[GOAL]
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
inst✝² : HasKernels C
X Y : C
inst✝¹ : Simple X
inst✝ : Simple Y
f : X ⟶ Y
I : IsIso f
⊢ f ≠ 0
[PROOFSTEP]
intro h
[GOAL]
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
inst✝² : HasKernels C
X Y : C
inst✝¹ : Simple X
inst✝ : Simple Y
f : X ⟶ Y
I : IsIso f
h : f = 0
⊢ False
[PROOFSTEP]
apply id_nonzero X
[GOAL]
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
inst✝² : HasKernels C
X Y : C
inst✝¹ : Simple X
inst✝ : Simple Y
f : X ⟶ Y
I : IsIso f
h : f = 0
⊢ 𝟙 X = 0
[PROOFSTEP]
simp only [← IsIso.hom_inv_id f, h, zero_comp]
[GOAL]
C : Type u_1
inst✝³ : Category.{?u.3173, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasKernels C
X : C
inst✝ : Simple X
⊢ DivisionRing (End X)
[PROOFSTEP]
classical exact
  {
    (inferInstance :
      Ring (End X)) with
    inv := fun f =>
      if h : f = 0 then 0
      else
        haveI := isIso_of_hom_simple h
        inv f
    exists_pair_ne := ⟨𝟙 X, 0, id_nonzero _⟩
    inv_zero := dif_pos rfl
    mul_inv_cancel := fun f h => by
      dsimp
      rw [dif_neg h]
      haveI := isIso_of_hom_simple h
      exact IsIso.inv_hom_id f }
[GOAL]
C : Type u_1
inst✝³ : Category.{?u.3173, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasKernels C
X : C
inst✝ : Simple X
⊢ DivisionRing (End X)
[PROOFSTEP]
exact
  {
    (inferInstance :
      Ring (End X)) with
    inv := fun f =>
      if h : f = 0 then 0
      else
        haveI := isIso_of_hom_simple h
        inv f
    exists_pair_ne := ⟨𝟙 X, 0, id_nonzero _⟩
    inv_zero := dif_pos rfl
    mul_inv_cancel := fun f h => by
      dsimp
      rw [dif_neg h]
      haveI := isIso_of_hom_simple h
      exact IsIso.inv_hom_id f }
[GOAL]
C : Type u_1
inst✝³ : Category.{?u.4991, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasKernels C
X : C
inst✝ : Simple X
src✝ : Ring (End X) := inferInstance
f : End X
h : f ≠ 0
⊢ f * f⁻¹ = 1
[PROOFSTEP]
dsimp
[GOAL]
C : Type u_1
inst✝³ : Category.{?u.4991, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasKernels C
X : C
inst✝ : Simple X
src✝ : Ring (End X) := inferInstance
f : End X
h : f ≠ 0
⊢ (if h : f = 0 then 0 else inv f) ≫ f = 𝟙 X
[PROOFSTEP]
rw [dif_neg h]
[GOAL]
C : Type u_1
inst✝³ : Category.{?u.4991, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasKernels C
X : C
inst✝ : Simple X
src✝ : Ring (End X) := inferInstance
f : End X
h : f ≠ 0
⊢ inv f ≫ f = 𝟙 X
[PROOFSTEP]
haveI := isIso_of_hom_simple h
[GOAL]
C : Type u_1
inst✝³ : Category.{?u.4991, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasKernels C
X : C
inst✝ : Simple X
src✝ : Ring (End X) := inferInstance
f : End X
h : f ≠ 0
this : IsIso f
⊢ inv f ≫ f = 𝟙 X
[PROOFSTEP]
exact IsIso.inv_hom_id f
