# Documentation

Mathlib.Algebra.Field.MinimalAxioms

# Minimal Axioms for a Field #

This file defines constructors to define a Field structure on a Type, while proving a minimum number of equalities.

## Main Definitions #

• Field.ofMinimalAxioms: Define a Field structure on a Type by proving a minimized set of axioms
@[reducible]
def Field.ofMinimalAxioms (K : Type u) [Add K] [Mul K] [Neg K] [Inv K] [Zero K] [One K] (add_assoc : ∀ (a b c : K), a + b + c = a + (b + c)) (zero_add : ∀ (a : K), 0 + a = a) (add_left_neg : ∀ (a : K), -a + a = 0) (mul_assoc : ∀ (a b c : K), a * b * c = a * (b * c)) (mul_comm : ∀ (a b : K), a * b = b * a) (one_mul : ∀ (a : K), 1 * a = a) (mul_inv_cancel : ∀ (a : K), a 0a * a⁻¹ = 1) (inv_zero : 0⁻¹ = 0) (left_distrib : ∀ (a b c : K), a * (b + c) = a * b + a * c) (exists_pair_ne : x y, x y) :

Define a Field structure on a Type by proving a minimized set of axioms. Note that this uses the default definitions for npow, nsmul, zsmul, div and sub See note [reducible non-instances].

Instances For