awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Basic (goal)
Goal:
K : Type u_1
V : Type u_2
inst : DivisionRing K
inst_1 : AddCommGroup V
inst_2 : Module K V
L : Type u_3
W : Type u_4
inst_3 : DivisionRing L
inst_4 : AddCommGroup W
inst_5 : Module L W
F : Type u_5
U : Type u_6
inst_6 : DivisionRing F
inst_7 : AddCommGroup U
inst_8 : Module F U
σ : K →+* L
τ : L →+* F
γ : K →+* F
inst_9 : RingHomCompTriple σ τ γ
f : V →ₛₗ[σ] W
hf : Function.Injective.{u_2 + 1, u_4 + 1} (α := V) (β := W) (⇑f : (a : V) → W)
g : W →ₛₗ[τ] U
hg : Function.Injective.{u_4 + 1, u_6 + 1} (α := W) (β := U) (⇑g : (a : W) → U)
hgf : Function.Injective.{u_2 + 1, u_6 + 1} (α := V) (β := U) (⇑(g ∘ₛₗ f) : (a : V) → U)
⊢ map (g ∘ₛₗ f) hgf = map g hg ∘ map f hf
have h : ∀ (x : ℙ K V) (v : { v : V // v ≠ 0 }),
  map (g ∘ₛₗ f) hgf (Quot.mk (⇑(projectivizationSetoid K V) : { v : V // v ≠ 0 } → { v : V // v ≠ 0 } → Prop) v) =
    (map g hg ∘ map f hf) (Quot.mk (⇑(projectivizationSetoid K V) : { v : V // v ≠ 0 } → { v : V // v ≠ 0 } → Prop) v)
Tactic: ext ⟨v⟩
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Basic (goal)
Goal:
K : Type u_1
V : Type u_2
inst : DivisionRing K
inst_1 : AddCommGroup V
inst_2 : Module K V
D D' : ℙ K V
⊢ LinearIndependent K ![D.rep, D'.rep] ↔ D ≠ D'
have rw : (∀ (a : K), a • D.rep ≠ D'.rep) ↔ D ≠ D'
Tactic: rw [LinearIndependent.pair_iff' (rep_nonzero _)]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Basic (goal)
Goal:
K : Type u_1
V : Type u_2
inst : DivisionRing K
inst_1 : AddCommGroup V
inst_2 : Module K V
D D' : ℙ K V
⊢ (∀ (a : K), a • D.rep ≠ D'.rep) ↔ D ≠ D'
have refine : ∀ (h : D ≠ D') (a : K) (hD : a • D.rep = D'.rep), D = D'
Tactic: refine ⟨fun h hD ↦ h 1 (by simp [hD]), fun h a hD ↦ h ?_⟩
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Basic (goal)
Goal:
K : Type u_1
V : Type u_2
inst : DivisionRing K
inst_1 : AddCommGroup V
inst_2 : Module K V
D D' : ℙ K V
h : D ≠ D'
a : K
hD : a • D.rep = D'.rep
⊢ D = D'
have rw : ∃ (a : Kˣ), a • D.rep = D'.rep
Tactic: rw [eq_comm, ← mk_rep D, ← mk_rep D', mk_eq_mk_iff]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Basic (goal)
Goal:
K : Type u_1
V : Type u_2
inst : DivisionRing K
inst_1 : AddCommGroup V
inst_2 : Module K V
D D' : ℙ K V
h : D ≠ D'
a : K
hD : a • D.rep = D'.rep
⊢ ∃ (a : Kˣ), a • D.rep = D'.rep
have suffices : a ≠ 0
Tactic: suffices a ≠ 0 by refine ⟨(Ne.isUnit this).unit, by simp [← hD]⟩
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Basic (goal)
Goal:
K : Type u_1
V : Type u_2
inst : DivisionRing K
inst_1 : AddCommGroup V
inst_2 : Module K V
D D' : ℙ K V
⊢ LinearIndepOn (ι := V) K id {D.rep, D'.rep}
have pos : ∀ (h : D = D'), LinearIndepOn (ι := V) K id {D.rep, D'.rep}
Tactic: by_cases h : D = D'
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Basic (goal)
Goal:
K : Type u_1
V : Type u_2
inst : DivisionRing K
inst_1 : AddCommGroup V
inst_2 : Module K V
D D' : ℙ K V
pos : ∀ (h : D = D'), LinearIndepOn (ι := V) K id {D.rep, D'.rep}
⊢ LinearIndepOn (ι := V) K id {D.rep, D'.rep}
have neg : ∀ (h : ¬D = D'), LinearIndepOn (ι := V) K id {D.rep, D'.rep}
Tactic: by_cases h : D = D'
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Basic (goal)
Goal:
K : Type u_1
V : Type u_2
inst : DivisionRing K
inst_1 : AddCommGroup V
inst_2 : Module K V
D D' : ℙ K V
h : ¬D = D'
⊢ LinearIndepOn (ι := V) K id {D.rep, D'.rep}
have neg : ∀ (h : LinearIndepOn K id (Set.range ![D'.rep, D.rep])), LinearIndepOn (ι := V) K id {D.rep, D'.rep}
Tactic: rw [← ne_eq, ← linearIndependent_pair_iff_ne, LinearIndependent.pair_symm_iff,
    ← linearIndepOn_id_range_iff] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Basic (goal)
Goal:
K : Type u_1
V : Type u_2
inst : DivisionRing K
inst_1 : AddCommGroup V
