[GOAL]
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
⊢ Independent f ↔ LinearIndependent K (Projectivization.rep ∘ f)
[PROOFSTEP]
refine' ⟨_, fun h => _⟩
[GOAL]
case refine'_1
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
⊢ Independent f → LinearIndependent K (Projectivization.rep ∘ f)
[PROOFSTEP]
rintro ⟨ff, hff, hh⟩
[GOAL]
case refine'_1.mk
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
ff : ι → V
hff : ∀ (i : ι), ff i ≠ 0
hh : LinearIndependent K ff
⊢ LinearIndependent K (Projectivization.rep ∘ fun i => mk K (ff i) (_ : ff i ≠ 0))
[PROOFSTEP]
choose a ha using fun i : ι => exists_smul_eq_mk_rep K (ff i) (hff i)
[GOAL]
case refine'_1.mk
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
ff : ι → V
hff : ∀ (i : ι), ff i ≠ 0
hh : LinearIndependent K ff
a : ι → Kˣ
ha : ∀ (i : ι), a i • ff i = Projectivization.rep (mk K (ff i) (_ : ff i ≠ 0))
⊢ LinearIndependent K (Projectivization.rep ∘ fun i => mk K (ff i) (_ : ff i ≠ 0))
[PROOFSTEP]
convert hh.units_smul a
[GOAL]
case h.e'_4
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
ff : ι → V
hff : ∀ (i : ι), ff i ≠ 0
hh : LinearIndependent K ff
a : ι → Kˣ
ha : ∀ (i : ι), a i • ff i = Projectivization.rep (mk K (ff i) (_ : ff i ≠ 0))
⊢ (Projectivization.rep ∘ fun i => mk K (ff i) (_ : ff i ≠ 0)) = a • ff
[PROOFSTEP]
ext i
[GOAL]
case h.e'_4.h
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
ff : ι → V
hff : ∀ (i : ι), ff i ≠ 0
hh : LinearIndependent K ff
a : ι → Kˣ
ha : ∀ (i : ι), a i • ff i = Projectivization.rep (mk K (ff i) (_ : ff i ≠ 0))
i : ι
⊢ (Projectivization.rep ∘ fun i => mk K (ff i) (_ : ff i ≠ 0)) i = (a • ff) i
[PROOFSTEP]
exact (ha i).symm
[GOAL]
case refine'_2
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
h : LinearIndependent K (Projectivization.rep ∘ f)
⊢ Independent f
[PROOFSTEP]
convert Independent.mk _ _ h
[GOAL]
case h.e'_7.h
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
h : LinearIndependent K (Projectivization.rep ∘ f)
x✝ : ι
⊢ f x✝ = mk K ((Projectivization.rep ∘ f) x✝) (_ : ?m.12147 x✝ ≠ 0)
[PROOFSTEP]
simp only [mk_rep, Function.comp_apply]
[GOAL]
case refine'_2
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
h : LinearIndependent K (Projectivization.rep ∘ f)
⊢ ∀ (i : ι), (Projectivization.rep ∘ f) i ≠ 0
[PROOFSTEP]
intro i
[GOAL]
case refine'_2
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
h : LinearIndependent K (Projectivization.rep ∘ f)
i : ι
⊢ (Projectivization.rep ∘ f) i ≠ 0
[PROOFSTEP]
apply rep_nonzero
[GOAL]
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
⊢ Independent f ↔ CompleteLattice.Independent fun i => Projectivization.submodule (f i)
[PROOFSTEP]
refine' ⟨_, fun h => _⟩
[GOAL]
case refine'_1
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
⊢ Independent f → CompleteLattice.Independent fun i => Projectivization.submodule (f i)
[PROOFSTEP]
rintro ⟨f, hf, hi⟩
[GOAL]
case refine'_1.mk
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → V
hf : ∀ (i : ι), f i ≠ 0
hi : LinearIndependent K f
⊢ CompleteLattice.Independent fun i => Projectivization.submodule ((fun i => mk K (f i) (_ : f i ≠ 0)) i)
[PROOFSTEP]
simp only [submodule_mk]
[GOAL]
case refine'_1.mk
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → V
hf : ∀ (i : ι), f i ≠ 0
hi : LinearIndependent K f
⊢ CompleteLattice.Independent fun i => Submodule.span K {f i}
[PROOFSTEP]
exact (CompleteLattice.independent_iff_linearIndependent_of_ne_zero (R := K) hf).mpr hi
[GOAL]
case refine'_2
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
h : CompleteLattice.Independent fun i => Projectivization.submodule (f i)
⊢ Independent f
[PROOFSTEP]
rw [independent_iff]
[GOAL]
case refine'_2
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
h : CompleteLattice.Independent fun i => Projectivization.submodule (f i)
⊢ LinearIndependent K (Projectivization.rep ∘ f)
[PROOFSTEP]
refine' h.linearIndependent (Projectivization.submodule ∘ f) (fun i => _) fun i => _
[GOAL]
case refine'_2.refine'_1
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
h : CompleteLattice.Independent fun i => Projectivization.submodule (f i)
i : ι
⊢ (Projectivization.rep ∘ f) i ∈ (Projectivization.submodule ∘ f) i
[PROOFSTEP]
simpa only [Function.comp_apply, submodule_eq] using Submodule.mem_span_singleton_self _
[GOAL]
