Module: Mathlib.FieldTheory.IsAlgClosed.Classification

Goal state before:
pp: case refine_1
R : Type u_1
L : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : Field L
inst✝² : Algebra R L
ι : Type u_4
v : ι → K
κ : Type u_5
w : κ → L
hv✝ : AlgebraicIndependent R v
hw✝ : AlgebraicIndependent R w
inst✝¹ : IsAlgClosed K
inst✝ : IsAlgClosed L
e : ι ≃ κ
hv : IsTranscendenceBasis R v
hw : IsTranscendenceBasis R w
this✝ : IsAlgClosure (↥(Algebra.adjoin R (Set.range v))) K := isAlgClosure_of_transcendence_basis v hv
this : IsAlgClosure (↥(Algebra.adjoin R (Set.range w))) L := isAlgClosure_of_transcendence_basis w hw
⊢ (MvPolynomial.rename ⇑e).comp (MvPolynomial.rename ⇑e.symm) = AlgHom.id R (MvPolynomial κ R)

pp: case refine_2
R : Type u_1
L : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : Field L
inst✝² : Algebra R L
ι : Type u_4
v : ι → K
κ : Type u_5
w : κ → L
hv✝ : AlgebraicIndependent R v
hw✝ : AlgebraicIndependent R w
inst✝¹ : IsAlgClosed K
inst✝ : IsAlgClosed L
e : ι ≃ κ
hv : IsTranscendenceBasis R v
hw : IsTranscendenceBasis R w
this✝ : IsAlgClosure (↥(Algebra.adjoin R (Set.range v))) K := isAlgClosure_of_transcendence_basis v hv
this : IsAlgClosure (↥(Algebra.adjoin R (Set.range w))) L := isAlgClosure_of_transcendence_basis w hw
⊢ (MvPolynomial.rename ⇑e.symm).comp (MvPolynomial.rename ⇑e) = AlgHom.id R (MvPolynomial ι R)

pp: case refine_3
R : Type u_1
L : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : Field L
inst✝² : Algebra R L
ι : Type u_4
v : ι → K
κ : Type u_5
w : κ → L
hv✝ : AlgebraicIndependent R v
hw✝ : AlgebraicIndependent R w
inst✝¹ : IsAlgClosed K
inst✝ : IsAlgClosed L
e : ι ≃ κ
hv : IsTranscendenceBasis R v
hw : IsTranscendenceBasis R w
this✝ : IsAlgClosure (↥(Algebra.adjoin R (Set.range v))) K := isAlgClosure_of_transcendence_basis v hv
this : IsAlgClosure (↥(Algebra.adjoin R (Set.range w))) L := isAlgClosure_of_transcendence_basis w hw
⊢ MvPolynomial κ R ≃+* ↥(Algebra.adjoin R (Set.range w))

Tactic:
· ext; simp

Goal state after:
case refine_2
R : Type u_1
L : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : Field L
inst✝² : Algebra R L
ι : Type u_4
v : ι → K
κ : Type u_5
w : κ → L
hv✝ : AlgebraicIndependent R v
hw✝ : AlgebraicIndependent R w
inst✝¹ : IsAlgClosed K
inst✝ : IsAlgClosed L
e : ι ≃ κ
hv : IsTranscendenceBasis R v
hw : IsTranscendenceBasis R w
this✝ : IsAlgClosure (↥(Algebra.adjoin R (Set.range v))) K := isAlgClosure_of_transcendence_basis v hv
this : IsAlgClosure (↥(Algebra.adjoin R (Set.range w))) L := isAlgClosure_of_transcendence_basis w hw
⊢ (MvPolynomial.rename ⇑e.symm).comp (MvPolynomial.rename ⇑e) = AlgHom.id R (MvPolynomial ι R)

case refine_3
R : Type u_1
L : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : Field L
inst✝² : Algebra R L
ι : Type u_4
v : ι → K
κ : Type u_5
w : κ → L
hv✝ : AlgebraicIndependent R v
hw✝ : AlgebraicIndependent R w
inst✝¹ : IsAlgClosed K
inst✝ : IsAlgClosed L
e : ι ≃ κ
hv : IsTranscendenceBasis R v
hw : IsTranscendenceBasis R w
this✝ : IsAlgClosure (↥(Algebra.adjoin R (Set.range v))) K := isAlgClosure_of_transcendence_basis v hv
this : IsAlgClosure (↥(Algebra.adjoin R (Set.range w))) L := isAlgClosure_of_transcendence_basis w hw
⊢ MvPolynomial κ R ≃+* ↥(Algebra.adjoin R (Set.range w))