-- Module: Mathlib.LinearAlgebra.LinearIndependent.Lemmas
Goal:
ι : Type u'
R : Type u_2
M : Type u_4
inst : Ring R
inst_1 : AddCommGroup M
inst_2 : Module R M
v : ι → M
rcases :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (hImaximal : ∀ (t : Set ι), I ⊆ t → LinearIndepOn R v t → I = t),
    ∃ (s : Set ι),
      LinearIndepOn R v s ∧ ∀ i ∉ s, ∃ (a : R), a ≠ 0 ∧ Membership.mem (γ := Submodule R M) (span R (v '' s)) (a • v i)
right right₁ :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (hImaximal : ∀ (t : Set ι), I ⊆ t → LinearIndepOn R v t → I = t)
    (i : ι) (hi : i ∉ I), ∃ (a : R), a ≠ 0 ∧ Membership.mem (γ := Submodule R M) (span R (v '' I)) (a • v i)
right₂ :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (i : ι) (hi : i ∉ I)
    (hImaximal : LinearIndepOn R v (I ∪ {i}) → I = I ∪ {i}),
    ∃ (a : R), a ≠ 0 ∧ Membership.mem (γ := Submodule R M) (span R (v '' I)) (a • v i)
right₃ :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (i : ι) (hi : i ∉ I),
    let J := I ∪ {i};
    ∀ (hImaximal : LinearIndepOn R v J → I = J) (hJ : J = I ∪ {i}),
      ∃ (a : R), a ≠ 0 ∧ Membership.mem (γ := Submodule R M) (span R (v '' I)) (a • v i)
right₄ :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (i : ι) (hi : i ∉ I),
    let J := I ∪ {i};
    ∀ (hImaximal : LinearIndepOn R v J → I = J) (hJ : J = I ∪ {i}) (memJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I),
      ∃ (a : R), a ≠ 0 ∧ Membership.mem (γ := Submodule R M) (span R (v '' I)) (a • v i)
right₅ :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (i : ι) (hi : i ∉ I),
    let J := I ∪ {i};
    ∀ (hImaximal : LinearIndepOn R v J → I = J) (hJ : J = I ∪ {i}) (memJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I)
      (hiJ : i ∈ J), ∃ (a : R), a ≠ 0 ∧ Membership.mem (γ := Submodule R M) (span R (v '' I)) (a • v i)
right₆ :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (i : ι) (hi : i ∉ I),
    let J := I ∪ {i};
    ∀ (hImaximal : LinearIndepOn R v J → I = J) (hJ : J = I ∪ {i}) (memJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I)
      (hiJ : i ∈ J) (h : ¬LinearIndepOn R v J),
      ∃ (a : R), a ≠ 0 ∧ Membership.mem (γ := Submodule R M) (span R (v '' I)) (a • v i)
right₇ :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (i : ι) (hi : i ∉ I),
    let J := I ∪ {i};
    ∀ (hImaximal : LinearIndepOn R v J → I = J) (hJ : J = I ∪ {i}) (memJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I)
      (hiJ : i ∈ J) (h : ¬LinearIndepOn R v J) (f : ι →₀ R)
      (supp_f : Membership.mem (γ := Submodule R (ι →₀ R)) (Finsupp.supported R R J) f)
      (sum_f :
        Eq.{u_4 + 1} (α := M) (∑ i ∈ f.support, HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) i : R) (v i)) 0)
      (f_ne : f ≠ 0), ∃ (a : R), a ≠ 0 ∧ Membership.mem (γ := Submodule R M) (span R (v '' I)) (a • v i)
right₈ :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (i : ι) (hi : i ∉ I),
    let J := I ∪ {i};
    ∀ (hImaximal : LinearIndepOn R v J → I = J) (hJ : J = I ∪ {i}) (memJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I)
      (hiJ : i ∈ J) (h : ¬LinearIndepOn R v J) (f : ι →₀ R)
      (supp_f : Membership.mem (γ := Submodule R (ι →₀ R)) (Finsupp.supported R R J) f)
      (sum_f :
        Eq.{u_4 + 1} (α := M) (∑ i ∈ f.support, HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) i : R) (v i)) 0)
      (f_ne : f ≠ 0) (hfi : Ne.{u_2 + 1} (α := R) ((f : (a : ι) → R) i : R) (0 : R)),
      ∃ (a : R), a ≠ 0 ∧ Membership.mem (γ := Submodule R M) (span R (v '' I)) (a • v i)
right₉ :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (i : ι) (hi : i ∉ I),
    let J := I ∪ {i};
    ∀ (hImaximal : LinearIndepOn R v J → I = J) (hJ : J = I ∪ {i}) (memJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I)
      (hiJ : i ∈ J) (h : ¬LinearIndepOn R v J) (f : ι →₀ R)
      (supp_f : Membership.mem (γ := Submodule R (ι →₀ R)) (Finsupp.supported R R J) f)
      (sum_f :
        Eq.{u_4 + 1} (α := M) (∑ i ∈ f.support, HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) i : R) (v i)) 0)
      (f_ne : f ≠ 0) (hfi : Ne.{u_2 + 1} (α := R) ((f : (a : ι) → R) i : R) (0 : R)),
      Membership.mem (γ := Submodule R M) (span R (v '' I))
        (HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) i : R) (v i))
right₁₀ :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (i : ι) (hi : i ∉ I),
    let J := I ∪ {i};
    ∀ (hImaximal : LinearIndepOn R v J → I = J) (hJ : J = I ∪ {i}) (memJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I)
