Documentation

Mathlib.Algebra.CharZero.Lemmas

Characteristic zero (additional theorems) #

A ring R is called of characteristic zero if every natural number n is non-zero when considered as an element of R. Since this definition doesn't mention the multiplicative structure of R except for the existence of 1 in this file characteristic zero is defined for additive monoids with 1.

Main statements #

source
@[simp]
theorem Nat.castEmbedding_apply {R : Type u_1} [AddMonoidWithOne R] [CharZero R] :
∀ (a : ℕ), Nat.castEmbedding a = ↑a
source
def Nat.castEmbedding {R : Type u_1} [AddMonoidWithOne R] [CharZero R] :
ℕ ↪ R

Nat.cast as an embedding into monoids of characteristic 0.

Equations
  • Nat.castEmbedding = { toFun := Nat.cast, inj' := ⋯ }
Instances For
    source
    @[simp]
    theorem Nat.cast_pow_eq_one {R : Type u_2} [Semiring R] [CharZero R] (q : ℕ) (n : ℕ) (hn : n ≠ 0) :
    ↑q ^ n = 1 ↔ q = 1
    source
    @[simp]
    theorem Nat.cast_div_charZero {k : Type u_2} [DivisionSemiring k] [CharZero k] {m : ℕ} {n : ℕ} (n_dvd : n ∣ m) :
    ↑(m / n) = ↑m / ↑n
    source
    instance CharZero.NeZero.two {M : Type u_2} [AddMonoidWithOne M] [CharZero M] :
    NeZero 2
    Equations
    • ⋯ = ⋯
    source
    theorem Function.support_natCast {α : Type u_1} {M : Type u_2} [AddMonoidWithOne M] [CharZero M] {n : ℕ} (hn : n ≠ 0) :
    Function.support ↑n = Set.univ
    source
    theorem Function.mulSupport_natCast {α : Type u_1} {M : Type u_2} [AddMonoidWithOne M] [CharZero M] {n : ℕ} (hn : n ≠ 1) :
    Function.mulSupport ↑n = Set.univ
    source
    @[simp]
    theorem add_self_eq_zero {R : Type u_1} [NonAssocSemiring R] [NoZeroDivisors R] [CharZero R] {a : R} :
    a + a = 0 ↔ a = 0
    source
    @[simp]
    theorem bit0_eq_zero {R : Type u_1} [NonAssocSemiring R] [NoZeroDivisors R] [CharZero R] {a : R} :
    bit0 a = 0 ↔ a = 0
    source
    @[simp]
    theorem zero_eq_bit0 {R : Type u_1} [NonAssocSemiring R] [NoZeroDivisors R] [CharZero R] {a : R} :
    0 = bit0 a ↔ a = 0
    source
    theorem bit0_ne_zero {R : Type u_1} [NonAssocSemiring R] [NoZeroDivisors R] [CharZero R] {a : R} :
    bit0 a ≠ 0 ↔ a ≠ 0
    source
    theorem zero_ne_bit0 {R : Type u_1} [NonAssocSemiring R] [NoZeroDivisors R] [CharZero R] {a : R} :
    0 ≠ bit0 a ↔ a ≠ 0
    source
    @[simp]
    theorem neg_eq_self_iff {R : Type u_1} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] {a : R} :
    -a = a ↔ a = 0
    source
    @[simp]
    theorem eq_neg_self_iff {R : Type u_1} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] {a : R} :
    a = -a ↔ a = 0
    source
    theorem nat_mul_inj {R : Type u_1} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] {n : ℕ} {a : R} {b : R} (h : ↑n * a = ↑n * b) :
    n = 0 ∨ a = b
    source
    theorem nat_mul_inj' {R : Type u_1} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] {n : ℕ} {a : R} {b : R} (h : ↑n * a = ↑n * b) (w : n ≠ 0) :
    a = b
    source
    theorem bit0_injective {R : Type u_1} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] :
    Function.Injective bit0
    source
    theorem bit1_injective {R : Type u_1} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] :
    Function.Injective bit1
    source
    @[simp]
    theorem bit0_eq_bit0 {R : Type u_1} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] {a : R} {b : R} :
    bit0 a = bit0 b ↔ a = b
    source
    @[simp]
    theorem bit1_eq_bit1 {R : Type u_1} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] {a : R} {b : R} :
    bit1 a = bit1 b ↔ a = b
    source
    @[simp]
    theorem bit1_eq_one {R : Type u_1} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] {a : R} :
    bit1 a = 1 ↔ a = 0
    source
    @[simp]
    theorem one_eq_bit1 {R : Type u_1} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] {a : R} :
    1 = bit1 a ↔ a = 0
    source
    @[simp]
    theorem half_add_self {R : Type u_1} [DivisionRing R] [CharZero R] (a : R) :
    (a + a) / 2 = a
    source
    @[simp]
    theorem add_halves' {R : Type u_1} [DivisionRing R] [CharZero R] (a : R) :
    a / 2 + a / 2 = a
    source
    theorem sub_half {R : Type u_1} [DivisionRing R] [CharZero R] (a : R) :
    a - a / 2 = a / 2
    source
    theorem half_sub {R : Type u_1} [DivisionRing R] [CharZero R] (a : R) :
    a / 2 - a = -(a / 2)
    source
    instance WithTop.instCharZeroWithTopAddMonoidWithOne {R : Type u_1} [AddMonoidWithOne R] [CharZero R] :
    CharZero (WithTop R)
    Equations
    • ⋯ = ⋯
    source
    instance WithBot.instCharZeroWithBotAddMonoidWithOne {R : Type u_1} [AddMonoidWithOne R] [CharZero R] :
    CharZero (WithBot R)
    Equations
    • ⋯ = ⋯
    source
    theorem RingHom.charZero {R : Type u_1} {S : Type u_2} [NonAssocSemiring R] [NonAssocSemiring S] (ϕ : R →+* S) [hS : CharZero S] :
    CharZero R
    source
    theorem RingHom.charZero_iff {R : Type u_1} {S : Type u_2} [NonAssocSemiring R] [NonAssocSemiring S] {ϕ : R →+* S} (hϕ : Function.Injective ⇑ϕ) :
    CharZero R ↔ CharZero S
    source
    theorem RingHom.injective_nat {R : Type u_1} [NonAssocSemiring R] (f : ℕ →+* R) [CharZero R] :
    Function.Injective ⇑f
    source
    @[simp]
    theorem units_ne_neg_self {R : Type u_1} [Ring R] [CharZero R] (u : Rˣ) :
    u ≠ -u
    source
    @[simp]
    theorem neg_units_ne_self {R : Type u_1} [Ring R] [CharZero R] (u : Rˣ) :
    -u ≠ u