Module: Mathlib.RingTheory.Trace.Basic

Goal state before:
pp: K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
E : Type u_7
inst✝¹ : Field E
inst✝ : Algebra K E
pb : PowerBasis K L
hE : (Polynomial.map (algebraMap K E) (minpoly K pb.gen)).Splits
hfx : IsSeparable K pb.gen
this✝ : DecidableEq E := Classical.decEq E
this : Fintype (L →ₐ[K] E) := PowerBasis.AlgHom.fintype pb
⊢ ((minpoly K pb.gen).aroots E).sum = ∑ x ∈ ((minpoly K pb.gen).aroots E).toFinset, ?g x

pp: case f
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
E : Type u_7
inst✝¹ : Field E
inst✝ : Algebra K E
pb : PowerBasis K L
hE : (Polynomial.map (algebraMap K E) (minpoly K pb.gen)).Splits
hfx : IsSeparable K pb.gen
this✝ : DecidableEq E := ⋯
this : Fintype (L →ₐ[K] E) := ⋯
⊢ { x // x ∈ (minpoly K pb.gen).aroots E } → E

pp: case g
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
E : Type u_7
inst✝¹ : Field E
inst✝ : Algebra K E
pb : PowerBasis K L
hE : (Polynomial.map (algebraMap K E) (minpoly K pb.gen)).Splits
hfx : IsSeparable K pb.gen
this✝ : DecidableEq E := ⋯
this : Fintype (L →ₐ[K] E) := ⋯
⊢ E → E

pp: case hfg
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
E : Type u_7
inst✝¹ : Field E
inst✝ : Algebra K E
pb : PowerBasis K L
hE : (Polynomial.map (algebraMap K E) (minpoly K pb.gen)).Splits
hfx : IsSeparable K pb.gen
this✝ : DecidableEq E := Classical.decEq E
this : Fintype (L →ₐ[K] E) := PowerBasis.AlgHom.fintype pb
⊢ ∀ (x : { x // x ∈ (minpoly K pb.gen).aroots E }), ?f x = ?g ↑x

pp: case h
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
E : Type u_7
inst✝¹ : Field E
inst✝ : Algebra K E
pb : PowerBasis K L
hE : (Polynomial.map (algebraMap K E) (minpoly K pb.gen)).Splits
hfx : IsSeparable K pb.gen
this✝ : DecidableEq E := Classical.decEq E
this : Fintype (L →ₐ[K] E) := PowerBasis.AlgHom.fintype pb
⊢ ∀ (x : L →ₐ[K] E), x pb.gen = ?f (pb.liftEquiv' x)

Tactic:
Finset.sum_eq_multiset_sum,

Goal state after:
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
E : Type u_7
inst✝¹ : Field E
inst✝ : Algebra K E
pb : PowerBasis K L
hE : (Polynomial.map (algebraMap K E) (minpoly K pb.gen)).Splits
hfx : IsSeparable K pb.gen
this✝ : DecidableEq E := Classical.decEq E
this : Fintype (L →ₐ[K] E) := PowerBasis.AlgHom.fintype pb
⊢ ((minpoly K pb.gen).aroots E).sum = (Multiset.map ?f ((minpoly K pb.gen).aroots E).toFinset.val).sum

case f
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
E : Type u_7
inst✝¹ : Field E
inst✝ : Algebra K E
pb : PowerBasis K L
hE : (Polynomial.map (algebraMap K E) (minpoly K pb.gen)).Splits
hfx : IsSeparable K pb.gen
this✝ : DecidableEq E := ⋯
this : Fintype (L →ₐ[K] E) := ⋯
⊢ E → E

case f
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
E : Type u_7
inst✝¹ : Field E
inst✝ : Algebra K E
pb : PowerBasis K L
hE : (Polynomial.map (algebraMap K E) (minpoly K pb.gen)).Splits
hfx : IsSeparable K pb.gen
this✝ : DecidableEq E := ⋯
this : Fintype (L →ₐ[K] E) := ⋯
⊢ { x // x ∈ (minpoly K pb.gen).aroots E } → E

case hfg
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
E : Type u_7
inst✝¹ : Field E
inst✝ : Algebra K E
pb : PowerBasis K L
hE : (Polynomial.map (algebraMap K E) (minpoly K pb.gen)).Splits
hfx : IsSeparable K pb.gen
this✝ : DecidableEq E := Classical.decEq E
this : Fintype (L →ₐ[K] E) := PowerBasis.AlgHom.fintype pb
⊢ ∀ (x : { x // x ∈ (minpoly K pb.gen).aroots E }), ?f✝ x = ?f ↑x

case h
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
E : Type u_7
inst✝¹ : Field E
inst✝ : Algebra K E
pb : PowerBasis K L
hE : (Polynomial.map (algebraMap K E) (minpoly K pb.gen)).Splits
hfx : IsSeparable K pb.gen
this✝ : DecidableEq E := Classical.decEq E
this : Fintype (L →ₐ[K] E) := PowerBasis.AlgHom.fintype pb
⊢ ∀ (x : L →ₐ[K] E), x pb.gen = ?f✝ (pb.liftEquiv' x)