Affine Group of One Dimension is Group/Proof 1

Proof
Taking the group axioms in turn:

$\text G 0$: Closure
Let $f_{ab}, f_{cd} \in \map {\operatorname {Af}_1} \R$.

Then:

By the field axioms, $a c \in \R_{\ne 0}$ and $a d + b \in \R$.

Thus $f_{ab} \circ f_{cd} \in \map {\operatorname {Af}_1} \R$ and so $\map {\operatorname {Af}_1} \R$ is closed.

$\text G 1$: Associativity
From Composition of Mappings is Associative, it follows directly that $\circ$ is associative on $\map {\operatorname {Af}_1} \R$.

$\text G 2$: Identity
By Identity of Affine Group of One Dimension, $\map {\operatorname {Af}_1} \R$ has $f_{1, 0}$ as an identity element.

$\text G 3$: Inverses
By Inverse in Affine Group of One Dimension, every element $f_{a b}$ of $\map {\operatorname {Af}_1} \R$ has an inverse $f_{c d}$ where $c = \dfrac 1 a$ and $d = \dfrac {-b} a$.

All the group axioms are thus seen to be fulfilled, and so $\map {\operatorname {Af}_1} \R$ is a group.