inst_2 : Module K V
D D' : ℙ K V
h : ¬D = D'
neg : ∀ (h : LinearIndepOn K id (Set.range ![D'.rep, D.rep])), LinearIndepOn (ι := V) K id {D.rep, D'.rep}
⊢ LinearIndepOn (ι := V) K id {D.rep, D'.rep}
have neg₁ : ∀ (h : LinearIndependent K ![D'.rep, D.rep]), Function.Injective ![D'.rep, D.rep]
Tactic: rw [← ne_eq, ← linearIndependent_pair_iff_ne, LinearIndependent.pair_symm_iff,
    ← linearIndepOn_id_range_iff] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
b : { v : V // v ≠ 0 }
⊢ Eq.{u_1 + 1} (α := Subgroup kˣ) (MulAction.stabilizer kˣ b) ⊥
have rw : Eq.{u_1 + 1} (α := Subgroup kˣ) (MulAction.stabilizer kˣ (↑b : V)) ⊥
Tactic: rw [(Units.nonZeroSubMul k V).stabilizer_of_subMul,
      Module.stabilizer_units_eq_bot_of_ne_zero k b.property]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
b : { v : V // v ≠ 0 }
rw : Eq.{u_1 + 1} (α := Subgroup kˣ) (MulAction.stabilizer kˣ (↑b : V)) ⊥
⊢ Eq.{u_1 + 1} (α := Subgroup kˣ) (MulAction.stabilizer kˣ b) ⊥
have rw₁ : Eq.{u_1 + 1} (α := Subgroup kˣ) ⊥ ⊥
Tactic: rw [(Units.nonZeroSubMul k V).stabilizer_of_subMul,
      Module.stabilizer_units_eq_bot_of_ne_zero k b.property]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : Subsingleton V
⊢ IsEmpty.{u_2 + 1} (ℙ k V)
have this : IsEmpty.{max 1 (u_2 + 1)} { v : V // v ≠ 0 }
Tactic: have : IsEmpty { v : V // v ≠ 0 } := ⟨fun v ↦ v.2 (Subsingleton.elim v.1 0)⟩
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : Finite k
⊢ Finite.{u_2 + 1} (ℙ k V) ↔ Finite V
have refine : ∀ (h : Finite.{u_2 + 1} (ℙ k V)), Finite V
Tactic: refine ⟨fun h ↦ ?_, fun h ↦ inferInstance⟩
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : Finite k
h : Finite.{u_2 + 1} (ℙ k V)
e : { v : V // v ≠ 0 } ≃ Prod.{u_2, u_1} (ℙ k V) kˣ := nonZeroEquivProjectivizationProdUnits k V
⊢ Finite V
have this : Finite.{max 1 (u_2 + 1)} { v : V // v ≠ 0 }
Tactic: have : Finite { v : V // v ≠ 0 } := Finite.of_equiv _ e.symm
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : Finite k
⊢ Finite.{u_2 + 1} (ℙ k V) ↔ Finite V
have refine : ∀ (h : Finite.{u_2 + 1} (ℙ k V)), Finite V
Tactic: classical
  refine ⟨fun h ↦ ?_, fun h ↦ inferInstance⟩
  let e := nonZeroEquivProjectivizationProdUnits k V
  have : Finite { v : V // v ≠ 0 } := Finite.of_equiv _ e.symm
  let eq : { v : V // v ≠ 0 } ⊕ Unit ≃ V :=
    ⟨(Sum.elim Subtype.val (fun _ ↦ 0)), fun v ↦ if h : v = 0 then Sum.inr () else Sum.inl ⟨v, h⟩,
      by intro x; aesop, by intro x; aesop⟩
  exact Finite.of_equiv _ eq
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : Finite k
refine : ∀ (h : Finite.{u_2 + 1} (ℙ k V)), Finite V
refine₁ : ∀ (h : Finite.{u_2 + 1} (ℙ k V)), Finite V
⊢ Finite.{u_2 + 1} (ℙ k V) ↔ Finite V
have refine₂ : ∀ (h : Finite.{u_2 + 1} (ℙ k V)) (this : Finite.{max 1 (u_2 + 1)} { v : V // v ≠ 0 }), Finite V
Tactic: classical
  refine ⟨fun h ↦ ?_, fun h ↦ inferInstance⟩
  let e := nonZeroEquivProjectivizationProdUnits k V
  have : Finite { v : V // v ≠ 0 } := Finite.of_equiv _ e.symm
  let eq : { v : V // v ≠ 0 } ⊕ Unit ≃ V :=
    ⟨(Sum.elim Subtype.val (fun _ ↦ 0)), fun v ↦ if h : v = 0 then Sum.inr () else Sum.inl ⟨v, h⟩,
      by intro x; aesop, by intro x; aesop⟩
  exact Finite.of_equiv _ eq
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have a : Nontrivial V
Tactic: nontriviality V
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h : Infinite k
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have this : Infinite V
Tactic: have : Infinite V := Module.Free.infinite k V
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Infinite V
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have this : ¬Finite.{u_2 + 1} (ℙ k V) ↔ ¬Finite V
Tactic: have := not_iff_not.mpr (finite_iff_of_finite k V)
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Infinite V
