-- Mathlib/Analysis/Complex/Polynomial/Basic.lean
theorem card_complex_roots_eq_card_real_add_card_not_gal_inv (p : ℚ[X]) :
    (p.rootSet ℂ).toFinset.card =
      (p.rootSet ℝ).toFinset.card +
        (galActionHom p ℂ (restrict p ℂ
        (AlgEquiv.restrictScalars ℚ Complex.conjAe))).support.card

Proof state:
case neg.refine_2.refine_2.refine_2
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
hb : ∀ (z : ℂ), z ∈ b ↔ (aeval z) p = 0 ∧ z.im = 0
⊢ a.card = b.card + c.card

Tactic:
have hc0 :
  ∀ w : p.rootSet ℂ, galActionHom p ℂ (restrict p ℂ (Complex.conjAe.restrictScalars ℚ)) w = w ↔
      w.val.im = 0 := by
  intro w
  rw [Subtype.ext_iff, galActionHom_restrict]
  exact Complex.conj_eq_iff_im

Result:
case neg.refine_2.refine_2.refine_2
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
hb : ∀ (z : ℂ), z ∈ b ↔ (aeval z) p = 0 ∧ z.im = 0
hc0 :
  ∀ (w : ↑(p.rootSet ℂ)), ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))) w = w ↔ (↑w).im = 0
⊢ a.card = b.card + c.card

Proof state:
case a
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
z : ℂ
hz1 : (aeval z) p = 0
hz2 : z.im = 0
⊢ ((IsScalarTower.toAlgHom ℚ ℝ ℂ) z.re).im = 0

Tactic:
rfl

Result:
no goals

Proof state:
p : ℚ[X]
hp : p = 0
⊢ IsEmpty ↑(p.rootSet ℂ)

Tactic:
rw [hp, rootSet_zero]

Result:
p : ℚ[X]
hp : p = 0
⊢ IsEmpty ↑∅

Proof state:
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
z : ℂ
⊢ z ∈ a ↔ (aeval z) p = 0

Tactic:
rw [Set.mem_toFinset, mem_rootSet_of_ne hp]

Result:
no goals

Proof state:
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
⊢ ∀ (z : ℂ), z ∈ b ↔ (aeval z) p = 0 ∧ z.im = 0

Tactic:
intro z

Result:
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
z : ℂ
⊢ z ∈ b ↔ (aeval z) p = 0 ∧ z.im = 0

Proof state:
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
z : ℂ
hz1 : (aeval z) p = 0
hz2 : z.im = 0
⊢ (IsScalarTower.toAlgHom ℚ ℝ ℂ) z.re = z

Tactic:
ext

Result:
case a
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
z : ℂ
hz1 : (aeval z) p = 0
hz2 : z.im = 0
⊢ ((IsScalarTower.toAlgHom ℚ ℝ ℂ) z.re).re = z.re

case a
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
z : ℂ
hz1 : (aeval z) p = 0
hz2 : z.im = 0
⊢ ((IsScalarTower.toAlgHom ℚ ℝ ℂ) z.re).im = z.im

Proof state:
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
hb : ∀ (z : ℂ), z ∈ b ↔ (aeval z) p = 0 ∧ z.im = 0
hc0 :
  ∀ (w : ↑(p.rootSet ℂ)), ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))) w = w ↔ (↑w).im = 0
z : ℂ
⊢ (∃ a ∈ ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support, ↑a = z) ↔
    (aeval z) p = 0 ∧ z.im ≠ 0

Tactic:
constructor

Result:
case mp
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
hb : ∀ (z : ℂ), z ∈ b ↔ (aeval z) p = 0 ∧ z.im = 0
hc0 :
  ∀ (w : ↑(p.rootSet ℂ)), ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))) w = w ↔ (↑w).im = 0
z : ℂ
⊢ (∃ a ∈ ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support, ↑a = z) →
    (aeval z) p = 0 ∧ z.im ≠ 0

case mpr
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
hb : ∀ (z : ℂ), z ∈ b ↔ (aeval z) p = 0 ∧ z.im = 0
hc0 :
  ∀ (w : ↑(p.rootSet ℂ)), ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))) w = w ↔ (↑w).im = 0
z : ℂ
⊢ (aeval z) p = 0 ∧ z.im ≠ 0 →
    ∃ a ∈ ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support, ↑a = z

Proof state:
case pos
p : ℚ[X]
hp : p = 0
this : IsEmpty ↑(p.rootSet ℂ)
⊢ (p.rootSet ℂ).toFinset.card =
    (p.rootSet ℝ).toFinset.card + ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support.card

Tactic:
simp_rw [(galActionHom p ℂ _).support.eq_empty_of_isEmpty, hp, rootSet_zero,
  Set.toFinset_empty, Finset.card_empty]

Result:
no goals