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[Merged by Bors] - perf: speed up galActionHom_bijective_of_prime_degree #10608

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[Merged by Bors] - perf: speed up galActionHom_bijective_of_prime_degree #10608

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7 changes: 3 additions & 4 deletions Mathlib/Analysis/Complex/Polynomial.lean
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Expand Up @@ -128,8 +128,6 @@ theorem galActionHom_bijective_of_prime_degree {p : ℚ[X]} (p_irr : Irreducible
rw [Multiset.toFinset_card_of_nodup, ← natDegree_eq_card_roots]
· exact IsAlgClosed.splits_codomain p
· exact nodup_roots ((separable_map (algebraMap ℚ ℂ)).mpr p_irr.separable)
have h2 : Fintype.card p.Gal = Fintype.card (galActionHom p ℂ).range :=
Fintype.card_congr (MonoidHom.ofInjective (galActionHom_injective p ℂ)).toEquiv
let conj' := restrict p ℂ (Complex.conjAe.restrictScalars ℚ)
refine'
⟨galActionHom_injective p ℂ, fun x =>
Expand All @@ -138,8 +136,9 @@ theorem galActionHom_bijective_of_prime_degree {p : ℚ[X]} (p_irr : Irreducible
apply Equiv.Perm.subgroup_eq_top_of_swap_mem
· rwa [h1]
· rw [h1]
convert prime_degree_dvd_card p_irr p_deg using 1
convert h2.symm
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
· exact ⟨conj', rfl⟩
· rw [← Equiv.Perm.card_support_eq_two]
apply Nat.add_left_cancel
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5 changes: 5 additions & 0 deletions Mathlib/SetTheory/Cardinal/Finite.lean
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Expand Up @@ -42,6 +42,11 @@ theorem card_eq_fintype_card [Fintype α] : Nat.card α = Fintype.card α :=
mk_toNat_eq_card
#align nat.card_eq_fintype_card Nat.card_eq_fintype_card

/-- Because this theorem takes `Fintype α` as a non-instance argument, it can be used in particular
when `Fintype.card` ends up with different instance than the one found by inference -/
theorem _root_.Fintype.card_eq_nat_card {_ : Fintype α} : Fintype.card α = Nat.card α :=
mk_toNat_eq_card.symm

lemma card_eq_finsetCard (s : Finset α) : Nat.card s = s.card := by
simp only [Nat.card_eq_fintype_card, Fintype.card_coe]

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