File: Mathlib/Algebra/Ring/Divisibility/Basic.lean

full_name: map_dvd_iff
code: theorem map_dvd_iff {a b} : f a ∣ f b ↔ a ∣ b 
goal: f a ∣ f b ↔ a ∣ b

full_name: MulEquiv.decompositionMonoid
code: theorem MulEquiv.decompositionMonoid [DecompositionMonoid β] : DecompositionMonoid α 
goal: DecompositionMonoid α

full_name: dvd_add
code: theorem dvd_add [LeftDistribClass α] {a b c : α} (h₁ : a ∣ b) (h₂ : a ∣ c) : a ∣ b + c 
goal: a ∣ b + c

full_name: min_pow_dvd_add
code: lemma min_pow_dvd_add (ha : c ^ m ∣ a) (hb : c ^ n ∣ b) : c ^ min m n ∣ a + b 
goal: c ^ min m n ∣ a + b

full_name: Dvd.dvd.linear_comb
code: theorem Dvd.dvd.linear_comb {d x y : α} (hdx : d ∣ x) (hdy : d ∣ y) (a b : α) : d ∣ a * x + b * y 
goal: d ∣ a * x + b * y

full_name: dvd_neg
code: /-- An element `a` of a semigroup with a distributive negation divides the negation of an element
`b` iff `a` divides `b`. -/
@[simp]
theorem dvd_neg : a ∣ -b ↔ a ∣ b 
goal: a ∣ -b ↔ a ∣ b

full_name: neg_dvd
code: /-- The negation of an element `a` of a semigroup with a distributive negation divides another
element `b` iff `a` divides `b`. -/
@[simp]
theorem neg_dvd : -a ∣ b ↔ a ∣ b 
goal: -a ∣ b ↔ a ∣ b

full_name: dvd_sub
code: theorem dvd_sub (h₁ : a ∣ b) (h₂ : a ∣ c) : a ∣ b - c 
goal: a ∣ b - c

full_name: dvd_add_left
code: /-- If an element `a` divides another element `c` in a ring, `a` divides the sum of another element
`b` with `c` iff `a` divides `b`. -/
theorem dvd_add_left (h : a ∣ c) : a ∣ b + c ↔ a ∣ b 
goal: a ∣ b + c ↔ a ∣ b

full_name: dvd_add_right
code: /-- If an element `a` divides another element `b` in a ring, `a` divides the sum of `b` and another
element `c` iff `a` divides `c`. -/
theorem dvd_add_right (h : a ∣ b) : a ∣ b + c ↔ a ∣ c 
goal: a ∣ b + c ↔ a ∣ c

full_name: dvd_sub_left
code: /-- If an element `a` divides another element `c` in a ring, `a` divides the difference of another
element `b` with `c` iff `a` divides `b`. -/
theorem dvd_sub_left (h : a ∣ c) : a ∣ b - c ↔ a ∣ b 
goal: a ∣ b - c ↔ a ∣ b

full_name: dvd_sub_right
code: /-- If an element `a` divides another element `b` in a ring, `a` divides the difference of `b` and
another element `c` iff `a` divides `c`. -/
theorem dvd_sub_right (h : a ∣ b) : a ∣ b - c ↔ a ∣ c 
goal: a ∣ b - c ↔ a ∣ c

full_name: dvd_iff_dvd_of_dvd_sub
code: theorem dvd_iff_dvd_of_dvd_sub (h : a ∣ b - c) : a ∣ b ↔ a ∣ c 
goal: a ∣ b ↔ a ∣ c

full_name: dvd_sub_comm
code: theorem dvd_sub_comm : a ∣ b - c ↔ a ∣ c - b 
goal: a ∣ b - c ↔ a ∣ c - b

full_name: dvd_add_self_left
code: /-- An element a divides the sum a + b if and only if a divides b. -/
@[simp]
theorem dvd_add_self_left {a b : α} : a ∣ a + b ↔ a ∣ b 
goal: a ∣ a + b ↔ a ∣ b

full_name: dvd_add_self_right
code: /-- An element a divides the sum b + a if and only if a divides b. -/
@[simp]
theorem dvd_add_self_right {a b : α} : a ∣ b + a ↔ a ∣ b 
goal: a ∣ b + a ↔ a ∣ b

full_name: dvd_sub_self_left
code: /-- An element `a` divides the difference `a - b` if and only if `a` divides `b`. -/
@[simp]
theorem dvd_sub_self_left : a ∣ a - b ↔ a ∣ b 
goal: a ∣ a - b ↔ a ∣ b

full_name: dvd_sub_self_right
code: /-- An element `a` divides the difference `b - a` if and only if `a` divides `b`. -/
@[simp]
theorem dvd_sub_self_right : a ∣ b - a ↔ a ∣ b 
goal: a ∣ b - a ↔ a ∣ b

full_name: dvd_mul_sub_mul
code: theorem dvd_mul_sub_mul {k a b x y : α} (hab : k ∣ a - b) (hxy : k ∣ x - y) :
    k ∣ a * x - b * y 
goal: k ∣ a * x - b * y