I don't understand why this proof is valid. The proof just swaps the left and right hand side of the equality. Is this a valid method of proof for equalities? If so, what's the logical intuition behind it?
Theorem: $0⋅a = 0$ for any integer $a$.
Proof: Add zero!
$\begin{align} 0 + \color{Green}{0⋅a} = 0⋅a & = (0 + 0)⋅a \\ & = 0⋅a + \color{Green}{0⋅a} \end{align}$ by using the identity property twice and then the distributive law. Use cancellation on the first and last piece to get $0 = 0⋅a$
The problem is not algebra. I don't understand why transforming $0\cdot a =0$ to $0=0\cdot a$ proves this. I mean to prove $a=b$, we transform it to $b=a$, why is it a proof method. What logic does it depend on?