[GOAL]
C : Type u_1
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Preadditive C
𝕜 : Type u_2
inst✝⁴ : DivisionRing 𝕜
inst✝³ : HasKernels C
inst✝² : Linear 𝕜 C
X Y : C
inst✝¹ : Simple X
inst✝ : Simple Y
h : (X ≅ Y) → False
f : X ⟶ Y
⊢ f = 0
[PROOFSTEP]
have p := not_congr (isIso_iff_nonzero f)
[GOAL]
C : Type u_1
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Preadditive C
𝕜 : Type u_2
inst✝⁴ : DivisionRing 𝕜
inst✝³ : HasKernels C
inst✝² : Linear 𝕜 C
X Y : C
inst✝¹ : Simple X
inst✝ : Simple Y
h : (X ≅ Y) → False
f : X ⟶ Y
p : ¬IsIso f ↔ ¬f ≠ 0
⊢ f = 0
[PROOFSTEP]
simp only [Classical.not_not, Ne.def] at p 
[GOAL]
C : Type u_1
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Preadditive C
𝕜 : Type u_2
inst✝⁴ : DivisionRing 𝕜
inst✝³ : HasKernels C
inst✝² : Linear 𝕜 C
X Y : C
inst✝¹ : Simple X
inst✝ : Simple Y
h : (X ≅ Y) → False
f : X ⟶ Y
p : ¬IsIso f ↔ f = 0
⊢ f = 0
[PROOFSTEP]
refine' p.mp fun _ => h (asIso f)
[GOAL]
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
𝕜 : Type u_2
inst✝² : Field 𝕜
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Linear 𝕜 C
X : C
isIso_iff_nonzero : ∀ (f : X ⟶ X), IsIso f ↔ f ≠ 0
I : FiniteDimensional 𝕜 (X ⟶ X)
⊢ finrank 𝕜 (X ⟶ X) = 1
[PROOFSTEP]
have id_nonzero := (isIso_iff_nonzero (𝟙 X)).mp (by infer_instance)
[GOAL]
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
𝕜 : Type u_2
inst✝² : Field 𝕜
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Linear 𝕜 C
X : C
isIso_iff_nonzero : ∀ (f : X ⟶ X), IsIso f ↔ f ≠ 0
I : FiniteDimensional 𝕜 (X ⟶ X)
⊢ IsIso (𝟙 X)
[PROOFSTEP]
infer_instance
[GOAL]
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
𝕜 : Type u_2
inst✝² : Field 𝕜
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Linear 𝕜 C
X : C
isIso_iff_nonzero : ∀ (f : X ⟶ X), IsIso f ↔ f ≠ 0
I : FiniteDimensional 𝕜 (X ⟶ X)
id_nonzero : 𝟙 X ≠ 0
⊢ finrank 𝕜 (X ⟶ X) = 1
[PROOFSTEP]
refine' finrank_eq_one (𝟙 X) id_nonzero _
[GOAL]
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
𝕜 : Type u_2
inst✝² : Field 𝕜
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Linear 𝕜 C
X : C
isIso_iff_nonzero : ∀ (f : X ⟶ X), IsIso f ↔ f ≠ 0
I : FiniteDimensional 𝕜 (X ⟶ X)
id_nonzero : 𝟙 X ≠ 0
⊢ ∀ (w : X ⟶ X), ∃ c, c • 𝟙 X = w
[PROOFSTEP]
intro f
[GOAL]
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
𝕜 : Type u_2
inst✝² : Field 𝕜
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Linear 𝕜 C
X : C
isIso_iff_nonzero : ∀ (f : X ⟶ X), IsIso f ↔ f ≠ 0
I : FiniteDimensional 𝕜 (X ⟶ X)
id_nonzero : 𝟙 X ≠ 0
f : X ⟶ X
⊢ ∃ c, c • 𝟙 X = f
[PROOFSTEP]
have : Nontrivial (End X) := nontrivial_of_ne _ _ id_nonzero
[GOAL]
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
𝕜 : Type u_2
inst✝² : Field 𝕜
