Provide reasons (mostly vector space properties) as justification for each of the six steps of the following

Question:

Provide reasons (mostly vector space properties) as justification for each of the six steps of the following proof.
Theorem
For any vector u ∈ Cm, 0u = 0:
Proof: Let u ∈ Cm.
0 = 0u + (-0u)
= (0 + 0)u + (-0u)
= (0u + 0u) + (-0u)
= 0u + (0u + (-0u))
= 0u + 0
= 0u