The property of the sum and product of ordered numbers

Question

Let $\mu, \nu, \mu^{\prime}, \nu^{\prime}$ be ordinal numbers, respectively.
We have the property of summation like below:

$$ \nu < \nu^{\prime} \to \mu + \nu < \mu + \nu^{\prime}$$ $$ \mu < \mu^{\prime} \to \mu + \nu \leqq \mu^{\prime} + \nu$$

Why is the first expression $<$ and the second $\leqq$?
Where does this difference come from?

I have found several questions similar to this question, but no general proof for these two equations.

1. Ordinal numbers addition property: $b<c$ implies $b+a \le c+a$
   
2. Ordinal multiplication property: $\alpha<\beta$ implies $\alpha\gamma\le\beta\gamma$

Answer 1

The second inequality isn't strict cause we have, e.g. $$0+\omega=1+\omega.$$

To see the first is strict, observe that by definition, $$ \beta + (\alpha + 1) = (\beta + \alpha) + 1 > \beta + \alpha.$$