I found the following on Wikipedia, on the page for Inequalities:
If $a<b$ and $c<d$ then $a+c < b+d$.
It references Intermediate Algebra, but I don't see this specific property there.
Is there a name for this particular property? Is there a proof for this property from the other properties?
This holds in any ordered field (or more generally, partially ordered group); the only property we need to take advantage of is translation invariance and transitivity. That is, the properties that $$a<b\Leftrightarrow a+c<b+c$$ $$a<b \text{ and } b<c\Rightarrow a<c$$
Starting with $$a<b$$ we can, using translation invariance, add a constant to both sides: $$a+c<b+c$$ We can, using translation invariance, we can also establish, from $c<d$ translated by $b$ get that $$b+c<b+d$$ and using transitivity gives $$a+c<b+d$$
I don't know about the name.
But proof: yes. If $p$ and $q$ are positive numbers, then $p + q$ is a positive number; this follows from a commonly used postulate of the real number system. We can take $a < b$ to mean $b - a$ is positive.
Now let $p = b - a$ and $q = d - c$. Both are positive hence $p + q = (b + d) - (a + c)$ is as well. Therefore
$$a < b \ \text{ and } \ c < d \ \ \Longrightarrow \ \ a + c < b + d$$
