Non-zero divisors

In this file we define the submonoid nonZeroDivisors of a MonoidWithZero.

Notations

This file declares the notation R⁰ for the submonoid of non-zero-divisors of R, in the locale nonZeroDivisors.

Use the statement open nonZeroDivisors to access this notation in your own code.

def nonZeroDivisors (R : Type u_1) [MonoidWithZero R] : Submonoid R

The submonoid of non-zero-divisors of a MonoidWithZero R.

Equations

• One or more equations did not get rendered due to their size.

def nonZeroDivisors.«term_⁰» : Lean.TrailingParserDescr

The notation for the submonoid of non-zerodivisors.

Equations

• nonZeroDivisors.«term_⁰» = Lean.ParserDescr.trailingNode `nonZeroDivisors.term_⁰ 9000 0 (Lean.ParserDescr.symbol "⁰")

theorem mem_nonZeroDivisors_iff {M : Type u_1} [MonoidWithZero M] {r : M} : r ∈ nonZeroDivisors M ↔ ∀ (x : M), x * r = 0 → x = 0

theorem mul_right_mem_nonZeroDivisors_eq_zero_iff {M : Type u_1} [MonoidWithZero M] {x : M} {r : M} (hr : r ∈ nonZeroDivisors M) : x * r = 0 ↔ x = 0

@[simp]

theorem mul_right_coe_nonZeroDivisors_eq_zero_iff {M : Type u_1} [MonoidWithZero M] {x : M} {c : { x // x ∈ nonZeroDivisors M }} : x * ↑c = 0 ↔ x = 0

theorem mul_left_mem_nonZeroDivisors_eq_zero_iff {M₁ : Type u_3} [CommMonoidWithZero M₁] {r : M₁} {x : M₁} (hr : r ∈ nonZeroDivisors M₁) : r * x = 0 ↔ x = 0

@[simp]

theorem mul_left_coe_nonZeroDivisors_eq_zero_iff {M₁ : Type u_3} [CommMonoidWithZero M₁] {c : { x // x ∈ nonZeroDivisors M₁ }} {x : M₁} : ↑c * x = 0 ↔ x = 0

theorem mul_cancel_right_mem_nonZeroDivisors {R : Type u_4} [Ring R] {x : R} {y : R} {r : R} (hr : r ∈ nonZeroDivisors R) : x * r = y * r ↔ x = y

theorem mul_cancel_right_coe_nonZeroDivisors {R : Type u_4} [Ring R] {x : R} {y : R} {c : { x // x ∈ nonZeroDivisors R }} : x * ↑c = y * ↑c ↔ x = y

@[simp]

theorem mul_cancel_left_mem_nonZeroDivisors {R' : Type u_5} [CommRing R'] {x : R'} {y : R'} {r : R'} (hr : r ∈ nonZeroDivisors R') : r * x = r * y ↔ x = y

theorem mul_cancel_left_coe_nonZeroDivisors {R' : Type u_5} [CommRing R'] {x : R'} {y : R'} {c : { x // x ∈ nonZeroDivisors R' }} : ↑c * x = ↑c * y ↔ x = y

theorem nonZeroDivisors.ne_zero {M : Type u_1} [MonoidWithZero M] [Nontrivial M] {x : M} (hx : x ∈ nonZeroDivisors M) : x ≠ 0

theorem nonZeroDivisors.coe_ne_zero {M : Type u_1} [MonoidWithZero M] [Nontrivial M] (x : { x // x ∈ nonZeroDivisors M }) : ↑x ≠ 0

theorem mul_mem_nonZeroDivisors {M₁ : Type u_3} [CommMonoidWithZero M₁] {a : M₁} {b : M₁} : a * b ∈ nonZeroDivisors M₁ ↔ a ∈ nonZeroDivisors M₁ ∧ b ∈ nonZeroDivisors M₁

theorem isUnit_of_mem_nonZeroDivisors {G₀ : Type u_7} [GroupWithZero G₀] {x : G₀} (hx : x ∈ nonZeroDivisors G₀) : IsUnit x

theorem eq_zero_of_ne_zero_of_mul_right_eq_zero {M : Type u_1} [MonoidWithZero M] [NoZeroDivisors M] {x : M} {y : M} (hnx : x ≠ 0) (hxy : y * x = 0) : y = 0

theorem eq_zero_of_ne_zero_of_mul_left_eq_zero {M : Type u_1} [MonoidWithZero M] [NoZeroDivisors M] {x : M} {y : M} (hnx : x ≠ 0) (hxy : x * y = 0) : y = 0

theorem mem_nonZeroDivisors_of_ne_zero {M : Type u_1} [MonoidWithZero M] [NoZeroDivisors M] {x : M} (hx : x ≠ 0) : x ∈ nonZeroDivisors M

theorem mem_nonZeroDivisors_iff_ne_zero {M : Type u_1} [MonoidWithZero M] [NoZeroDivisors M] [Nontrivial M] {x : M} : x ∈ nonZeroDivisors M ↔ x ≠ 0

theorem map_ne_zero_of_mem_nonZeroDivisors {M : Type u_1} {M' : Type u_2} {F : Type u_6} [MonoidWithZero M] [MonoidWithZero M'] [Nontrivial M] [ZeroHomClass F M M'] (g : F) (hg : Function.Injective ↑g) {x : M} (h : x ∈ nonZeroDivisors M) : ↑g x ≠ 0

theorem map_mem_nonZeroDivisors {M : Type u_1} {M' : Type u_2} {F : Type u_6} [MonoidWithZero M] [MonoidWithZero M'] [Nontrivial M] [NoZeroDivisors M'] [ZeroHomClass F M M'] (g : F) (hg : Function.Injective ↑g) {x : M} (h : x ∈ nonZeroDivisors M) : ↑g x ∈ nonZeroDivisors M'

theorem le_nonZeroDivisors_of_noZeroDivisors {M : Type u_1} [MonoidWithZero M] [NoZeroDivisors M] {S : Submonoid M} (hS : ¬0 ∈ S) : S ≤ nonZeroDivisors M

theorem powers_le_nonZeroDivisors_of_noZeroDivisors {M : Type u_1} [MonoidWithZero M] [NoZeroDivisors M] {a : M} (ha : a ≠ 0) : Submonoid.powers a ≤ nonZeroDivisors M

theorem map_le_nonZeroDivisors_of_injective {M : Type u_1} {M' : Type u_2} {F : Type u_6} [MonoidWithZero M] [MonoidWithZero M'] [NoZeroDivisors M'] [MonoidWithZeroHomClass F M M'] (f : F) (hf : Function.Injective ↑f) {S : Submonoid M} (hS : S ≤ nonZeroDivisors M) : Submonoid.map f S ≤ nonZeroDivisors M'

theorem nonZeroDivisors_le_comap_nonZeroDivisors_of_injective {M : Type u_1} {M' : Type u_2} {F : Type u_6} [MonoidWithZero M] [MonoidWithZero M'] [NoZeroDivisors M'] [MonoidWithZeroHomClass F M M'] (f : F) (hf : Function.Injective ↑f) : nonZeroDivisors M ≤ Submonoid.comap f (nonZeroDivisors M')

theorem prod_zero_iff_exists_zero {M₁ : Type u_3} [CommMonoidWithZero M₁] [NoZeroDivisors M₁] [Nontrivial M₁] {s : Multiset M₁} : Multiset.prod s = 0 ↔ ∃ r x, r = 0