this : ¬Finite.{u_2 + 1} (ℙ k V) ↔ ¬Finite V
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have this₁ : Infinite.{u_2 + 1} (ℙ k V) ↔ Infinite V
Tactic: push Not at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Infinite V
this : Infinite.{u_2 + 1} (ℙ k V) ↔ Infinite V
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have this₁ : Infinite.{u_2 + 1} (ℙ k V)
Tactic: have : Infinite (ℙ k V) := by rwa [this]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Finite V
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have inl : ∀ (this : Fintype V), Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
Tactic: haveI : Fintype V := Fintype.ofFinite V
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Finite V
this : Fintype V
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have inl : ∀ (this : Fintype (ℙ k V)), Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
Tactic: haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Finite V
this_1 : Fintype V
this : Fintype (ℙ k V)
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have inl : ∀ (this : Fintype k), Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
Tactic: haveI : Fintype k := Fintype.ofFinite k
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Finite V
this_1 : Fintype V
this_2 : Fintype (ℙ k V)
this : Fintype k
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have hV : Fintype.card.{u_2} { v : V // v ≠ 0 } = Fintype.card V - 1
Tactic: have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Finite V
this_1 : Fintype V
this_2 : Fintype (ℙ k V)
this : Fintype k
hV : Fintype.card.{u_2} { v : V // v ≠ 0 } = Fintype.card V - 1
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have inl : Fintype.card.{u_2} { v : V // v ≠ 0 } = Fintype.card.{u_2} (ℙ k V) * Fintype.card.{u_1} kˣ
Tactic: simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Finite V
this_1 : Fintype V
this_2 : Fintype (ℙ k V)
this : Fintype k
hV : Fintype.card.{u_2} { v : V // v ≠ 0 } = Fintype.card V - 1
⊢ Fintype.card.{u_2} { v : V // v ≠ 0 } = Fintype.card.{u_2} (ℙ k V) * Fintype.card.{u_1} kˣ
have inl : Fintype.card (Prod.{u_2, u_1} (ℙ k V) kˣ) = Fintype.card.{u_2} (ℙ k V) * Fintype.card.{u_1} kˣ
Tactic: rw [Fintype.card_congr (nonZeroEquivProjectivizationProdUnits k V), Fintype.card_prod]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Finite V
this_1 : Fintype V
this_2 : Fintype (ℙ k V)
this : Fintype k
hV : Fintype.card.{u_2} { v : V // v ≠ 0 } = Fintype.card V - 1
inl : Fintype.card (Prod.{u_2, u_1} (ℙ k V) kˣ) = Fintype.card.{u_2} (ℙ k V) * Fintype.card.{u_1} kˣ
⊢ Fintype.card.{u_2} { v : V // v ≠ 0 } = Fintype.card.{u_2} (ℙ k V) * Fintype.card.{u_1} kˣ
have inl₁ : Fintype.card.{u_2} (ℙ k V) * Fintype.card.{u_1} kˣ = Fintype.card.{u_2} (ℙ k V) * Fintype.card.{u_1} kˣ
Tactic: rw [Fintype.card_congr (nonZeroEquivProjectivizationProdUnits k V), Fintype.card_prod]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Finite V
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have inl : ∀ (this : Fintype V), Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
Tactic: classical
  haveI : Fintype V := Fintype.ofFinite V
  haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)
  haveI : Fintype k := Fintype.ofFinite k
  have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp
  simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]
  rw [Fintype.card_congr (nonZeroEquivProjectivizationProdUnits k V), Fintype.card_prod]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Finite V
inl inl₁ inl₂ : ∀ (this : Fintype V), Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have inl₃ : ∀ (this : Fintype V) (this : Fintype (ℙ k V)), Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
