Proof for checking correctness of the multiplication result

Question

Yesterday I was reading a topic about arithmetic and accidentally came across with an interesting way to verify the multiplication of two numbers.

For example, let's say i have $$123\times 456=56088$$ and i doubt with this answer, so i want to verify that the result is correct. So, i have to show the implication : "If the digits sum of the multiplier times the digits sum of the multiplicand is equal to the digits sum of the result, then the result is correct" is true.

1. Digits sum of the multiplier $123$ $\Rightarrow \, 1+2+3=6$
2. Digits sum of the multiplicand $456$ $\Rightarrow \, 4+5+6=15$ $\Rightarrow 1+5 = 6$
3. Digits sum of the product of those digits $\Rightarrow 6\times 6=36$ $\Rightarrow 3+6=9$
4. Digits sum of the result product $56088$ $\Rightarrow 5+6+0+8+8=27$ $\Rightarrow 2+7=9$
5. Since $9=9$, then the product of those two numbers is correct.

I searched for proof about this to convince me so i can use it whenever i need it, but i got nothing. Have you ever seen the proof of this statement formally?