      (hiJ : i ∈ J) (h : ¬LinearIndepOn R v J) (f : ι →₀ R)
      (supp_f : Membership.mem (γ := Submodule R (ι →₀ R)) (Finsupp.supported R R J) f)
      (sum_f :
        Eq.{u_4 + 1} (α := M) (∑ i ∈ f.support, HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) i : R) (v i)) 0)
      (f_ne : f ≠ 0) (hfi : Ne.{u_2 + 1} (α := R) ((f : (a : ι) → R) i : R) (0 : R)) (hfi' : i ∈ f.support),
      Membership.mem (γ := Submodule R M) (span R (v '' I))
        (HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) i : R) (v i))
right₁₁ :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (i : ι) (hi : i ∉ I),
    let J := I ∪ {i};
    ∀ (hImaximal : LinearIndepOn R v J → I = J) (hJ : J = I ∪ {i}) (memJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I)
      (hiJ : i ∈ J) (h : ¬LinearIndepOn R v J) (f : ι →₀ R)
      (supp_f : Membership.mem (γ := Submodule R (ι →₀ R)) (Finsupp.supported R R J) f)
      (sum_f :
        HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) i : R) (v i) =
          -∑ x ∈ f.support.erase i, HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) x : R) (v x))
      (f_ne : f ≠ 0) (hfi : Ne.{u_2 + 1} (α := R) ((f : (a : ι) → R) i : R) (0 : R)) (hfi' : i ∈ f.support),
      Membership.mem (γ := Submodule R M) (span R (v '' I))
        (HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) i : R) (v i))
right₁₂ :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (i : ι) (hi : i ∉ I),
    let J := I ∪ {i};
    ∀ (hImaximal : LinearIndepOn R v J → I = J) (hJ : J = I ∪ {i}) (memJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I)
      (hiJ : i ∈ J) (h : ¬LinearIndepOn R v J) (f : ι →₀ R)
      (supp_f : Membership.mem (γ := Submodule R (ι →₀ R)) (Finsupp.supported R R J) f)
      (sum_f :
        HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) i : R) (v i) =
          -∑ x ∈ f.support.erase i, HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) x : R) (v x))
      (f_ne : f ≠ 0) (hfi : Ne.{u_2 + 1} (α := R) ((f : (a : ι) → R) i : R) (0 : R)) (hfi' : i ∈ f.support),
      Membership.mem (γ := Submodule R M) (span R (v '' I))
        (-∑ x ∈ f.support.erase i, HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) x : R) (v x))
right₁₃ :
  ∀ (I : Set ι) (hIlinind : LinearIndepOn R v I) (i : ι) (hi : i ∉ I),
    let J := I ∪ {i};
    ∀ (hImaximal : LinearIndepOn R v J → I = J) (hJ : J = I ∪ {i}) (memJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I)
      (hiJ : i ∈ J) (h : ¬LinearIndepOn R v J) (f : ι →₀ R)
      (supp_f : Membership.mem (γ := Submodule R (ι →₀ R)) (Finsupp.supported R R J) f)
      (sum_f :
        HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) i : R) (v i) =
          -∑ x ∈ f.support.erase i, HSMul.hSMul.{u_2, u_4, u_4} (α := R) ((f : (a : ι) → R) x : R) (v x))
      (f_ne : f ≠ 0) (hfi : Ne.{u_2 + 1} (α := R) ((f : (a : ι) → R) i : R) (0 : R)) (hfi' : i ∈ f.support) (c : ι)
      (hc : c ∈ f.support.erase i), c ∈ I
⊢ ∃ (s : Set ι),
    LinearIndepOn R v s ∧ ∀ i ∉ s, ∃ (a : R), a ≠ 0 ∧ Membership.mem (γ := Submodule R M) (span R (v '' s)) (a • v i)

Tactic:
classical
    rcases exists_maximal_linearIndepOn' R v with ⟨I, hIlinind, hImaximal⟩
    use I, hIlinind
    intro i hi
    specialize hImaximal (I ∪ {i}) (by simp)
    set J := I ∪ {i} with hJ
    have memJ : ∀ {x}, x ∈ J ↔ x = i ∨ x ∈ I := by simp [hJ]
    have hiJ : i ∈ J := by simp [J]
    have h := by
      refine mt hImaximal ?_
      · intro h2
        rw [h2] at hi
        exact absurd hiJ hi
    obtain ⟨f, supp_f, sum_f, f_ne⟩ := linearDepOn_iff.mp h
    have hfi : f i ≠ 0 := by
      contrapose hIlinind
      refine linearDepOn_iff.mpr ⟨f, ?_, sum_f, f_ne⟩
      simp only [Finsupp.mem_supported, hJ] at supp_f ⊢
      rintro x hx
      refine (memJ.mp (supp_f hx)).resolve_left ?_
      rintro rfl
      exact (Finsupp.mem_support_iff.mp hx) hIlinind
    use f i, hfi
    have hfi' : i ∈ f.support := Finsupp.mem_support_iff.mpr hfi
    rw [← Finset.insert_erase hfi', Finset.sum_insert (Finset.notMem_erase _ _),
      add_eq_zero_iff_eq_neg] at sum_f
    rw [sum_f]
    refine neg_mem (sum_mem fun c hc => smul_mem _ _ (subset_span ⟨c, ?_, rfl⟩))
    exact (memJ.mp (supp_f (Finset.erase_subset _ _ hc))).resolve_left (Finset.ne_of_mem_erase hc)

Premises used: add_eq_zero_iff_eq_neg, exists_maximal_linearIndepOn', Finset.notMem_erase, Finset.sum_insert