Gyrogroup implies left-inverse property loop

Statement

Suppose $(G,*)$ is a gyrogroup with neutral element $e$ and inverse map ${}^{-1}$. Then, $(G,*)$ is a left inverse property loop with the same neutral element $e$ and where the left inverse map is the same as the inverse map.

Proof

Given: A gyrogroup $(G,*)$, elements $a,b \in G$.

To prove: There exists unique $u \in G$ such that $a * u = b$ and there exists unique $y \in G$ such that $v * a = b$ (together, these prove it's a loop). Further, $u = a^{-1} * b$ (this shows the left inverse property).

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 $\operatorname{gyr}[e,x]$ is the identity map for all $x \in G$. We note that, for any $y \in G$, $e * (x * y) = x * y = (e * x) * y$. By uniqueness of $\operatorname{gyr}[e,x]y$, we get $\operatorname{gyr}[e,x]y = y$.
2 Suppose $x,z \in G$ are such that $z * x = e$. Then, $\operatorname{gyr}[z,x]$ is the identity map. gyroassociativity Step (1) By the left loop property, $\operatorname{gyr}[z,x] = \operatorname{gyr}[z * x,x] = \operatorname{gyr}[e,x]$ which is the identity map by Step (1).
3 $\operatorname{gyr}[a^{-1},a]$ is equal to the identity map. Step (2)
4 We have $a * u = b \implies a^{-1} * (a * u) = a^{-1} * b$. This simplifies to $u = a^{-1} * b$. Step (3) By Step (3), we obtain that $a^{-1} * (a * u) = (a^{-1} * a) * u$ and thus it simplifies to $e * u = u$.
5 $v * a = b \implies \operatorname{gyr}[v,a] = \operatorname{gyr}[b,a]$ left loop property we use that $\operatorname{gyr}[v,a] = \operatorname{gyr}[v * a,a]$
6 We have $v * a = b \implies v = b * \operatorname{gyr}[b,a](a^{-1})$. In particular, $v$ exists and is uniquely determined by $a,b$. Step (5) Consider the gyroassociativity of $v,a,a^{-1}$: $v * (a * a^{-1}) = (v * a) * \operatorname{gyr}[v,a]a^{-1}$. Simplifying the left side gives $v = (v * a) * \operatorname{gyr}[v,a]a^{-1}$. Now use Step (5) and we are done.

Steps (4) and (6) complete the proof.