Proof of Addition on ℝ Is an Associative and Commutative Binary Operation

By Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (i), $+$ is a binary operation on the set $\mathbb{R}$, so Associative and Commutative Binary Operations, and Neutral Elements applies to it with $X=\mathbb{R}$. By Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (iii), $(u+v)+w=u+(v+w)$ for all $u,v,w\in\mathbb{R}$; this is the condition of Associative and Commutative Binary Operations, and Neutral Elements (i) for $+$, which gives (i). By Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (ii), $u+v=v+u$ for all $u,v\in\mathbb{R}$; this is the condition of Associative and Commutative Binary Operations, and Neutral Elements (ii) for $+$, which gives (ii).