Tactic: classical
  haveI : Fintype V := Fintype.ofFinite V
  haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)
  haveI : Fintype k := Fintype.ofFinite k
  have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp
  simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]
  rw [Fintype.card_congr (nonZeroEquivProjectivizationProdUnits k V), Fintype.card_prod]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Finite V
inl inl₁ inl₂ : ∀ (this : Fintype V), Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
inl₃ : ∀ (this : Fintype V) (this : Fintype (ℙ k V)), Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have inl₄ : ∀ (this : Fintype V) (this : Fintype (ℙ k V)) (this : Fintype k),
  Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
Tactic: classical
  haveI : Fintype V := Fintype.ofFinite V
  haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)
  haveI : Fintype k := Fintype.ofFinite k
  have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp
  simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]
  rw [Fintype.card_congr (nonZeroEquivProjectivizationProdUnits k V), Fintype.card_prod]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Finite V
inl inl₁ inl₂ : ∀ (this : Fintype V), Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
inl₃ : ∀ (this : Fintype V) (this : Fintype (ℙ k V)), Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
inl₄ :
  ∀ (this : Fintype V) (this : Fintype (ℙ k V)) (this : Fintype k),
    Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have inl₅ : ∀ (this : Fintype V) (this_1 : Fintype (ℙ k V)) (this_2 : Fintype k)
  (hV : Fintype.card.{u_2} { v : V // v ≠ 0 } = Fintype.card V - 1),
  Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
Tactic: classical
  haveI : Fintype V := Fintype.ofFinite V
  haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)
  haveI : Fintype k := Fintype.ofFinite k
  have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp
  simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]
  rw [Fintype.card_congr (nonZeroEquivProjectivizationProdUnits k V), Fintype.card_prod]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h_2 : Finite k
h : Finite V
inl inl₁ inl₂ : ∀ (this : Fintype V), Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
inl₃ : ∀ (this : Fintype V) (this : Fintype (ℙ k V)), Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
inl₄ :
  ∀ (this : Fintype V) (this : Fintype (ℙ k V)) (this : Fintype k),
    Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
inl₅ :
  ∀ (this : Fintype V) (this_1 : Fintype (ℙ k V)) (this_2 : Fintype k)
    (hV : Fintype.card.{u_2} { v : V // v ≠ 0 } = Fintype.card V - 1),
    Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have inl₆ : ∀ (this : Fintype V) (this_1 : Fintype (ℙ k V)) (this_2 : Fintype k)
  (hV : Fintype.card.{u_2} { v : V // v ≠ 0 } = Fintype.card V - 1),
  Fintype.card.{u_2} { v : V // v ≠ 0 } = Fintype.card.{u_2} (ℙ k V) * Fintype.card.{u_1} kˣ
Tactic: classical
  haveI : Fintype V := Fintype.ofFinite V
  haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)
  haveI : Fintype k := Fintype.ofFinite k
  have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp
  simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]
  rw [Fintype.card_congr (nonZeroEquivProjectivizationProdUnits k V), Fintype.card_prod]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h_1 : Nontrivial V
h : Finite k
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have x : Finite V ∨ Infinite V
Tactic: cases finite_or_infinite V with
  | inr h =>
    have := not_iff_not.mpr (finite_iff_of_finite k V)
    push Not at this
    have : Infinite (ℙ k V) := by rwa [this]
    simp
  | inl h =>
  classical
  haveI : Fintype V := Fintype.ofFinite V
  haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)
  haveI : Fintype k := Fintype.ofFinite k
  have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp
  simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]
  rw [Fintype.card_congr (nonZeroEquivProjectivizationProdUnits k V), Fintype.card_prod]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
h : Nontrivial V
⊢ Nat.card V - 1 = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1)
have x : Finite k ∨ Infinite k
Tactic: cases finite_or_infinite k with
  | inr h =>
    have : Infinite V := Module.Free.infinite k V
    simp
  | inl h =>
  cases finite_or_infinite V with
  | inr h =>
    have := not_iff_not.mpr (finite_iff_of_finite k V)
    push Not at this
    have : Infinite (ℙ k V) := by rwa [this]
    simp
  | inl h =>
  classical
  haveI : Fintype V := Fintype.ofFinite V
  haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)
  haveI : Fintype k := Fintype.ofFinite k
  have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp
  simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]
  rw [Fintype.card_congr (nonZeroEquivProjectivizationProdUnits k V), Fintype.card_prod]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : Finite V
⊢ Nat.card V = Nat.card.{u_2} (ℙ k V) * (Nat.card k - 1) + 1
have rw : Nat.card V = Nat.card V - 1 + 1
Tactic: rw [← card k V]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : DivisionRing k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : Finite V
⊢ Nat.card V = Nat.card V - 1 + 1
have this : Nat.card V > 0
Tactic: have : Nat.card V > 0 := Nat.card_pos
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (hypothesis)
Goal:
k : Type u_1
V : Type u_2
inst : Field k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : Finite k
⊢ Nat.card.{u_2} (ℙ k V) = (Nat.card V - 1) / (Nat.card k - 1)
have this : 1 < Nat.card k
Tactic: have : 1 < Nat.card k := Finite.one_lt_card
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Field k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : Finite k
this : 1 < Nat.card k
⊢ Nat.card.{u_2} (ℙ k V) = (Nat.card V - 1) / (Nat.card k - 1)
have H : 0 < Nat.card k - 1
Tactic: rw [card k, Nat.mul_div_cancel]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Field k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : Finite k
n : ℕ
h : Module.finrank k V = n
⊢ Nat.card.{u_2} (ℙ k V) = ∑ i ∈ Finset.range n, Nat.card k ^ i
have inr : ∀
  (this :
    ∀ (k : Type u_1) (V : Type u_2) [inst : Field k] [inst_4 : AddCommGroup V] [inst_5 : Module k V] [Finite k] {n : ℕ}
      (h : Module.finrank k V = n) (hf : Finite V), Nat.card.{u_2} (ℙ k V) = ∑ i ∈ Finset.range n, Nat.card k ^ i)
  (hf : ¬Finite V), Nat.card.{u_2} (ℙ k V) = ∑ i ∈ Finset.range n, Nat.card k ^ i
Tactic: wlog hf : Finite V
-- Lean 4 proof step from Mathlib.LinearAlgebra.Projectivization.Cardinality (goal)
Goal:
k : Type u_1
V : Type u_2
inst : Field k
inst_1 : AddCommGroup V
inst_2 : Module k V
inst_3 : Finite k
n : ℕ
h : Module.finrank k V = n
inr :
  ∀
    (this :
      ∀ (k : Type u_1) (V : Type u_2) [inst : Field k] [inst_4 : AddCommGroup V] [inst_5 : Module k V] [Finite k]
        {n : ℕ} (h : Module.finrank k V = n) (hf : Finite V),
        Nat.card.{u_2} (ℙ k V) = ∑ i ∈ Finset.range n, Nat.card k ^ i)
    (hf : ¬Finite V), Nat.card.{u_2} (ℙ k V) = ∑ i ∈ Finset.range n, Nat.card k ^ i
⊢ Nat.card.{u_2} (ℙ k V) = ∑ i ∈ Finset.range n, Nat.card k ^ i
have wlog : ∀ (k : Type u_1) (V : Type u_2) [inst : Field k] [inst_4 : AddCommGroup V] [inst_5 : Module k V] [Finite k] {n : ℕ}
  (h : Module.finrank k V = n) (hf : Finite V), Nat.card.{u_2} (ℙ k V) = ∑ i ∈ Finset.range n, Nat.card k ^ i
Tactic: wlog hf : Finite V
Cleared hypotheses: inst_3, n, h