/-- Swapping elements in `σ` with equal values in `v` result in zero if the swap has no effect
on the quotient. -/
theorem domCoprod.summand_eq_zero_of_smul_invariant (a : Mᵢ [Λ^ιa]→ₗ[R'] N₁)
    (b : Mᵢ [Λ^ιb]→ₗ[R'] N₂) (σ : Perm.ModSumCongr ιa ιb) {v : Sum ιa ιb → Mᵢ}
    {i j : Sum ιa ιb} (hv : v i = v j) (hij : i ≠ j) :
    swap i j • σ = σ → domCoprod.summand a b σ v = 0 := by
  refine Quotient.inductionOn' σ fun σ => ?_
  dsimp only [Quotient.liftOn'_mk'', Quotient.map'_mk'', MultilinearMap.smul_apply,
    MultilinearMap.domDomCongr_apply, MultilinearMap.domCoprod_apply, domCoprod.summand]
  intro hσ
  cases' hi : σ⁻¹ i with val val <;> cases' hj : σ⁻¹ j with val_1 val_1 <;>
    rw [Perm.inv_eq_iff_eq] at hi hj <;> substs hi hj <;> revert val val_1
  -- Porting note: `on_goal` is not available in `case _ | _ =>`, so the proof gets tedious.
  -- the term pairs with and cancels another term
  case inl.inr =>
    intro i' j' _ _ hσ
    obtain ⟨⟨sl, sr⟩, hσ⟩ := QuotientGroup.leftRel_apply.mp (Quotient.exact' hσ)
    replace hσ := Equiv.congr_fun hσ (Sum.inl i')
    dsimp only at hσ
    rw [smul_eq_mul, ← mul_swap_eq_swap_mul, mul_inv_rev, swap_inv, inv_mul_cancel_right] at hσ
    simp at hσ
  

Proof state:
case inl.inl
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 : Mᵢ [Λ^ιa]→ₗ[R'] N₁
b : Mᵢ [Λ^ιb]→ₗ[R'] N₂
σ✝ : Perm.ModSumCongr ιa ιb
v : ιa ⊕ ιb → Mᵢ
σ : Perm (ιa ⊕ ιb)
⊢ ∀ (val val_1 : ιa),
    v (σ (Sum.inl val)) = v (σ (Sum.inl val_1)) →
      σ (Sum.inl val) ≠ σ (Sum.inl val_1) →
        swap (σ (Sum.inl val)) (σ (Sum.inl val_1)) • Quotient.mk'' σ = Quotient.mk'' σ →
          (Perm.sign σ • (↑a fun i => v (σ (Sum.inl i))) ⊗ₜ[R'] ↑b fun i => v (σ (Sum.inr i))) = 0
case inr.inl
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 : Mᵢ [Λ^ιa]→ₗ[R'] N₁
b : Mᵢ [Λ^ιb]→ₗ[R'] N₂
σ✝ : Perm.ModSumCongr ιa ιb
v : ιa ⊕ ιb → Mᵢ
σ : Perm (ιa ⊕ ιb)
⊢ ∀ (val : ιb) (val_1 : ιa),
    v (σ (Sum.inr val)) = v (σ (Sum.inl val_1)) →
      σ (Sum.inr val) ≠ σ (Sum.inl val_1) →
        swap (σ (Sum.inr val)) (σ (Sum.inl val_1)) • Quotient.mk'' σ = Quotient.mk'' σ →
          (Perm.sign σ • (↑a fun i => v (σ (Sum.inl i))) ⊗ₜ[R'] ↑b fun i => v (σ (Sum.inr i))) = 0
case inr.inr
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 : Mᵢ [Λ^ιa]→ₗ[R'] N₁
b : Mᵢ [Λ^ιb]→ₗ[R'] N₂
σ✝ : Perm.ModSumCongr ιa ιb
v : ιa ⊕ ιb → Mᵢ
σ : Perm (ιa ⊕ ιb)
⊢ ∀ (val val_1 : ιb),
    v (σ (Sum.inr val)) = v (σ (Sum.inr val_1)) →
      σ (Sum.inr val) ≠ σ (Sum.inr val_1) →
        swap (σ (Sum.inr val)) (σ (Sum.inr val_1)) • Quotient.mk'' σ = Quotient.mk'' σ →
          (Perm.sign σ • (↑a fun i => v (σ (Sum.inl i))) ⊗ₜ[R'] ↑b fun i => v (σ (Sum.inr i))) = 0

Next tactic:
case inr.inl =>
    intro i' j' _ _ hσ
    obtain ⟨⟨sl, sr⟩, hσ⟩ := QuotientGroup.leftRel_apply.mp (Quotient.exact' hσ)
    replace hσ := Equiv.congr_fun hσ (Sum.inr i')
    dsimp only at hσ
    rw [smul_eq_mul, ← mul_swap_eq_swap_mul, mul_inv_rev, swap_inv, inv_mul_cancel_right] at hσ
    simp at hσ