-- Mathlib/LinearAlgebra/Basis/Cardinality.lean
theorem union_support_maximal_linearIndependent_eq_range_basis {ι : Type w} (b : Basis ι R M)
    {κ : Type w'} (v : κ → M) (ind : LinearIndependent R v) (m : ind.Maximal) :
    ⋃ k, ((b.repr (v k)).support : Set ι) = Set.univ

Proof state:
case intro
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
repr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearCombination R v) l)) i = 0
⊢ False

Tactic:
let v' (o : Option κ) : M := o.elim (b i) v

Result:
case intro
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
repr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearCombination R v) l)) i = 0
v' : Option κ → M := fun o => o.elim (b i) v
⊢ False

Proof state:
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
repr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearCombination R v) l)) i = 0
v' : Option κ → M := fun o => o.elim (b i) v
r : range v ⊆ range v'
r' : b i ∉ range v
r'' : range v ≠ range v'
l l' : Option κ →₀ R
z :
  l none • none.elim (b i) v + (linearCombination R v) l.some =
    l' none • none.elim (b i) v + (linearCombination R v) l'.some
a : κ
j : ι
⊢ (b.repr ((linearCombination R v) l.some)) j = (b.repr ((linearCombination R v) l'.some)) j

Tactic:
obtain rfl | ne := eq_or_ne i j

Result:
case inl
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
repr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearCombination R v) l)) i = 0
v' : Option κ → M := fun o => o.elim (b i) v
r : range v ⊆ range v'
r' : b i ∉ range v
r'' : range v ≠ range v'
l l' : Option κ →₀ R
z :
  l none • none.elim (b i) v + (linearCombination R v) l.some =
    l' none • none.elim (b i) v + (linearCombination R v) l'.some
a : κ
⊢ (b.repr ((linearCombination R v) l.some)) i = (b.repr ((linearCombination R v) l'.some)) i

case inr
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
repr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearCombination R v) l)) i = 0
v' : Option κ → M := fun o => o.elim (b i) v
r : range v ⊆ range v'
r' : b i ∉ range v
r'' : range v ≠ range v'
l l' : Option κ →₀ R
z :
  l none • none.elim (b i) v + (linearCombination R v) l.some =
    l' none • none.elim (b i) v + (linearCombination R v) l'.some
a : κ
j : ι
ne : i ≠ j
⊢ (b.repr ((linearCombination R v) l.some)) j = (b.repr ((linearCombination R v) l'.some)) j

Proof state:
case i.h.some
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
repr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearCombination R v) l)) i = 0
v' : Option κ → M := fun o => o.elim (b i) v
r : range v ⊆ range v'
r' : b i ∉ range v
r'' : range v ≠ range v'
l l' : Option κ →₀ R
z :
  l none • none.elim (b i) v + (linearCombination R v) l.some =
    l' none • none.elim (b i) v + (linearCombination R v) l'.some
a : κ
l₁ : l.some = l'.some
⊢ l (Option.some a) = l' (Option.some a)

Tactic:
exact DFunLike.congr_fun l₁ a

Result:
no goals

Proof state:
case intro
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
⊢ False

Tactic:
have repr_eq_zero (l) : b.repr (linearCombination R v l) i = 0 := by
  simp [linearCombination_apply, Finsupp.sum, w]

Result:
case intro
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
repr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearCombination R v) l)) i = 0
⊢ False

Proof state:
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
h : ∃ a, ∀ (x : κ), (b.repr (v x)) a = 0
⊢ False

Tactic:
obtain ⟨i, w⟩ := h

Result:
case intro
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
⊢ False

Proof state:
case intro
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
repr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearCombination R v) l)) i = 0
v' : Option κ → M := fun o => o.elim (b i) v
k : κ
⊢ v k ∈ range v'

Tactic:
exact ⟨some k, rfl⟩

Result:
no goals

Proof state:
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
h : ¬⋃ k, ↑(b.repr (v k)).support = Set.univ
⊢ False

Tactic:
simp only [← Ne.eq_def, ne_univ_iff_exists_not_mem, mem_iUnion, not_exists_not,
  Finsupp.mem_support_iff, Finset.mem_coe] at h

Result:
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
h : ∃ a, ∀ (x : κ), (b.repr (v x)) a = 0
⊢ False

Proof state:
case intro
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
repr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearCombination R v) l)) i = 0
v' : Option κ → M := fun o => o.elim (b i) v
⊢ False

Tactic:
have r : range v ⊆ range v' := by rintro - ⟨k, rfl⟩; exact ⟨some k, rfl⟩

Result:
case intro
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Nontrivial R
inst✝ : Module R M
ι : Type w
b : Basis ι R M
κ : Type w'
v : κ → M
ind : LinearIndependent R v
m : ind.Maximal
i : ι
w : ∀ (x : κ), (b.repr (v x)) i = 0
repr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearCombination R v) l)) i = 0
v' : Option κ → M := fun o => o.elim (b i) v
r : range v ⊆ range v'
⊢ False