R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
h : IsNilpotent a
⊢ IsNilpotent ((LinearMap.mulLeft R - LinearMap.mulRight R) a)
R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
h : IsNilpotent a
⊢ IsNilpotent (↑(LieAlgebra.ad R A) a)
rw [LieAlgebra.ad_eq_lmul_left_sub_lmul_right]
R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
h : IsNilpotent a
hl : IsNilpotent (LinearMap.mulLeft R a)
⊢ IsNilpotent ((LinearMap.mulLeft R - LinearMap.mulRight R) a)
R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
h : IsNilpotent a
⊢ IsNilpotent ((LinearMap.mulLeft R - LinearMap.mulRight R) a)
have hl : IsNilpotent (LinearMap.mulLeft R a) := by rwa [LinearMap.isNilpotent_mulLeft_iff]
R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
h : IsNilpotent a
hl : IsNilpotent (LinearMap.mulLeft R a)
hr : IsNilpotent (LinearMap.mulRight R a)
⊢ IsNilpotent ((LinearMap.mulLeft R - LinearMap.mulRight R) a)
R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
h : IsNilpotent a
hl : IsNilpotent (LinearMap.mulLeft R a)
⊢ IsNilpotent ((LinearMap.mulLeft R - LinearMap.mulRight R) a)
have hr : IsNilpotent (LinearMap.mulRight R a) := by rwa [LinearMap.isNilpotent_mulRight_iff]
R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
h : IsNilpotent a
hl : IsNilpotent (LinearMap.mulLeft R a)
hr : IsNilpotent (LinearMap.mulRight R a)
this : Commute (LinearMap.mulLeft R a) (LinearMap.mulRight R a)
⊢ IsNilpotent ((LinearMap.mulLeft R - LinearMap.mulRight R) a)
R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
h : IsNilpotent a
hl : IsNilpotent (LinearMap.mulLeft R a)
hr : IsNilpotent (LinearMap.mulRight R a)
⊢ IsNilpotent ((LinearMap.mulLeft R - LinearMap.mulRight R) a)
have := @LinearMap.commute_mulLeft_right R A _ _ _ _ _ a a
no goals
R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
h : IsNilpotent a
hl : IsNilpotent (LinearMap.mulLeft R a)
hr : IsNilpotent (LinearMap.mulRight R a)
this : Commute (LinearMap.mulLeft R a) (LinearMap.mulRight R a)
⊢ IsNilpotent ((LinearMap.mulLeft R - LinearMap.mulRight R) a)
exact this.isNilpotent_sub hl hr
no goals
R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
h : IsNilpotent a
⊢ IsNilpotent (LinearMap.mulLeft R a)
rwa [LinearMap.isNilpotent_mulLeft_iff]
no goals
R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
h : IsNilpotent a
hl : IsNilpotent (LinearMap.mulLeft R a)
⊢ IsNilpotent (LinearMap.mulRight R a)
rwa [LinearMap.isNilpotent_mulRight_iff]