Documentation

Mathlib.Algebra.NeZero

NeZero typeclass #

We create a typeclass NeZero n which carries around the fact that (n : R) ≠ 0.

Main declarations #

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class NeZero {R : Type u_1} [Zero R] (n : R) :
Prop
  • out : n ≠ 0

    The proposition that n is not zero.

A type-class version of n ≠ 0.

Instances
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    theorem NeZero.ne {R : Type u_1} [Zero R] (n : R) [h : NeZero n] :
    n ≠ 0
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    theorem NeZero.ne' {R : Type u_1} [Zero R] (n : R) [h : NeZero n] :
    0 ≠ n
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    theorem neZero_iff {R : Type u_1} [Zero R] {n : R} :
    NeZero n ↔ n ≠ 0
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    theorem not_neZero {R : Type u_1} [Zero R] {n : R} :
    ¬NeZero n ↔ n = 0
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    theorem eq_zero_or_neZero {α : Type u_1} [Zero α] (a : α) :
    a = 0 ∨ NeZero a
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    @[simp]
    theorem zero_ne_one {α : Type u_1} [Zero α] [One α] [NeZero 1] :
    0 ≠ 1
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    @[simp]
    theorem one_ne_zero {α : Type u_1} [Zero α] [One α] [NeZero 1] :
    1 ≠ 0
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    theorem ne_zero_of_eq_one {α : Type u_1} [Zero α] [One α] [NeZero 1] {a : α} (h : a = 1) :
    a ≠ 0
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    theorem two_ne_zero {α : Type u_1} [Zero α] [OfNat α 2] [NeZero 2] :
    2 ≠ 0
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    theorem three_ne_zero {α : Type u_1} [Zero α] [OfNat α 3] [NeZero 3] :
    3 ≠ 0
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    theorem four_ne_zero {α : Type u_1} [Zero α] [OfNat α 4] [NeZero 4] :
    4 ≠ 0
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    theorem zero_ne_one' (α : Type u_1) [Zero α] [One α] [NeZero 1] :
    0 ≠ 1
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    theorem one_ne_zero' (α : Type u_1) [Zero α] [One α] [NeZero 1] :
    1 ≠ 0
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    theorem two_ne_zero' (α : Type u_1) [Zero α] [OfNat α 2] [NeZero 2] :
    2 ≠ 0
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    theorem three_ne_zero' (α : Type u_1) [Zero α] [OfNat α 3] [NeZero 3] :
    3 ≠ 0
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    theorem four_ne_zero' (α : Type u_1) [Zero α] [OfNat α 4] [NeZero 4] :
    4 ≠ 0
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    instance NeZero.succ {n : ℕ} :
    NeZero (n + 1)
    Equations
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    theorem NeZero.of_pos {M : Type u_1} {x : M} [Preorder M] [Zero M] (h : 0 < x) :
    NeZero x