/-- Computing the `MultilinearMap.alternatization` of the `MultilinearMap.domCoprod` is the same
as computing the `AlternatingMap.domCoprod` of the `MultilinearMap.alternatization`s.
-/
theorem MultilinearMap.domCoprod_alternization [DecidableEq ιa] [DecidableEq ιb]
    (a : MultilinearMap R' (fun _ : ιa => Mᵢ) N₁) (b : MultilinearMap R' (fun _ : ιb => Mᵢ) N₂) :
    MultilinearMap.alternatization (MultilinearMap.domCoprod a b) =
      a.alternatization.domCoprod (MultilinearMap.alternatization b) := by
  apply coe_multilinearMap_injective
  rw [domCoprod_coe, MultilinearMap.alternatization_coe,
    Finset.sum_partition (QuotientGroup.leftRel (Perm.sumCongrHom ιa ιb).range)]
  congr 1
  ext1 σ
  refine Quotient.inductionOn' σ fun σ => ?_
  -- unfold the quotient mess left by `Finset.sum_partition`
  -- Porting note: Was `conv in .. => ..`.
  erw
    [@Finset.filter_congr _ _ (fun a => @Quotient.decidableEq _ _
      (QuotientGroup.leftRelDecidable (MonoidHom.range (Perm.sumCongrHom ιa ιb)))
      (Quotient.mk (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb))) a)
      (Quotient.mk'' σ)) _ (s := Finset.univ)
    fun x _ => QuotientGroup.eq' (s := MonoidHom.range (Perm.sumCongrHom ιa ιb)) (a := x) (b := σ)]
  -- eliminate a multiplication
  rw [← Finset.map_univ_equiv (Equiv.mulLeft σ), Finset.filter_map, Finset.sum_map]
  simp_rw [Equiv.coe_toEmbedding, Equiv.coe_mulLeft, (· ∘ ·), mul_inv_rev, inv_mul_cancel_right,
    Subgroup.inv_mem_iff, MonoidHom.mem_range, Finset.univ_filter_exists,
    Finset.sum_image (Perm.sumCongrHom_injective.injOn _)]
  -- now we're ready to clean up the RHS, pulling out the summation
  rw [domCoprod.summand_mk'', MultilinearMap.domCoprod_alternization_coe, ← Finset.sum_product',
    Finset.univ_product_univ, ← MultilinearMap.domDomCongrEquiv_apply, _root_.map_sum,
    Finset.smul_sum]
  congr 1
  ext1 ⟨al, ar⟩
  dsimp only
  -- pull out the pair of smuls on the RHS, by rewriting to `_ →ₗ[ℤ] _` and back
  rw [← AddEquiv.coe_toAddMonoidHom, ← AddMonoidHom.coe_toIntLinearMap, LinearMap.map_smul_of_tower,
    LinearMap.map_smul_of_tower, AddMonoidHom.coe_toIntLinearMap, AddEquiv.coe_toAddMonoidHom,
    MultilinearMap.domDomCongrEquiv_apply]
  -- pick up the pieces
  

Proof state:
case a.e_f.h.e_f.h.mk
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : MultilinearMap R' (fun x => Mᵢ) N₁
b : MultilinearMap R' (fun x => Mᵢ) N₂
σ✝ : Quotient (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb)))
σ : Perm (ιa ⊕ ιb)
al : Perm ιa
ar : Perm ιb
⊢ Perm.sign (σ * (Perm.sumCongrHom ιa ιb) (al, ar)) •
      domDomCongr (σ * (Perm.sumCongrHom ιa ιb) (al, ar)) (domCoprod a b) =
    Perm.sign σ • Perm.sign al • Perm.sign ar • domDomCongr σ (domCoprod (domDomCongr al a) (domDomCongr ar b))

Next tactic:
rw [MultilinearMap.domDomCongr_mul, Perm.sign_mul, Perm.sumCongrHom_apply,
    MultilinearMap.domCoprod_domDomCongr_sumCongr, Perm.sign_sumCongr, mul_smul, mul_smul]