The product of any number and zero is zero 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! 

$$\begin{align} 0 + \color{Green}{0⋅a} = 0⋅a &amp; = (0 + 0)⋅a  \\ &amp; = 0⋅a + \color{Green}{0⋅a} \end{align}$$ 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?
you want to prove 

$$0\cdot a = 0$$

assume, where x is any number

$$0\cdot a = x \tag{1}$$ 

We know that

$$0\cdot a = (0+0)\cdot a$$

so

$$(0+0)\cdot a = x$$

$$0\cdot a + 0\cdot a = x\tag{2}$$

subtracting $(2)-(1)$

$$(0\cdot a + 0\cdot a) - (0\cdot a) = x-x = 0$$

that is 

$$0\cdot a + 0\cdot a = 0\cdot a$$

so

$$0\cdot a = 0\cdot a - 0\cdot a$$

as a result

$$0\cdot a = 0$$
if you want to prove 0.a=0 you need not to prove it through algebra.
consider following 2.4=8(dot means multiply)
what actually we have done is 2+2+2+2=8
multiplication itself do not have any significance
it can be said as short form of addition or shortcut of addition for adding large no.
so what will be value of 0.4=?
it is 0+0+0+0+0=0
similarly 0.a is 0+0+0...a