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The distributive property of real numbers states that $“$for all $a, b, c \in \mathbb{R}$, we've $a⋅(b + c) = a⋅b + a⋅c$ and $(b + c)⋅a = b⋅a + c⋅a”$. How to prove this field property of real numbers? Is there any rigorous proof of this? Why was this property accepted as an axiom? It doesn't seem trivial to me like other axioms of real numbers. Was the multiplication between two negative real numbers or a negative real number & a positive real number defined before setting up the distributive property of real numbers as an axiomatic property of real numbers?

Basically, I want to know the proof of the following property of $\mathbb{R}$:‌ $“(-a)⋅(-b + c) = (-a)⋅(-b) + (-a)⋅c$ and $(-b + c)⋅(-a) = (-b)⋅(-a) + c⋅(-a)$ where $-a$ and $-b$ are any negative real number and $c$ is any positive real number$”$.

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  • $\begingroup$ To answer just one of your questions, properties of this kind aren't called "axioms" because they are trivial. Axioms are simply properties that are defined as true for a certain mathematical object, but they can be as complex or as trivial as you want, depending on what it is you're defining. In this case you will see the distributive property described as an "axiom" because the distributive property is one of the ring axioms, that is, in order for something to be a ring then multiplication must distribute over addition (among other things). $\endgroup$
    – SeraPhim
    Commented Apr 26, 2021 at 17:14

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This depends on how you define $\mathbb{R}$ and addition. In the most formal of treatments, you start by formalizing integer arithmetic, then rational arithmetic, and then real arithmetic. To go from rationals to reals, read about Dedekind cuts--any introductory analysis book (ie, Baby Rudin) will have some information about them. A set theory book will teach you how to go from sets to defining integers/rationals and there arithmetic.

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