Theorem: minpoly_algHom_toLinearMap
File: Mathlib/FieldTheory/Minpoly/Field.lean

tactic: have : orderOf σ = orderOf (AlgEquiv.algHomUnitsEquiv _ _ hσ.unit) := by
  rw [← MonoidHom.coe_coe, orderOf_injective (AlgEquiv.algHomUnitsEquiv K L)
    (AlgEquiv.algHomUnitsEquiv K L).injective, ← orderOf_units, IsOfFinOrder.val_unit]
state_before: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L →ₐ[K] L
hσ : IsOfFinOrder σ
⊢ minpoly K σ.toLinearMap = X ^ orderOf σ - C 1
state_after: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L →ₐ[K] L
hσ : IsOfFinOrder σ
this : orderOf σ = orderOf ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit)
⊢ minpoly K σ.toLinearMap = X ^ orderOf σ - C 1

tactic: rw [this, ← minpoly_algEquiv_toLinearMap]
state_before: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L →ₐ[K] L
hσ : IsOfFinOrder σ
this : orderOf σ = orderOf ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit)
⊢ minpoly K σ.toLinearMap = X ^ orderOf σ - C 1
state_after: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L →ₐ[K] L
hσ : IsOfFinOrder σ
this : orderOf σ = orderOf ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit)
⊢ minpoly K σ.toLinearMap = minpoly K ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit).toLinearMap

case hσ
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L →ₐ[K] L
hσ : IsOfFinOrder σ
this : orderOf σ = orderOf ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit)
⊢ IsOfFinOrder ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit)

tactic: rw [← MonoidHom.coe_coe, orderOf_injective (AlgEquiv.algHomUnitsEquiv K L)
  (AlgEquiv.algHomUnitsEquiv K L).injective, ← orderOf_units, IsOfFinOrder.val_unit]
state_before: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L →ₐ[K] L
hσ : IsOfFinOrder σ
⊢ orderOf σ = orderOf ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit)
state_after: no goals

tactic: apply congr_arg
state_before: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L →ₐ[K] L
hσ : IsOfFinOrder σ
this : orderOf σ = orderOf ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit)
⊢ minpoly K σ.toLinearMap = minpoly K ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit).toLinearMap
state_after: case h
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L →ₐ[K] L
hσ : IsOfFinOrder σ
this : orderOf σ = orderOf ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit)
⊢ σ.toLinearMap = ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit).toLinearMap

tactic: ext
state_before: case h
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L →ₐ[K] L
hσ : IsOfFinOrder σ
this : orderOf σ = orderOf ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit)
⊢ σ.toLinearMap = ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit).toLinearMap
state_after: case h.h
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L →ₐ[K] L
hσ : IsOfFinOrder σ
this : orderOf σ = orderOf ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit)
x✝ : L
⊢ σ.toLinearMap x✝ = ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit).toLinearMap x✝

tactic: simp
state_before: case h.h
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L →ₐ[K] L
hσ : IsOfFinOrder σ
this : orderOf σ = orderOf ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit)
x✝ : L
⊢ σ.toLinearMap x✝ = ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit).toLinearMap x✝
state_after: no goals

tactic: rwa [← orderOf_pos_iff, ← this, orderOf_pos_iff]
state_before: case hσ
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : CommRing L
inst✝¹ : IsDomain L
inst✝ : Algebra K L
σ : L →ₐ[K] L
hσ : IsOfFinOrder σ
this : orderOf σ = orderOf ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit)
⊢ IsOfFinOrder ((AlgEquiv.algHomUnitsEquiv K L) hσ.unit)
state_after: no goals