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Linear 𝕜 C
X : C
isIso_iff_nonzero : ∀ (f : X ⟶ X), IsIso f ↔ f ≠ 0
I : FiniteDimensional 𝕜 (X ⟶ X)
id_nonzero : 𝟙 X ≠ 0
f : X ⟶ X
this : Nontrivial (End X)
⊢ ∃ c, c • 𝟙 X = f
[PROOFSTEP]
have : FiniteDimensional 𝕜 (End X) := I
[GOAL]
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
𝕜 : Type u_2
inst✝² : Field 𝕜
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Linear 𝕜 C
X : C
isIso_iff_nonzero : ∀ (f : X ⟶ X), IsIso f ↔ f ≠ 0
I : FiniteDimensional 𝕜 (X ⟶ X)
id_nonzero : 𝟙 X ≠ 0
f : X ⟶ X
this✝ : Nontrivial (End X)
this : FiniteDimensional 𝕜 (End X)
⊢ ∃ c, c • 𝟙 X = f
[PROOFSTEP]
obtain ⟨c, nu⟩ := spectrum.nonempty_of_isAlgClosed_of_finiteDimensional 𝕜 (End.of f)
[GOAL]
case intro
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
𝕜 : Type u_2
inst✝² : Field 𝕜
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Linear 𝕜 C
X : C
isIso_iff_nonzero : ∀ (f : X ⟶ X), IsIso f ↔ f ≠ 0
I : FiniteDimensional 𝕜 (X ⟶ X)
id_nonzero : 𝟙 X ≠ 0
f : X ⟶ X
this✝ : Nontrivial (End X)
this : FiniteDimensional 𝕜 (End X)
c : 𝕜
nu : c ∈ spectrum 𝕜 (End.of f)
⊢ ∃ c, c • 𝟙 X = f
[PROOFSTEP]
use c
[GOAL]
case h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
𝕜 : Type u_2
inst✝² : Field 𝕜
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Linear 𝕜 C
X : C
isIso_iff_nonzero : ∀ (f : X ⟶ X), IsIso f ↔ f ≠ 0
I : FiniteDimensional 𝕜 (X ⟶ X)
id_nonzero : 𝟙 X ≠ 0
f : X ⟶ X
this✝ : Nontrivial (End X)
this : FiniteDimensional 𝕜 (End X)
c : 𝕜
nu : c ∈ spectrum 𝕜 (End.of f)
⊢ c • 𝟙 X = f
[PROOFSTEP]
rw [spectrum.mem_iff, IsUnit.sub_iff, isUnit_iff_isIso, isIso_iff_nonzero, Ne.def, Classical.not_not, sub_eq_zero,
  Algebra.algebraMap_eq_smul_one] at nu 
[GOAL]
case h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
𝕜 : Type u_2
inst✝² : Field 𝕜
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Linear 𝕜 C
X : C
isIso_iff_nonzero : ∀ (f : X ⟶ X), IsIso f ↔ f ≠ 0
I : FiniteDimensional 𝕜 (X ⟶ X)
id_nonzero : 𝟙 X ≠ 0
f : X ⟶ X
this✝ : Nontrivial (End X)
this : FiniteDimensional 𝕜 (End X)
c : 𝕜
nu : End.of f = c • 1
⊢ c • 𝟙 X = f
[PROOFSTEP]
exact nu.symm
[GOAL]
C : Type u_1
inst✝⁶ : Category.{?u.16907, u_1} C
inst✝⁵ : Preadditive C
𝕜 : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : IsAlgClosed 𝕜
inst✝² : Linear 𝕜 C
inst✝¹ : HasKernels C
X : C
inst✝ : Simple X
I : FiniteDimensional 𝕜 (X ⟶ X)
⊢ Field (End X)
[PROOFSTEP]
classical exact
  { (inferInstance : DivisionRing (End X)) with
    mul_comm := fun f g => by
      obtain ⟨c, rfl⟩ := endomorphism_simple_eq_smul_id 𝕜 f
      obtain ⟨d, rfl⟩ := endomorphism_simple_eq_smul_id 𝕜 g
      simp [← mul_smul, mul_comm c d] }
[GOAL]
C : Type u_1
inst✝⁶ : Category.{?u.16907, u_1} C
