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

tactic: refine le_antisymm ?_ ?_
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✝ : PerfectField K
α : Weight K (↥H) L
⊢ ↑(corootSpace ⇑α) = span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}
state_after: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
⊢ ↑(corootSpace ⇑α) ≤ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}

case refine_2
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✝ : PerfectField K
α : Weight K (↥H) L
⊢ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)} ≤ ↑(corootSpace ⇑α)

tactic: intro ⟨x, hx⟩ hx'
state_before: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
⊢ ↑(corootSpace ⇑α) ≤ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}
state_after: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : ⟨x, hx⟩ ∈ ↑(corootSpace ⇑α)
⊢ ⟨x, hx⟩ ∈ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}

tactic: have : {⁅y, z⁆ | (y ∈ rootSpace H α) (z ∈ rootSpace H (-α))} ⊆
    K ∙ ((cartanEquivDual H).symm α : L) := by
  rintro - ⟨e, heα, f, hfα, rfl⟩
  rw [lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg heα hfα, SetLike.mem_coe,
    Submodule.mem_span_singleton]
  exact ⟨killingForm K L e f, rfl⟩
state_before: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : ⟨x, hx⟩ ∈ ↑(corootSpace ⇑α)
⊢ ⟨x, hx⟩ ∈ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}
state_after: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : ⟨x, hx⟩ ∈ ↑(corootSpace ⇑α)
this :
  {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x} ⊆
    ↑(span K {↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))})
⊢ ⟨x, hx⟩ ∈ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}

tactic: simp only [LieSubmodule.mem_coeSubmodule, mem_corootSpace] at hx'
state_before: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : ⟨x, hx⟩ ∈ ↑(corootSpace ⇑α)
this :
  {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x} ⊆
    ↑(span K {↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))})
⊢ ⟨x, hx⟩ ∈ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}
state_after: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
this :
  {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x} ⊆
    ↑(span K {↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))})
hx' : x ∈ span K {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x}
⊢ ⟨x, hx⟩ ∈ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}

tactic: replace this := Submodule.span_mono this hx'
state_before: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
this :
  {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x} ⊆
    ↑(span K {↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))})
hx' : x ∈ span K {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x}
⊢ ⟨x, hx⟩ ∈ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}
state_after: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : x ∈ span K {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x}
this : x ∈ span K ↑(span K {↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))})
⊢ ⟨x, hx⟩ ∈ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}

tactic: rw [Submodule.span_span] at this
state_before: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : x ∈ span K {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x}
this : x ∈ span K ↑(span K {↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))})
⊢ ⟨x, hx⟩ ∈ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}
state_after: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : x ∈ span K {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x}
this : x ∈ span K {↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))}
⊢ ⟨x, hx⟩ ∈ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}

tactic: rw [Submodule.mem_span_singleton] at this ⊢
state_before: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : x ∈ span K {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x}
this : x ∈ span K {↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))}
⊢ ⟨x, hx⟩ ∈ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)}
state_after: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : x ∈ span K {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x}
this : ∃ a, a • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) = x
⊢ ∃ a, a • (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α) = ⟨x, hx⟩

tactic: obtain ⟨t, rfl⟩ := this
state_before: case refine_1
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : x ∈ span K {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x}
this : ∃ a, a • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) = x
⊢ ∃ a, a • (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α) = ⟨x, hx⟩
state_after: case refine_1.intro
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✝ : PerfectField K
α : Weight K (↥H) L
t : K
hx : t • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) ∈ H
hx' :
  t • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) ∈
    span K {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x}
⊢ ∃ a,
    a • (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α) =
      ⟨t • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)), hx⟩

tactic: use t
state_before: case refine_1.intro
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✝ : PerfectField K
α : Weight K (↥H) L
t : K
hx : t • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) ∈ H
hx' :
  t • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) ∈
    span K {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x}
⊢ ∃ a,
    a • (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α) =
      ⟨t • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)), hx⟩
state_after: case h
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✝ : PerfectField K
α : Weight K (↥H) L
t : K
hx : t • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) ∈ H
hx' :
  t • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) ∈
    span K {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x}
⊢ t • (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α) =
    ⟨t • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)), hx⟩

tactic: simp [Subtype.ext_iff]
state_before: case h
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✝ : PerfectField K
α : Weight K (↥H) L
t : K
hx : t • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) ∈ H
hx' :
  t • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) ∈
    span K {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x}
⊢ t • (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α) =
    ⟨t • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)), hx⟩
state_after: no goals

tactic: rintro - ⟨e, heα, f, hfα, rfl⟩
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : ⟨x, hx⟩ ∈ ↑(corootSpace ⇑α)
⊢ {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = x} ⊆
    ↑(span K {↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))})
state_after: case intro.intro.intro.intro
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : ⟨x, hx⟩ ∈ ↑(corootSpace ⇑α)
e : L
heα : e ∈ rootSpace H ⇑α
f : L
hfα : f ∈ rootSpace H (-⇑α)
⊢ ⁅e, f⁆ ∈ ↑(span K {↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))})

tactic: rw [lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg heα hfα, SetLike.mem_coe,
  Submodule.mem_span_singleton]
state_before: case intro.intro.intro.intro
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : ⟨x, hx⟩ ∈ ↑(corootSpace ⇑α)
e : L
heα : e ∈ rootSpace H ⇑α
f : L
hfα : f ∈ rootSpace H (-⇑α)
⊢ ⁅e, f⁆ ∈ ↑(span K {↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))})
state_after: case intro.intro.intro.intro
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : ⟨x, hx⟩ ∈ ↑(corootSpace ⇑α)
e : L
heα : e ∈ rootSpace H ⇑α
f : L
hfα : f ∈ rootSpace H (-⇑α)
⊢ ∃ a,
    a • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) =
      ((killingForm K L) e) f • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))

tactic: exact ⟨killingForm K L e f, rfl⟩
state_before: case intro.intro.intro.intro
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✝ : PerfectField K
α : Weight K (↥H) L
x : L
hx : x ∈ H
hx' : ⟨x, hx⟩ ∈ ↑(corootSpace ⇑α)
e : L
heα : e ∈ rootSpace H ⇑α
f : L
hfα : f ∈ rootSpace H (-⇑α)
⊢ ∃ a,
    a • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) =
      ((killingForm K L) e) f • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))
state_after: no goals

tactic: simp only [Submodule.span_singleton_le_iff_mem, LieSubmodule.mem_coeSubmodule]
state_before: case refine_2
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✝ : PerfectField K
α : Weight K (↥H) L
⊢ span K {(cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)} ≤ ↑(corootSpace ⇑α)
state_after: case refine_2
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✝ : PerfectField K
α : Weight K (↥H) L
⊢ (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α) ∈ corootSpace ⇑α

tactic: exact cartanEquivDual_symm_apply_mem_corootSpace K L H α
state_before: case refine_2
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✝ : PerfectField K
α : Weight K (↥H) L
⊢ (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α) ∈ corootSpace ⇑α
state_after: no goals