/-- The dependent version of `MultilinearMap.domDomCongrLinearEquiv`. -/
@[simps apply symm_apply]
def domDomCongrLinearEquiv' {ι' : Type*} (σ : ι ≃ ι') :
    MultilinearMap R M₁ M₂ ≃ₗ[S] MultilinearMap R (fun i => M₁ (σ.symm i)) M₂ where
  toFun f :=
    { toFun := f ∘ (σ.piCongrLeft' M₁).symm
      map_add' := fun m i => by
        letI := σ.decidableEq
        rw [← σ.apply_symm_apply i]
        intro x y
        simp only [comp_apply, piCongrLeft'_symm_update, f.map_add]
      map_smul' := fun m i c => by
        letI := σ.decidableEq
        rw [← σ.apply_symm_apply i]
        intro x
        simp only [Function.comp, piCongrLeft'_symm_update, f.map_smul] }
  invFun f :=
    { toFun := f ∘ σ.piCongrLeft' M₁
      map_add' := fun m i => by
        letI := σ.symm.decidableEq
        rw [← σ.symm_apply_apply i]
        intro x y
        simp only [comp_apply, piCongrLeft'_update, f.map_add]
      map_smul' := fun m i c => by
        letI := σ.symm.decidableEq
        rw [← σ.symm_apply_apply i]
        intro x
        simp only [Function.comp, piCongrLeft'_update, f.map_smul] }
  map_add' f₁ f₂ := by
    ext
    

Proof state:
case H
R : Type uR
S : Type uS
ι : Type uι
n : ℕ
M : Fin (Nat.succ n) → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝⁸ : Semiring R
inst✝⁷ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁶ : (i : ι) → Module R (M₁ i)
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Semiring S
inst✝² : Module S M₂
inst✝¹ : Module R M₂
inst✝ : SMulCommClass R S M₂
ι' : Type u_1
σ : ι ≃ ι'
f₁ f₂ : MultilinearMap R M₁ M₂
x✝ : (i : ι') → M₁ (σ.symm i)
⊢ ((fun f =>
          { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm,
            map_add' :=
              (_ :
                ∀ [inst : DecidableEq ι'] (m : (i : ι') → M₁ (σ.symm i)) (i : ι') (x y : M₁ (σ.symm i)),
                  (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i (x + y)) =
                    (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i x) +
                      (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i y)),
            map_smul' :=
              (_ :
                ∀ [inst : DecidableEq ι'] (m : (i : ι') → M₁ (σ.symm i)) (i : ι') (c : R) (x : M₁ (σ.symm i)),
                  (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i (c • x)) =
                    c • (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i x)) })
        (f₁ + f₂))
      x✝ =
    ((fun f =>
            { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm,
              map_add' :=
                (_ :
                  ∀ [inst : DecidableEq ι'] (m : (i : ι') → M₁ (σ.symm i)) (i : ι') (x y : M₁ (σ.symm i)),
                    (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i (x + y)) =
                      (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i x) +
                        (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i y)),
              map_smul' :=
                (_ :
                  ∀ [inst : DecidableEq ι'] (m : (i : ι') → M₁ (σ.symm i)) (i : ι') (c : R) (x : M₁ (σ.symm i)),
                    (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i (c • x)) =
                      c • (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i x)) })
          f₁ +
        (fun f =>
            { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm,
              map_add' :=
                (_ :
                  ∀ [inst : DecidableEq ι'] (m : (i : ι') → M₁ (σ.symm i)) (i : ι') (x y : M₁ (σ.symm i)),
                    (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i (x + y)) =
                      (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i x) +
                        (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i y)),
              map_smul' :=
                (_ :
                  ∀ [inst : DecidableEq ι'] (m : (i : ι') → M₁ (σ.symm i)) (i : ι') (c : R) (x : M₁ (σ.symm i)),
                    (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i (c • x)) =
                      c • (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm) (update m i x)) })
          f₂)
      x✝

Next tactic:
simp only [Function.comp, coe_mk, add_apply]