case refine'_2.refine'_2
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
h : CompleteLattice.Independent fun i => Projectivization.submodule (f i)
i : ι
⊢ (Projectivization.rep ∘ f) i ≠ 0
[PROOFSTEP]
exact rep_nonzero (f i)
[GOAL]
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
⊢ Dependent f ↔ ¬LinearIndependent K (Projectivization.rep ∘ f)
[PROOFSTEP]
refine' ⟨_, fun h => _⟩
[GOAL]
case refine'_1
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
⊢ Dependent f → ¬LinearIndependent K (Projectivization.rep ∘ f)
[PROOFSTEP]
rintro ⟨ff, hff, hh1⟩
[GOAL]
case refine'_1.mk
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
ff : ι → V
hff : ∀ (i : ι), ff i ≠ 0
hh1 : ¬LinearIndependent K ff
⊢ ¬LinearIndependent K (Projectivization.rep ∘ fun i => mk K (ff i) (_ : ff i ≠ 0))
[PROOFSTEP]
contrapose! hh1
[GOAL]
case refine'_1.mk
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
ff : ι → V
hff : ∀ (i : ι), ff i ≠ 0
hh1 : LinearIndependent K (Projectivization.rep ∘ fun i => mk K (ff i) (_ : ff i ≠ 0))
⊢ LinearIndependent K ff
[PROOFSTEP]
choose a ha using fun i : ι => exists_smul_eq_mk_rep K (ff i) (hff i)
[GOAL]
case refine'_1.mk
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
ff : ι → V
hff : ∀ (i : ι), ff i ≠ 0
hh1 : LinearIndependent K (Projectivization.rep ∘ fun i => mk K (ff i) (_ : ff i ≠ 0))
a : ι → Kˣ
ha : ∀ (i : ι), a i • ff i = Projectivization.rep (mk K (ff i) (_ : ff i ≠ 0))
⊢ LinearIndependent K ff
[PROOFSTEP]
convert hh1.units_smul a⁻¹
[GOAL]
case h.e'_4
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
ff : ι → V
hff : ∀ (i : ι), ff i ≠ 0
hh1 : LinearIndependent K (Projectivization.rep ∘ fun i => mk K (ff i) (_ : ff i ≠ 0))
a : ι → Kˣ
ha : ∀ (i : ι), a i • ff i = Projectivization.rep (mk K (ff i) (_ : ff i ≠ 0))
⊢ ff = a⁻¹ • Projectivization.rep ∘ fun i => mk K (ff i) (_ : ff i ≠ 0)
[PROOFSTEP]
ext i
[GOAL]
case h.e'_4.h
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
ff : ι → V
hff : ∀ (i : ι), ff i ≠ 0
hh1 : LinearIndependent K (Projectivization.rep ∘ fun i => mk K (ff i) (_ : ff i ≠ 0))
a : ι → Kˣ
ha : ∀ (i : ι), a i • ff i = Projectivization.rep (mk K (ff i) (_ : ff i ≠ 0))
i : ι
⊢ ff i = (a⁻¹ • Projectivization.rep ∘ fun i => mk K (ff i) (_ : ff i ≠ 0)) i
[PROOFSTEP]
simp only [← ha, inv_smul_smul, Pi.smul_apply', Pi.inv_apply, Function.comp_apply]
[GOAL]
case refine'_2
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
h : ¬LinearIndependent K (Projectivization.rep ∘ f)
⊢ Dependent f
[PROOFSTEP]
convert Dependent.mk _ _ h
[GOAL]
case h.e'_7.h
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
h : ¬LinearIndependent K (Projectivization.rep ∘ f)
x✝ : ι
⊢ f x✝ = mk K ((Projectivization.rep ∘ f) x✝) (_ : ?m.30442 x✝ ≠ 0)
[PROOFSTEP]
simp only [mk_rep, Function.comp_apply]
[GOAL]
case refine'_2
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
h : ¬LinearIndependent K (Projectivization.rep ∘ f)
⊢ ∀ (i : ι), (Projectivization.rep ∘ f) i ≠ 0
[PROOFSTEP]
exact fun i => rep_nonzero (f i)
[GOAL]
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
⊢ Dependent f ↔ ¬Independent f
[PROOFSTEP]
rw [dependent_iff, independent_iff]
[GOAL]
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
⊢ Independent f ↔ ¬Dependent f
[PROOFSTEP]
rw [dependent_iff_not_independent, Classical.not_not]
[GOAL]
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
u v : ℙ K V
⊢ Dependent ![u, v] ↔ u = v
[PROOFSTEP]
rw [dependent_iff_not_independent, independent_iff, linearIndependent_fin2, Function.comp_apply, Matrix.cons_val_one,
  Matrix.head_cons, Ne.def]
[GOAL]
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
u v : ℙ K V
⊢ ¬(¬Projectivization.rep v = 0 ∧ ∀ (a : K), a • Projectivization.rep v ≠ (Projectivization.rep ∘ ![u, v]) 0) ↔ u = v
[PROOFSTEP]
simp only [Matrix.cons_val_zero, not_and, not_forall, Classical.not_not, Function.comp_apply, ←
  mk_eq_mk_iff' K _ _ (rep_nonzero u) (rep_nonzero v), mk_rep, imp_iff_right_iff]
[GOAL]
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
u v : ℙ K V
⊢ ¬Projectivization.rep v = 0 ∨ u = v
[PROOFSTEP]
exact Or.inl (rep_nonzero v)
[GOAL]
ι : Type u_1
K : Type u_2
V : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : ι → ℙ K V
u v : ℙ K V
⊢ Independent ![u, v] ↔ u ≠ v
[PROOFSTEP]
rw [independent_iff_not_dependent, dependent_pair_iff_eq u v]