inst✝⁵ : Preadditive C
𝕜 : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : IsAlgClosed 𝕜
inst✝² : Linear 𝕜 C
inst✝¹ : HasKernels C
X : C
inst✝ : Simple X
I : FiniteDimensional 𝕜 (X ⟶ X)
⊢ Field (End X)
[PROOFSTEP]
exact
  { (inferInstance : DivisionRing (End X)) with
    mul_comm := fun f g => by
      obtain ⟨c, rfl⟩ := endomorphism_simple_eq_smul_id 𝕜 f
      obtain ⟨d, rfl⟩ := endomorphism_simple_eq_smul_id 𝕜 g
      simp [← mul_smul, mul_comm c d] }
[GOAL]
C : Type u_1
inst✝⁶ : Category.{?u.16907, u_1} C
inst✝⁵ : Preadditive C
𝕜 : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : IsAlgClosed 𝕜
inst✝² : Linear 𝕜 C
inst✝¹ : HasKernels C
X : C
inst✝ : Simple X
I : FiniteDimensional 𝕜 (X ⟶ X)
src✝ : DivisionRing (End X) := inferInstance
f g : End X
⊢ f * g = g * f
[PROOFSTEP]
obtain ⟨c, rfl⟩ := endomorphism_simple_eq_smul_id 𝕜 f
[GOAL]
case intro
C : Type u_1
inst✝⁶ : Category.{?u.16907, u_1} C
inst✝⁵ : Preadditive C
𝕜 : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : IsAlgClosed 𝕜
inst✝² : Linear 𝕜 C
inst✝¹ : HasKernels C
X : C
inst✝ : Simple X
I : FiniteDimensional 𝕜 (X ⟶ X)
src✝ : DivisionRing (End X) := inferInstance
g : End X
c : 𝕜
⊢ c • 𝟙 X * g = g * c • 𝟙 X
[PROOFSTEP]
obtain ⟨d, rfl⟩ := endomorphism_simple_eq_smul_id 𝕜 g
[GOAL]
case intro.intro
C : Type u_1
inst✝⁶ : Category.{?u.16907, u_1} C
inst✝⁵ : Preadditive C
𝕜 : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : IsAlgClosed 𝕜
inst✝² : Linear 𝕜 C
inst✝¹ : HasKernels C
X : C
inst✝ : Simple X
I : FiniteDimensional 𝕜 (X ⟶ X)
src✝ : DivisionRing (End X) := inferInstance
c d : 𝕜
⊢ c • 𝟙 X * d • 𝟙 X = d • 𝟙 X * c • 𝟙 X
[PROOFSTEP]
simp [← mul_smul, mul_comm c d]
[GOAL]
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : FiniteDimensional 𝕜 (X ⟶ X)
inst✝¹ : Simple X
inst✝ : Simple Y
⊢ finrank 𝕜 (X ⟶ Y) ≤ 1
[PROOFSTEP]
obtain (h | h) := subsingleton_or_nontrivial (X ⟶ Y)
[GOAL]
case inl
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : FiniteDimensional 𝕜 (X ⟶ X)
inst✝¹ : Simple X
inst✝ : Simple Y
h : Subsingleton (X ⟶ Y)
⊢ finrank 𝕜 (X ⟶ Y) ≤ 1
[PROOFSTEP]
rw [finrank_zero_of_subsingleton]
[GOAL]
case inl
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : FiniteDimensional 𝕜 (X ⟶ X)
inst✝¹ : Simple X
inst✝ : Simple Y
h : Subsingleton (X ⟶ Y)
⊢ 0 ≤ 1
[PROOFSTEP]
exact zero_le_one
[GOAL]
case inr
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : FiniteDimensional 𝕜 (X ⟶ X)
inst✝¹ : Simple X
inst✝ : Simple Y
h : Nontrivial (X ⟶ Y)
⊢ finrank 𝕜 (X ⟶ Y) ≤ 1
[PROOFSTEP]
obtain ⟨f, nz⟩ := (nontrivial_iff_exists_ne 0).mp h
[GOAL]
case inr.intro
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : FiniteDimensional 𝕜 (X ⟶ X)
inst✝¹ : Simple X
inst✝ : Simple Y
h : Nontrivial (X ⟶ Y)
