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

tactic: rw [← span_weight_eq_top K L H]
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : IsKilling K L
⊢ span K (Weight.toLinear K (↥H) L '' {α | α.IsNonZero}) = ⊤
state_after: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : IsKilling K L
⊢ span K (Weight.toLinear K (↥H) L '' {α | α.IsNonZero}) = span K (range (Weight.toLinear K (↥H) L))

tactic: refine le_antisymm (Submodule.span_mono <| by simp) ?_
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : IsKilling K L
⊢ span K (Weight.toLinear K (↥H) L '' {α | α.IsNonZero}) = span K (range (Weight.toLinear K (↥H) L))
state_after: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : IsKilling K L
⊢ span K (range (Weight.toLinear K (↥H) L)) ≤ span K (Weight.toLinear K (↥H) L '' {α | α.IsNonZero})

tactic: suffices range (Weight.toLinear K H L) ⊆
  insert 0 ({α : Weight K H L | α.IsNonZero}.image (Weight.toLinear K H L)) by
  simpa only [Submodule.span_insert_zero] using Submodule.span_mono this
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : IsKilling K L
⊢ span K (range (Weight.toLinear K (↥H) L)) ≤ span K (Weight.toLinear K (↥H) L '' {α | α.IsNonZero})
state_after: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : IsKilling K L
⊢ range (Weight.toLinear K (↥H) L) ⊆ insert 0 (Weight.toLinear K (↥H) L '' {α | α.IsNonZero})

tactic: rintro - ⟨α, rfl⟩
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : IsKilling K L
⊢ range (Weight.toLinear K (↥H) L) ⊆ insert 0 (Weight.toLinear K (↥H) L '' {α | α.IsNonZero})
state_after: case intro
R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : IsKilling K L
α : Weight K (↥H) L
⊢ Weight.toLinear K (↥H) L α ∈ insert 0 (Weight.toLinear K (↥H) L '' {α | α.IsNonZero})

tactic: simp only [mem_insert_iff, Weight.coe_toLinear_eq_zero_iff, mem_image, mem_setOf_eq]
state_before: case intro
R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : IsKilling K L
α : Weight K (↥H) L
⊢ Weight.toLinear K (↥H) L α ∈ insert 0 (Weight.toLinear K (↥H) L '' {α | α.IsNonZero})
state_after: case intro
R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : IsKilling K L
α : Weight K (↥H) L
⊢ α.IsZero ∨ ∃ x, x.IsNonZero ∧ Weight.toLinear K (↥H) L x = Weight.toLinear K (↥H) L α

tactic: tauto
state_before: case intro
R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : IsKilling K L
α : Weight K (↥H) L
⊢ α.IsZero ∨ ∃ x, x.IsNonZero ∧ Weight.toLinear K (↥H) L x = Weight.toLinear K (↥H) L α
state_after: no goals

tactic: simp
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : IsKilling K L
⊢ Weight.toLinear K (↥H) L '' {α | α.IsNonZero} ⊆ range (Weight.toLinear K (↥H) L)
state_after: no goals

tactic: simpa only [Submodule.span_insert_zero] using Submodule.span_mono this
state_before: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : IsKilling K L
this : range (Weight.toLinear K (↥H) L) ⊆ insert 0 (Weight.toLinear K (↥H) L '' {α | α.IsNonZero})
⊢ span K (range (Weight.toLinear K (↥H) L)) ≤ span K (Weight.toLinear K (↥H) L '' {α | α.IsNonZero})
state_after: no goals