-- Mathlib/FieldTheory/PurelyInseparable/PerfectClosure.lean
theorem LinearIndependent.map_pow_expChar_pow_of_fd_isSeparable
    [FiniteDimensional F E] [Algebra.IsSeparable F E]
    (h : LinearIndependent F v) : LinearIndependent F (v · ^ q ^ n)

Proof state:
case h.e'_4.h
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
b : Basis (↑ι') F E := Basis.extend h'
this : Fintype ↑ι' := FiniteDimensional.fintypeBasisIndex b
H : LinearIndependent F fun (x : ↑ι') => b x ^ q ^ n
f : ι → ↑ι' := fun i => ⟨v i, ⋯⟩
x✝ : ι
⊢ v x✝ ^ q ^ n = (Basis.extend h') (f x✝) ^ q ^ n

Tactic:
rw [Basis.extend_apply_self]

Result:
no goals

Proof state:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
b : Basis (↑ι') F E := Basis.extend h'
this : Fintype ↑ι' := FiniteDimensional.fintypeBasisIndex b
H : LinearIndependent F fun (x : ↑ι') => b x ^ q ^ n
f : ι → ↑ι' := fun i => ⟨v i, ⋯⟩
⊢ LinearIndependent F fun x => v x ^ q ^ n

Tactic:
convert H.comp f fun _ _ heq ↦ h.injective (by simpa only [f, Subtype.mk.injEq] using heq)

Result:
case h.e'_4.h
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
b : Basis (↑ι') F E := Basis.extend h'
this : Fintype ↑ι' := FiniteDimensional.fintypeBasisIndex b
H : LinearIndependent F fun (x : ↑ι') => b x ^ q ^ n
f : ι → ↑ι' := fun i => ⟨v i, ⋯⟩
x✝ : ι
⊢ v x✝ ^ q ^ n = ((fun x => b x ^ q ^ n) ∘ f) x✝

Proof state:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
⊢ LinearIndependent F fun x => v x ^ q ^ n

Tactic:
let ι' := h'.extend (Set.range v).subset_univ

Result:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
⊢ LinearIndependent F fun x => v x ^ q ^ n

Proof state:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
b : Basis (↑ι') F E := Basis.extend h'
⊢ LinearIndependent F fun x => v x ^ q ^ n

Tactic:
letI : Fintype ι' := FiniteDimensional.fintypeBasisIndex b

Result:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
b : Basis (↑ι') F E := Basis.extend h'
this : Fintype ↑ι' := FiniteDimensional.fintypeBasisIndex b
⊢ LinearIndependent F fun x => v x ^ q ^ n

Proof state:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
⊢ LinearIndependent F fun x => v x ^ q ^ n

Tactic:
let b : Basis ι' F E := Basis.extend h'

Result:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
b : Basis (↑ι') F E := Basis.extend h'
⊢ LinearIndependent F fun x => v x ^ q ^ n

Proof state:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
b : Basis (↑ι') F E := Basis.extend h'
this : Fintype ↑ι' := FiniteDimensional.fintypeBasisIndex b
H : LinearIndependent F fun (x : ↑ι') => b x ^ q ^ n
f : ι → ↑ι' := fun i => ⟨v i, ⋯⟩
x✝¹ x✝ : ι
heq : f x✝¹ = f x✝
⊢ v x✝¹ = v x✝

Tactic:
simpa only [f, Subtype.mk.injEq] using heq

Result:
no goals

Proof state:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
b : Basis (↑ι') F E := Basis.extend h'
this : Fintype ↑ι' := FiniteDimensional.fintypeBasisIndex b
H : LinearIndependent F fun (x : ↑ι') => b x ^ q ^ n
⊢ LinearIndependent F fun x => v x ^ q ^ n

Tactic:
let f (i : ι) : ι' := ⟨v i, h'.subset_extend _ ⟨i, rfl⟩⟩

Result:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
b : Basis (↑ι') F E := Basis.extend h'
this : Fintype ↑ι' := FiniteDimensional.fintypeBasisIndex b
H : LinearIndependent F fun (x : ↑ι') => b x ^ q ^ n
f : ι → ↑ι' := fun i => ⟨v i, ⋯⟩
⊢ LinearIndependent F fun x => v x ^ q ^ n

Proof state:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
b : Basis (↑ι') F E := Basis.extend h'
this : Fintype ↑ι' := FiniteDimensional.fintypeBasisIndex b
⊢ LinearIndependent F fun x => v x ^ q ^ n

Tactic:
have H := linearIndependent_of_top_le_span_of_card_eq_finrank
  (Field.span_map_pow_expChar_pow_eq_top_of_isSeparable q n b.span_eq).ge
  (Module.finrank_eq_card_basis b).symm

Result:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
b : Basis (↑ι') F E := Basis.extend h'
this : Fintype ↑ι' := FiniteDimensional.fintypeBasisIndex b
H : LinearIndependent F fun (x : ↑ι') => b x ^ q ^ n
⊢ LinearIndependent F fun x => v x ^ q ^ n

Proof state:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
⊢ LinearIndependent F fun x => v x ^ q ^ n

Tactic:
have h' := h.linearIndepOn_id

Result:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
⊢ LinearIndependent F fun x => v x ^ q ^ n