f : X ⟶ Y
nz : f ≠ 0
⊢ finrank 𝕜 (X ⟶ Y) ≤ 1
[PROOFSTEP]
haveI fi := (isIso_iff_nonzero f).mpr nz
[GOAL]
case inr.intro
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : FiniteDimensional 𝕜 (X ⟶ X)
inst✝¹ : Simple X
inst✝ : Simple Y
h : Nontrivial (X ⟶ Y)
f : X ⟶ Y
nz : f ≠ 0
fi : IsIso f
⊢ finrank 𝕜 (X ⟶ Y) ≤ 1
[PROOFSTEP]
refine' finrank_le_one f _
[GOAL]
case inr.intro
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : FiniteDimensional 𝕜 (X ⟶ X)
inst✝¹ : Simple X
inst✝ : Simple Y
h : Nontrivial (X ⟶ Y)
f : X ⟶ Y
nz : f ≠ 0
fi : IsIso f
⊢ ∀ (w : X ⟶ Y), ∃ c, c • f = w
[PROOFSTEP]
intro g
[GOAL]
case inr.intro
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : FiniteDimensional 𝕜 (X ⟶ X)
inst✝¹ : Simple X
inst✝ : Simple Y
h : Nontrivial (X ⟶ Y)
f : X ⟶ Y
nz : f ≠ 0
fi : IsIso f
g : X ⟶ Y
⊢ ∃ c, c • f = g
[PROOFSTEP]
obtain ⟨c, w⟩ := endomorphism_simple_eq_smul_id 𝕜 (g ≫ inv f)
[GOAL]
case inr.intro.intro
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : FiniteDimensional 𝕜 (X ⟶ X)
inst✝¹ : Simple X
inst✝ : Simple Y
h : Nontrivial (X ⟶ Y)
f : X ⟶ Y
nz : f ≠ 0
fi : IsIso f
g : X ⟶ Y
c : 𝕜
w : c • 𝟙 X = g ≫ inv f
⊢ ∃ c, c • f = g
[PROOFSTEP]
exact ⟨c, by simpa using w =≫ f⟩
[GOAL]
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : FiniteDimensional 𝕜 (X ⟶ X)
inst✝¹ : Simple X
inst✝ : Simple Y
h : Nontrivial (X ⟶ Y)
f : X ⟶ Y
nz : f ≠ 0
fi : IsIso f
g : X ⟶ Y
c : 𝕜
w : c • 𝟙 X = g ≫ inv f
⊢ c • f = g
[PROOFSTEP]
simpa using w =≫ f
[GOAL]
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
⊢ finrank 𝕜 (X ⟶ Y) = 1 ↔ Nonempty (X ≅ Y)
[PROOFSTEP]
fconstructor
[GOAL]
case mp
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
⊢ finrank 𝕜 (X ⟶ Y) = 1 → Nonempty (X ≅ Y)
[PROOFSTEP]
intro h
[GOAL]
case mp
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : finrank 𝕜 (X ⟶ Y) = 1
⊢ Nonempty (X ≅ Y)
[PROOFSTEP]
rw [finrank_eq_one_iff'] at h 
[GOAL]
case mp
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ∃ v _n, ∀ (w : X ⟶ Y), ∃ c, c • v = w
⊢ Nonempty (X ≅ Y)
[PROOFSTEP]
obtain ⟨f, nz, -⟩ := h
[GOAL]
case mp.intro.intro
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
f : X ⟶ Y
nz : f ≠ 0
⊢ Nonempty (X ≅ Y)
[PROOFSTEP]
rw [← isIso_iff_nonzero] at nz 
[GOAL]
case mp.intro.intro
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
f : X ⟶ Y
nz : IsIso f
⊢ Nonempty (X ≅ Y)
[PROOFSTEP]
exact ⟨asIso f⟩
[GOAL]
case mpr
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
⊢ Nonempty (X ≅ Y) → finrank 𝕜 (X ⟶ Y) = 1
[PROOFSTEP]
rintro ⟨f⟩
[GOAL]
case mpr.intro
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
f : X ≅ Y
⊢ finrank 𝕜 (X ⟶ Y) = 1
[PROOFSTEP]
have le_one := finrank_hom_simple_simple_le_one 𝕜 X Y
[GOAL]
case mpr.intro
