Theorem: Polynomial.Gal.galActionHom_bijective_of_prime_degree
File: Mathlib/Analysis/Complex/Polynomial.lean

tactic: let conj' := restrict p ℂ (Complex.conjAe.restrictScalars ℚ)
state_before: p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
⊢ Function.Bijective ⇑(galActionHom p ℂ)
state_after: p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
⊢ Function.Bijective ⇑(galActionHom p ℂ)

tactic: refine
  ⟨galActionHom_injective p ℂ, fun x =>
    (congr_arg (Membership.mem x) (show (galActionHom p ℂ).range = ⊤ from ?_)).mpr
      (Subgroup.mem_top x)⟩
state_before: p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
⊢ Function.Bijective ⇑(galActionHom p ℂ)
state_after: p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ (galActionHom p ℂ).range = ⊤

tactic: apply Equiv.Perm.subgroup_eq_top_of_swap_mem
state_before: p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ (galActionHom p ℂ).range = ⊤
state_after: case h0
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ Nat.Prime (Fintype.card ↑(p.rootSet ℂ))

case h1
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ Fintype.card ↑(p.rootSet ℂ) ∣ Fintype.card ↥(galActionHom p ℂ).range

case h2
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ ?τ ∈ (galActionHom p ℂ).range

case h3
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ ?τ.IsSwap

case τ
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ Equiv.Perm ↑(p.rootSet ℂ)

tactic: simp_rw [rootSet_def, Finset.coe_sort_coe, Fintype.card_coe]
state_before: p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
⊢ Fintype.card ↑(p.rootSet ℂ) = p.natDegree
state_after: p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
⊢ (p.aroots ℂ).toFinset.card = p.natDegree

tactic: rw [Multiset.toFinset_card_of_nodup, ← natDegree_eq_card_roots]
state_before: p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
⊢ (p.aroots ℂ).toFinset.card = p.natDegree
state_after: p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
⊢ Splits (algebraMap ℚ ℂ) p

p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
⊢ (p.aroots ℂ).Nodup

tactic: exact IsAlgClosed.splits_codomain p
state_before: p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
⊢ Splits (algebraMap ℚ ℂ) p
state_after: no goals

tactic: exact nodup_roots ((separable_map (algebraMap ℚ ℂ)).mpr p_irr.separable)
state_before: p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
⊢ (p.aroots ℂ).Nodup
state_after: no goals

tactic: rwa [h1]
state_before: case h0
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ Nat.Prime (Fintype.card ↑(p.rootSet ℂ))
state_after: no goals

tactic: rw [h1]
state_before: case h1
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ Fintype.card ↑(p.rootSet ℂ) ∣ Fintype.card ↥(galActionHom p ℂ).range
state_after: case h1
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ p.natDegree ∣ Fintype.card ↥(galActionHom p ℂ).range

tactic: simpa only [Fintype.card_eq_nat_card,
  Nat.card_congr (MonoidHom.ofInjective (galActionHom_injective p ℂ)).toEquiv.symm]
  using prime_degree_dvd_card p_irr p_deg
state_before: case h1
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ p.natDegree ∣ Fintype.card ↥(galActionHom p ℂ).range
state_after: no goals

tactic: exact ⟨conj', rfl⟩
state_before: case h2
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ ?τ ∈ (galActionHom p ℂ).range
state_after: no goals

tactic: rw [← Equiv.Perm.card_support_eq_two]
state_before: case h3
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ ((galActionHom p ℂ) conj').IsSwap
state_after: case h3
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ ((galActionHom p ℂ) conj').support.card = 2

tactic: apply Nat.add_left_cancel
state_before: case h3
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ ((galActionHom p ℂ) conj').support.card = 2
state_after: case h3.a
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ ?h3.n + ((galActionHom p ℂ) conj').support.card = ?h3.n + 2

case h3.n
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ ℕ

tactic: rw [← p_roots, ← Set.toFinset_card (rootSet p ℝ), ← Set.toFinset_card (rootSet p ℂ)]
state_before: case h3.a
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ ?h3.n + ((galActionHom p ℂ) conj').support.card = ?h3.n + 2

case h3.n
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ ℕ
state_after: case h3.a
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ (p.rootSet ℝ).toFinset.card + ((galActionHom p ℂ) conj').support.card = (p.rootSet ℂ).toFinset.card

tactic: exact (card_complex_roots_eq_card_real_add_card_not_gal_inv p).symm
state_before: case h3.a
p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
h1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree
conj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)
x : Equiv.Perm ↑(p.rootSet ℂ)
⊢ (p.rootSet ℝ).toFinset.card + ((galActionHom p ℂ) conj').support.card = (p.rootSet ℂ).toFinset.card
state_after: no goals