Theorem: LieAlgebra.IsKilling.traceForm_coroot
File: Mathlib/Algebra/Lie/Weights/Killing.lean

Proof steps:
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsTriangularizable K (↥H) L
inst✝¹ : IsKilling K L
inst✝ : CharZero K
α : Weight K (↥H) L
x : ↥H
⊢ ((traceForm K (↥H) L) (coroot α)) x = 2 • (α ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)))⁻¹ • α x
state_after: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsTriangularizable K (↥H) L
inst✝¹ : IsKilling K L
inst✝ : CharZero K
α : Weight K (↥H) L
x : ↥H
this : ((cartanEquivDual H) ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))) x = α x
⊢ ((traceForm K (↥H) L) (coroot α)) x = 2 • (α ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)))⁻¹ • α x
tactic: have : cartanEquivDual H ((cartanEquivDual H).symm α) x = α x := by
    rw [LinearEquiv.apply_symm_apply, Weight.toLinear_apply]
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsTriangularizable K (↥H) L
inst✝¹ : IsKilling K L
inst✝ : CharZero K
α : Weight K (↥H) L
x : ↥H
this : ((cartanEquivDual H) ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))) x = α x
⊢ ((traceForm K (↥H) L) (coroot α)) x = 2 • (α ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)))⁻¹ • α x
state_after: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsTriangularizable K (↥H) L
inst✝¹ : IsKilling K L
inst✝ : CharZero K
α : Weight K (↥H) L
x : ↥H
this : ((cartanEquivDual H) ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))) x = α x
⊢ 2 •
      (α ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)))⁻¹ •
        ((traceForm K (↥H) L) ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))) x =
    2 • (α ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)))⁻¹ • α x
tactic: rw [coroot, map_nsmul, map_smul, LinearMap.smul_apply, LinearMap.smul_apply]
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsTriangularizable K (↥H) L
inst✝¹ : IsKilling K L
inst✝ : CharZero K
α : Weight K (↥H) L
x : ↥H
this : ((cartanEquivDual H) ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))) x = α x
⊢ 2 •
      (α ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)))⁻¹ •
        ((traceForm K (↥H) L) ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))) x =
    2 • (α ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)))⁻¹ • α x
state_after: no goals
tactic: congr 2