-- Module: Mathlib.RingTheory.Algebraic.MvPolynomial
Goal:
σ : Type u_1
R : Type u_2
inst : CommRing R
i : σ
s : Set σ
h : i ∉ s
f : R[X]
g : MvPolynomial.{u_1, u_2} (↑s : Type u_1) R[X] →ₐ[R] MvPolynomial.{u_1, u_2} (↑s : Type u_1) R[X] :=
  mapAlgHom (Polynomial.aeval f)
u : (MvPolynomial.{u_1, u_2} (↑s : Type u_1) R)[X] →ₐ[R] MvPolynomial σ R :=
  (AlgHom.comp
        (AlgHom.comp (Subalgebra.val (supported R (insert i s)))
          (↑(AlgEquiv.trans
                (AlgEquiv.trans (AlgEquiv.symm (optionEquivRight R (↑s : Type u_1)))
                  (renameEquiv R (Set.subtypeInsertEquivOption (t := s) h).symm))
                (AlgEquiv.symm (supportedEquivMvPolynomial (insert i s)))) :
            MvPolynomial (↑s : Type u_1) R[X] →ₐ[R] ↥(supported R (insert i s))))
        g).comp
    (↑(AlgEquiv.trans (AlgEquiv.symm (optionEquivLeft R (↑s : Type u_1))) (optionEquivRight R (↑s : Type u_1))) :
      (MvPolynomial (↑s : Type u_1) R)[X] →ₐ[R] MvPolynomial (↑s : Type u_1) R[X])
v : (MvPolynomial.{u_1, u_2} (↑s : Type u_1) R)[X] →ₐ[R] MvPolynomial σ R :=
  AlgHom.comp
    (AlgHom.restrictScalars R (S := ↥(supported R s))
      (Polynomial.aeval ((Polynomial.aeval (X i) : (a : R[X]) → MvPolynomial σ R) f)))
    (↑(Polynomial.mapAlgEquiv (AlgEquiv.symm (supportedEquivMvPolynomial s))) :
      (MvPolynomial (↑s : Type u_1) R)[X] →ₐ[R] (↥(supported R s))[X])
hf :
  Function.Injective (α := (MvPolynomial.{u_1, u_2} (↑s : Type u_1) R)[X]) (β := MvPolynomial σ R)
    (⇑u : (a : (MvPolynomial (↑s : Type u_1) R)[X]) → MvPolynomial σ R)
h1 :
  Eq (α := R[X] →ₐ[R] MvPolynomial σ R) (Polynomial.aeval (X i))
    (AlgHom.comp
      (AlgHom.comp (Subalgebra.val (supported R (insert i s)))
        (↑(AlgEquiv.symm (supportedEquivMvPolynomial (insert i s))) :
          MvPolynomial (↑(insert i s) : Type u_1) R →ₐ[R] ↥(supported R (insert i s))))
      (Polynomial.aeval (X (Subtype.mk (p := fun (x : σ) ↦ x ∈ insert i s) i ⋯))))
⊢ Eq (α := MvPolynomial σ R)
    (((AlgHom.comp
                (AlgHom.comp (Subalgebra.val (supported R (insert i s)))
                  (↑(AlgEquiv.trans
                        (AlgEquiv.trans (AlgEquiv.symm (optionEquivRight R (↑s : Type u_1)))
                          (renameEquiv R (Set.subtypeInsertEquivOption (t := s) h).symm))
                        (AlgEquiv.symm (supportedEquivMvPolynomial (insert i s)))) :
                    MvPolynomial (↑s : Type u_1) R[X] →ₐ[R] ↥(supported R (insert i s))))
                (mapAlgHom (Polynomial.aeval f))).comp
            (↑(AlgEquiv.trans (AlgEquiv.symm (optionEquivLeft R (↑s : Type u_1)))
                  (optionEquivRight R (↑s : Type u_1))) :
              (MvPolynomial (↑s : Type u_1) R)[X] →ₐ[R] MvPolynomial (↑s : Type u_1) R[X]) :
          (a : (MvPolynomial (↑s : Type u_1) R)[X]) → MvPolynomial σ R)
        Polynomial.X :
      MvPolynomial σ R)
    (((AlgHom.comp
              (AlgHom.restrictScalars R (S := ↥(supported R s))
                (Polynomial.aeval ((Polynomial.aeval (X i) : (a : R[X]) → MvPolynomial σ R) f)))
              (↑(Polynomial.mapAlgEquiv (AlgEquiv.symm (supportedEquivMvPolynomial s))) :
                (MvPolynomial (↑s : Type u_1) R)[X] →ₐ[R] (↥(supported R s))[X]) :
            (MvPolynomial.{u_1, u_2} (↑s : Type u_1) R)[X] →ₐ[R] MvPolynomial σ R) :
          (a : (MvPolynomial (↑s : Type u_1) R)[X]) → MvPolynomial σ R)
        Polynomial.X :
      MvPolynomial σ R)

Tactic:
simp [Set.subtypeInsertEquivOption, h1, optionEquivLeft_symm_apply]

Premises used: MvPolynomial.optionEquivLeft_symm_apply, Set.subtypeInsertEquivOption