File: Mathlib/RingTheory/Nilpotent/Basic.lean

full_name: IsNilpotent.neg
code: theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) 
goal: IsNilpotent (-x)

full_name: isNilpotent_neg_iff
code: @[simp]
theorem isNilpotent_neg_iff [Ring R] : IsNilpotent (-x) ↔ IsNilpotent x 
goal: IsNilpotent (-x) ↔ IsNilpotent x

full_name: IsNilpotent.smul
code: lemma IsNilpotent.smul [MonoidWithZero R] [MonoidWithZero S] [MulActionWithZero R S]
    [SMulCommClass R S S] [IsScalarTower R S S] {a : S} (ha : IsNilpotent a) (t : R) :
    IsNilpotent (t • a) 
goal: IsNilpotent (t • a)

full_name: IsNilpotent.isUnit_sub_one
code: theorem IsNilpotent.isUnit_sub_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r - 1) 
goal: IsUnit (r - 1)

full_name: IsNilpotent.isUnit_one_sub
code: theorem IsNilpotent.isUnit_one_sub [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (1 - r) 
goal: IsUnit (1 - r)

full_name: IsNilpotent.isUnit_add_one
code: theorem IsNilpotent.isUnit_add_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r + 1) 
goal: IsUnit (r + 1)

full_name: IsNilpotent.isUnit_one_add
code: theorem IsNilpotent.isUnit_one_add [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (1 + r) 
goal: IsUnit (1 + r)

full_name: IsNilpotent.isUnit_add_left_of_commute
code: theorem IsNilpotent.isUnit_add_left_of_commute [Ring R] {r u : R}
    (hnil : IsNilpotent r) (hu : IsUnit u) (h_comm : Commute r u) :
    IsUnit (u + r) 
goal: IsUnit (u + r)

full_name: IsNilpotent.isUnit_add_right_of_commute
code: theorem IsNilpotent.isUnit_add_right_of_commute [Ring R] {r u : R}
    (hnil : IsNilpotent r) (hu : IsUnit u) (h_comm : Commute r u) :
    IsUnit (r + u) 
goal: IsUnit (r + u)

full_name: instIsReducedProd
code: instance [Zero R] [Pow R ℕ] [Zero S] [Pow S ℕ] [IsReduced R] [IsReduced S] : IsReduced (R × S) 
goal: IsReduced (R × S)

full_name: Prime.isRadical
code: theorem Prime.isRadical [CommMonoidWithZero R] {y : R} (hy : Prime y) : IsRadical y 
goal: IsRadical y

full_name: zero_isRadical_iff
code: theorem zero_isRadical_iff [MonoidWithZero R] : IsRadical (0 : R) ↔ IsReduced R 
goal: IsRadical 0 ↔ IsReduced R

full_name: isReduced_iff_pow_one_lt
code: theorem isReduced_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
    IsReduced R ↔ ∀ x : R, x ^ k = 0 → x = 0 
goal: IsReduced R ↔ ∀ (x : R), x ^ k = 0 → x = 0

full_name: IsRadical.of_dvd
code: theorem IsRadical.of_dvd [CancelCommMonoidWithZero R] {x y : R} (hy : IsRadical y) (h0 : y ≠ 0)
    (hxy : x ∣ y) : IsRadical x 
goal: IsRadical x

full_name: Commute.add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero
code: theorem add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero (h_comm : Commute x y) {m n k : ℕ}
    (hx : x ^ m = 0) (hy : y ^ n = 0) (h : m + n ≤ k + 1) :
    (x + y) ^ k = 0 
goal: (x + y) ^ k = 0

full_name: Commute.add_pow_add_eq_zero_of_pow_eq_zero
code: theorem add_pow_add_eq_zero_of_pow_eq_zero (h_comm : Commute x y) {m n : ℕ}
    (hx : x ^ m = 0) (hy : y ^ n = 0) :
    (x + y) ^ (m + n - 1) = 0 
goal: (x + y) ^ (m + n - 1) = 0

full_name: Commute.isNilpotent_add
code: theorem isNilpotent_add (h_comm : Commute x y) (hx : IsNilpotent x) (hy : IsNilpotent y) :
    IsNilpotent (x + y) 
goal: IsNilpotent (x + y)

full_name: Commute.isNilpotent_sum
code: protected lemma isNilpotent_sum {ι : Type*} {s : Finset ι} {f : ι → R}
    (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ i j, i ∈ s → j ∈ s → Commute (f i) (f j)) :
    IsNilpotent (∑ i ∈ s, f i) 
goal: IsNilpotent (∑ i ∈ s, f i)

full_name: Commute.isNilpotent_mul_left_iff
code: protected lemma isNilpotent_mul_left_iff (h_comm : Commute x y) (hy : y ∈ nonZeroDivisorsLeft R) :
    IsNilpotent (x * y) ↔ IsNilpotent x 
goal: IsNilpotent (x * y) ↔ IsNilpotent x

full_name: Commute.isNilpotent_mul_right_iff
code: protected lemma isNilpotent_mul_right_iff (h_comm : Commute x y) (hx : x ∈ nonZeroDivisorsRight R) :
    IsNilpotent (x * y) ↔ IsNilpotent y 
goal: IsNilpotent (x * y) ↔ IsNilpotent y

full_name: Commute.isNilpotent_sub
code: theorem isNilpotent_sub (h_comm : Commute x y) (hx : IsNilpotent x) (hy : IsNilpotent y) :
    IsNilpotent (x - y) 
goal: IsNilpotent (x - y)

full_name: isNilpotent_sum
code: lemma isNilpotent_sum {ι : Type*} {s : Finset ι} {f : ι → R}
    (hnp : ∀ i ∈ s, IsNilpotent (f i)) :
    IsNilpotent (∑ i ∈ s, f i) 
goal: IsNilpotent (∑ i ∈ s, f i)

full_name: NoZeroSMulDivisors.isReduced
code: lemma NoZeroSMulDivisors.isReduced (R M : Type*)
    [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [Nontrivial M] [NoZeroSMulDivisors R M] :
    IsReduced R 
goal: IsReduced R