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
f : X ≅ Y
le_one : finrank 𝕜 (X ⟶ Y) ≤ 1
⊢ finrank 𝕜 (X ⟶ Y) = 1
[PROOFSTEP]
have zero_lt : 0 < finrank 𝕜 (X ⟶ Y) :=
  finrank_pos_iff_exists_ne_zero.mpr ⟨f.hom, (isIso_iff_nonzero f.hom).mp inferInstance⟩
[GOAL]
case mpr.intro
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
f : X ≅ Y
le_one : finrank 𝕜 (X ⟶ Y) ≤ 1
zero_lt : 0 < finrank 𝕜 (X ⟶ Y)
⊢ finrank 𝕜 (X ⟶ Y) = 1
[PROOFSTEP]
linarith
[GOAL]
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
⊢ finrank 𝕜 (X ⟶ Y) = 0 ↔ IsEmpty (X ≅ Y)
[PROOFSTEP]
rw [← not_nonempty_iff, ← not_congr (finrank_hom_simple_simple_eq_one_iff 𝕜 X Y)]
[GOAL]
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
⊢ finrank 𝕜 (X ⟶ Y) = 0 ↔ ¬finrank 𝕜 (X ⟶ Y) = 1
[PROOFSTEP]
refine' ⟨fun h => by rw [h]; simp, fun h => _⟩
[GOAL]
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : finrank 𝕜 (X ⟶ Y) = 0
⊢ ¬finrank 𝕜 (X ⟶ Y) = 1
[PROOFSTEP]
rw [h]
[GOAL]
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : finrank 𝕜 (X ⟶ Y) = 0
⊢ ¬0 = 1
[PROOFSTEP]
simp
[GOAL]
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ¬finrank 𝕜 (X ⟶ Y) = 1
⊢ finrank 𝕜 (X ⟶ Y) = 0
[PROOFSTEP]
have := finrank_hom_simple_simple_le_one 𝕜 X Y
[GOAL]
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ¬finrank 𝕜 (X ⟶ Y) = 1
this : finrank 𝕜 (X ⟶ Y) ≤ 1
⊢ finrank 𝕜 (X ⟶ Y) = 0
[PROOFSTEP]
interval_cases finrank 𝕜 (X ⟶ Y)
[GOAL]
case «0»
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ¬0 = 1
this : 0 ≤ 1
⊢ 0 = 0
[PROOFSTEP]
rfl
[GOAL]
case «1»
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ¬1 = 1
this : 1 ≤ 1
⊢ 1 = 0
[PROOFSTEP]
exact False.elim (h rfl)
[GOAL]
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : ∀ (X Y : C), FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
⊢ finrank 𝕜 (X ⟶ Y) = if Nonempty (X ≅ Y) then 1 else 0
[PROOFSTEP]
split_ifs with h
[GOAL]
case pos
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : ∀ (X Y : C), FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : Nonempty (X ≅ Y)
⊢ finrank 𝕜 (X ⟶ Y) = 1
case neg
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : ∀ (X Y : C), FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ¬Nonempty (X ≅ Y)
⊢ finrank 𝕜 (X ⟶ Y) = 0
[PROOFSTEP]
exact (finrank_hom_simple_simple_eq_one_iff 𝕜 X Y).2 h
[GOAL]
case neg
C : Type u_1
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Preadditive C
𝕜 : Type u_2
inst✝⁶ : Field 𝕜
inst✝⁵ : IsAlgClosed 𝕜
inst✝⁴ : Linear 𝕜 C
inst✝³ : HasKernels C
X Y : C
inst✝² : ∀ (X Y : C), FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
h : ¬Nonempty (X ≅ Y)
⊢ finrank 𝕜 (X ⟶ Y) = 0
[PROOFSTEP]
exact (finrank_hom_simple_simple_eq_zero_iff 𝕜 X Y).2 (not_nonempty_iff.mp h)
