Is this proof of $a^n\times a=a^{n+1}$ correct?

Question

High-schooler here!

I am learning logic, and, consequently, I am trying to prove simple things. So I tried to prove $a^{n}\times a=a^{n+1}$:

Proof

Definitions

Definition of exponentiation (D1):

${\displaystyle a^{n}=\underbrace {a \times\cdots \times a}_{n}}$

Derivation

According to D1, we can define $a^{n}$ as:

1 — ${\displaystyle a^{n}=\underbrace {a \times\cdots \times a}_{n}}$

Now — using the idea that if a = b then ax = bx — we can multiply line 1 by $a$:

2 — ${\displaystyle a^{n}\times a=\underbrace {a \times\cdots \times a}_{n} \underbrace{\times a} _{1}}$

3 — ${\displaystyle a^{n}\times a=\underbrace {a \times\cdots \times a \times a} _{n+1}}\space\space$ By definition

4 — ${\displaystyle a^{n}\times a=\underbrace {a \times\cdots \times a \times a} _{n+1} = a^{n+1}}\space\space$ D1

Now we can use the transitive axiom (if a = b and b = c, then a = c) with line 4:

5 — $\therefore{\displaystyle a^{n}\times a=a^{n+1}}\space\space$

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I took D1 from https://en.wikipedia.org/wiki/Exponentiation.

The transitive axiom I found in https://www.assignmentpoint.com/science/mathematic/transitive-axiom.html.

Question

So I ask: is this proof correct? If so, how can I try to improve it?

Thank you very much for reading this.