Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: case neg
n : ℕ
IH :
  ∀ m < n,
    ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),
      f.Monic →
        g.Monic →
          f.Splits →
            g.Splits →
              g.natDegree ≤ f.natDegree →
                f.natDegree + g.natDegree = m →
                  f.resultant g = (Multiset.map (fun ij ↦ ij.1 - ij.2) (f.roots ×ˢ g.roots)).prod
K : Type u_3
inst✝ : Field K
f g : K[X]
hf : f.Monic
hg : g.Monic
hf' : f.Splits
hg' : g.Splits
hfg : g.natDegree ≤ f.natDegree
hN : f.natDegree + g.natDegree = n
hr✝ : ¬g ∣ f
r : K[X] := C (f %ₘ g).leadingCoeff⁻¹ * (f %ₘ g)
hr₀ : f %ₘ g ≠ 0
hrd : r.natDegree < g.natDegree
hr : r.Monic
L : Type u_3 := r.SplittingField
this :
  (algebraMap K L) (g.resultant (f %ₘ g) g.natDegree r.natDegree) =
    (algebraMap K L) ((f %ₘ g).leadingCoeff ^ g.natDegree) *
      (Multiset.map (fun ij ↦ ij.1 - ij.2) ((map (algebraMap K L) g).roots ×ˢ (map (algebraMap K L) r).roots)).prod
⊢ (algebraMap K L) ((-1) ^ (g.natDegree * (f.natDegree - r.natDegree))) *
        (algebraMap K L) (g.coeff g.natDegree ^ (f.natDegree - r.natDegree)) *
      ((algebraMap K L) ((-1) ^ (r.natDegree * g.natDegree)) *
        ((algebraMap K L) ((f %ₘ g).leadingCoeff ^ g.natDegree) *
          (Multiset.map (fun x ↦ eval x (map (algebraMap K r.SplittingField) r))
              (Multiset.map (⇑(algebraMap K L)) g.roots)).prod)) =
    (algebraMap K L) ((-1) ^ (g.roots.card * f.roots.card)) *
      (algebraMap K L) (Multiset.map (fun x ↦ eval x f) g.roots).prod

Tactic:
have : (g.roots.map (eval · f)).prod =
        (f %ₘ g).leadingCoeff ^ g.natDegree * (g.roots.map (eval · r)).prod := by
      trans (g.roots.map ((f %ₘ g).leadingCoeff * eval · r)).prod
      · congr 1
        refine Multiset.map_congr rfl ?_
        simp only [mem_roots', ne_eq, IsRoot.def, eval_mul, eval_C, leadingCoeff_eq_zero, hr₀,
          not_false_eq_true, mul_inv_cancel_left₀, and_imp, r]
        intro x hx hxg
        conv_lhs => rw [← f.modByMonic_add_div, eval_add, eval_mul, hxg, zero_mul, add_zero]
      · simp [hg'.natDegree_eq_card_roots]

Goal state after:
case neg
n : ℕ
IH :
  ∀ m < n,
    ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),
      f.Monic →
        g.Monic →
          f.Splits →
            g.Splits →
              g.natDegree ≤ f.natDegree →
                f.natDegree + g.natDegree = m →
                  f.resultant g = (Multiset.map (fun ij ↦ ij.1 - ij.2) (f.roots ×ˢ g.roots)).prod
K : Type u_3
inst✝ : Field K
f g : K[X]
hf : f.Monic
hg : g.Monic
hf' : f.Splits
hg' : g.Splits
hfg : g.natDegree ≤ f.natDegree
hN : f.natDegree + g.natDegree = n
hr✝ : ¬g ∣ f
r : K[X] := C (f %ₘ g).leadingCoeff⁻¹ * (f %ₘ g)
hr₀ : f %ₘ g ≠ 0
hrd : r.natDegree < g.natDegree
hr : r.Monic
L : Type u_3 := r.SplittingField
this✝ :
  (algebraMap K L) (g.resultant (f %ₘ g) g.natDegree r.natDegree) =
    (algebraMap K L) ((f %ₘ g).leadingCoeff ^ g.natDegree) *
      (Multiset.map (fun ij ↦ ij.1 - ij.2) ((map (algebraMap K L) g).roots ×ˢ (map (algebraMap K L) r).roots)).prod
this :
  (Multiset.map (fun x ↦ eval x f) g.roots).prod =
    (f %ₘ g).leadingCoeff ^ g.natDegree * (Multiset.map (fun x ↦ eval x r) g.roots).prod
⊢ (algebraMap K L) ((-1) ^ (g.natDegree * (f.natDegree - r.natDegree))) *
        (algebraMap K L) (g.coeff g.natDegree ^ (f.natDegree - r.natDegree)) *
      ((algebraMap K L) ((-1) ^ (r.natDegree * g.natDegree)) *
        ((algebraMap K L) ((f %ₘ g).leadingCoeff ^ g.natDegree) *
          (Multiset.map (fun x ↦ eval x (map (algebraMap K r.SplittingField) r))
              (Multiset.map (⇑(algebraMap K L)) g.roots)).prod)) =
    (algebraMap K L) ((-1) ^ (g.roots.card * f.roots.card)) *
      (algebraMap K L) (Multiset.map (fun x ↦ eval x f) g.roots).prod