Is there a law that you can add or multiply to both sides of an equation?

It seems that given a statement $a = b$, that $a + c = b + c$ is assumed also to be true.

Why isn't this an axiom of arithmetic, like the commutative law or associative law?

Or is it a consequence of some other axiom of arithmetic?

Thanks!

Edit: I understand the intuitive meaning of equality. Answers that stated that $a = b$ means they are the same number or object make sense but what I'm asking is if there is an explicit law of replacement that allows us to make this intuitive truth a valid mathematical deduction. For example is there an axiom of Peano's Axioms or some other axiomatic system that allows for adding or multiplying both sides of an equation by the same number?

In all the texts I've come across I've never seen an axiom that states if $a = b$ then $a + c = b + c$. I have however seen if $a < b$ then $a + c < b + c$. In my view $<$ and $=$ are similar so the absence of a definition for equality is strange.

If you are given that $$a = b$$ then you can always infer that $$f(a) = f(b)$$ for a function $f$. That's what it means to be a function. However, if you are given $$f(a) = f(b)$$ then you can't infer $$a = b$$ unless the function is injective (invertible) over a domain containing $a$ and $b$.

For your problem, $f(x) = x + c$.

This is a basic property of equality. An equation like $$a=b$$ means that $a$ and $b$ are different names for the same number. If you do something to $a$, and you do the same thing to $b$, you must get the same result because $a$ and $b$ were the same to begin with.

For example, how do we know that Samuel Clemens and Mark Twain were equal in height? Simple: Because they were the same person.

How do we know that $a+c$ and $b+c$ are equal numbers? Because $a$ and $b$ are the same number.