How do you prove the distributive property?

How do you prove the distributive property?

You don't. It only exists because of certain axioms we just assume to be true.

read spivak chapter 1.

also this

Depends, for what? What set we are talking about, and how did you define your elements and operations? Proving the distribuitive property for the naturals build with the Von Neumann construction is entirely different from showing the distribuitivity of the reals build from dedeking cuts (which is also different from showing the distribuitivity of reals as equivalence classes of Cauchy sequences. Which is also different of doing it for the rationals build as a field of fractions of the integers and also different from showing distribuitivity for Complex Numbers build as the quotient field of R by the ideal created by x^2+1. And that last one is different from showing it for reals build as tuples of reals).

For Peano Arithmetic you just gotta define sum as iterated successor operation and the product as iterated sums. Then just apply the definitions.

It show have been
>complex numbers build as tuples of reals

and not
>reals build as tuples of reals

...

>these are the kinds of people who save brainlet wojak images

Explain

IIRC Rudin proves all the field axioms when he constructs the reals. Go read him.

Rudin is a brainlet book though

It's just perfect for this thread then

Why do you think that?

Literally the stewart of algebra books

>now I have to delete all my brainlet memes
fugg

>Rudin
>Algebra
What did he mean by this?

I get that Stewart isn't exactly proof heavy but man it's a really good book for brainlets who just wanna get the hang of calculus. I'm not sure that's such a bad thing.

Anything that caters to brainlets is a cancer that needs to be eradicated

What's a good proof heavy calculus book?
>inb4 Spivak

maybe I'm just looking for a real analysis book

Not him, but R. Courant and E. Moise both wrote pretty good books. Courant's is old book however, and pretty hard to read due to that (he also has a lot of content, like some numerical stuff, ODEs, Fourier Series, a bit of physics, etc).
I have heard good things about T. Apostol's book, but I did not read it.

these wojacks are getting better.

Stewart is proof heavy. I was under the mistaken impression that it wasn't, like most of you, and I looked through it. It has proofs for almost everything

If the math is correct then why does it matter if it's simple?

Fuck off shills

The fundamental theory of arithmatic

It's not proof heavy. He lists theorems for important ideas and proves some of them, but it's only out of courtesy. Stewart is very much computation based

he lists ALL the relevant theorems, and he proves all but two: change of variables and stoke's. I invite you to read it if you don't believe me. it is computation based, though, and has a lot of that.