/-- 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
    simp only [Function.comp, coe_mk, add_apply]
  map_smul' c f := 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'
c : S
f : MultilinearMap R M₁ M₂
x✝ : (i : ι') → M₁ (σ.symm i)
⊢ (AddHom.toFun
        {
          toFun := 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)) },
          map_add' :=
            (_ :
              ∀ (f₁ f₂ : MultilinearMap R M₁ M₂),
                (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) =
                  (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) }
        (c • f))
      x✝ =
    ((RingHom.id S) c •
        AddHom.toFun
          {
            toFun := 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)) },
            map_add' :=
              (_ :
                ∀ (f₁ f₂ : MultilinearMap R M₁ M₂),
                  (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) =
                    (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) }
          f)
      x✝

Next tactic:
simp only [Function.comp, coe_mk, smul_apply, RingHom.id_apply]