⨆ (_ : ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) } ∈ s),
                      p ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) }) })
          x) =
    a

Next tactic:
rw [Finset.sum_subset support_mk_subset, ← hμ]
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        rfl
    · refine' ⟨DFinsupp.mk s _, _⟩
      · rintro ⟨i, hi⟩
        refine' ⟨μ i, _⟩
        rw [iSup_pos]
        · exact coe_mem _
        · exact hi
      simp only [DFinsupp.sum]
      rw [Finset.sum_subset support_mk_subset, ← hμ]
      

Proof state:
case mpr.intro.refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
⊢ ∑ x in s,
      ↑((DFinsupp.mk s fun i =>
            { val := ↑(μ ↑i),
              property :=
                (_ :
                  ↑(μ ↑i) ∈
                    ⨆ (_ : ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) } ∈ s),
                      p ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) }) })
          x) =
    ∑ i in s, ↑(μ i)
case mpr.intro.refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
⊢ ∀ x ∈ s,
    x ∉
        support
          (DFinsupp.mk s fun i =>
            { val := ↑(μ ↑i),
              property :=
                (_ :
                  ↑(μ ↑i) ∈
                    ⨆ (_ : ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) } ∈ s),
                      p ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) }) }) →
      ↑((DFinsupp.mk s fun i =>
              { val := ↑(μ ↑i),
                property :=
                  (_ :
                    ↑(μ ↑i) ∈
                      ⨆ (_ : ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) } ∈ s),
                        p ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) }) })
            x) =
        0

Next tactic:
exact Finset.sum_congr rfl fun x hx => congr_arg Subtype.val <| mk_of_mem hx
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        rfl
    · refine' ⟨DFinsupp.mk s _, _⟩
      · rintro ⟨i, hi⟩
        refine' ⟨μ i, _⟩
        rw [iSup_pos]
        · exact coe_mem _
        · exact hi
      simp only [DFinsupp.sum]
      rw [Finset.sum_subset support_mk_subset, ← hμ]
      exact Finset.sum_congr rfl fun x hx => congr_arg Subtype.val <| mk_of_mem hx
      · 

Proof state:
case mpr.intro.refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
⊢ ∀ x ∈ s,
    x ∉
        support
          (DFinsupp.mk s fun i =>
            { val := ↑(μ ↑i),
              property :=
                (_ :
                  ↑(μ ↑i) ∈
                    ⨆ (_ : ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) } ∈ s),
                      p ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) }) }) →
      ↑((DFinsupp.mk s fun i =>
              { val := ↑(μ ↑i),
                property :=
                  (_ :
                    ↑(μ ↑i) ∈
                      ⨆ (_ : ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) } ∈ s),
                        p ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) }) })
            x) =
        0

Next tactic:
intro x _ hx
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        rfl
    · refine' ⟨DFinsupp.mk s _, _⟩
      · rintro ⟨i, hi⟩
        refine' ⟨μ i, _⟩
        rw [iSup_pos]
        · exact coe_mem _
        · exact hi
      simp only [DFinsupp.sum]
      rw [Finset.sum_subset support_mk_subset, ← hμ]
      exact Finset.sum_congr rfl fun x hx => congr_arg Subtype.val <| mk_of_mem hx
      · intro x _ hx
        

Proof state:
case mpr.intro.refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
x : ι
a✝ : x ∈ s
hx :
  x ∉
    support
      (DFinsupp.mk s fun i =>
        { val := ↑(μ ↑i),
          property :=
            (_ :
              ↑(μ ↑i) ∈
                ⨆ (_ : ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) } ∈ s), p ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) }) })
⊢ ↑((DFinsupp.mk s fun i =>
          { val := ↑(μ ↑i),
            property :=
              (_ :
                ↑(μ ↑i) ∈
                  ⨆ (_ : ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) } ∈ s), p ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) }) })
        x) =
    0

Next tactic:
rw [mem_support_iff, not_ne_iff] at hx
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        rfl
    · refine' ⟨DFinsupp.mk s _, _⟩
      · rintro ⟨i, hi⟩
        refine' ⟨μ i, _⟩
        rw [iSup_pos]
        · exact coe_mem _
        · exact hi
      simp only [DFinsupp.sum]
      rw [Finset.sum_subset support_mk_subset, ← hμ]
      exact Finset.sum_congr rfl fun x hx => congr_arg Subtype.val <| mk_of_mem hx
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        

Proof state:
case mpr.intro.refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
x : ι
a✝ : x ∈ s
hx :
  (DFinsupp.mk s fun i =>
        { val := ↑(μ ↑i),
          property :=
            (_ :
              ↑(μ ↑i) ∈
                ⨆ (_ : ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) } ∈ s), p ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) }) })
      x =
    0
⊢ ↑((DFinsupp.mk s fun i =>
          { val := ↑(μ ↑i),
            property :=
              (_ :
                ↑(μ ↑i) ∈
                  ⨆ (_ : ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) } ∈ s), p ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) }) })
        x) =
    0

Next tactic:
rw [hx]
theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
    (a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i in s, (μ i : N)) = a := by
  classical
    rw [Submodule.mem_iSup_iff_exists_dfinsupp']
    constructor <;> rintro ⟨μ, hμ⟩
    · use fun i => ⟨μ i, (iSup_const_le : _ ≤ p i) (coe_mem <| μ i)⟩
      rw [← hμ]
      symm
      apply Finset.sum_subset
      · intro x
        contrapose
        intro hx
        rw [mem_support_iff, not_ne_iff]
        ext
        rw [coe_zero, ← mem_bot R]
        suffices : ⊥ = ⨆ (_ : x ∈ s), p x
        · exact this.symm ▸ coe_mem (μ x)
        exact (iSup_neg hx).symm
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        rfl
    · refine' ⟨DFinsupp.mk s _, _⟩
      · rintro ⟨i, hi⟩
        refine' ⟨μ i, _⟩
        rw [iSup_pos]
        · exact coe_mem _
        · exact hi
      simp only [DFinsupp.sum]
      rw [Finset.sum_subset support_mk_subset, ← hμ]
      exact Finset.sum_congr rfl fun x hx => congr_arg Subtype.val <| mk_of_mem hx
      · intro x _ hx
        rw [mem_support_iff, not_ne_iff] at hx
        rw [hx]
        

Proof state:
case mpr.intro.refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
M : ι → Type u_4
N : Type u_5
dec_ι : DecidableEq ι
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i in s, ↑(μ i) = a
x : ι
a✝ : x ∈ s
hx :
  (DFinsupp.mk s fun i =>
        { val := ↑(μ ↑i),
          property :=
            (_ :
              ↑(μ ↑i) ∈
                ⨆ (_ : ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) } ∈ s), p ↑{ val := ↑i, property := (_ : ↑i ∈ ↑s) }) })
      x =
    0
⊢ ↑0 = 0

Next tactic:
rfl