Proving a statement with inequality and min operator How can I prove/disprove the following statement.
$$\min\{m_1+m_2,p_1+p_2\} \geq \min\{m_1,p_1\} + \min\{m_2,p_2\}$$
Where $m_i$ and $p_i$ for $i=1,2$ are natural numbers.
My intuition says that this is true. Also, the right-hand expression is able to output $p_1+p_2$ or $m_1+m_2$ if indeed any of them is the minimum. However I am not able to formally prove this.
Disclaimer: This is not for a class. I am an engineer making my way the world of mathematics.

First note that $m \geq \min(m,p)$. Now suppose, $m_1+m_2 < p_1+p_2$, then $\min(m_1+m_2,p_1+p_2) =m_1+m_2 \geq \min(m_1,p_1) +\min(m_2,p_2)$. On the other hand, if $m_1+m_2 \geq p_1+p_2$, $\min(m_1+m_2,p_1+p_2) =p_1+p_2 \geq \min(m_1,p_1) +\min(m_2,p_2)$. 
