I'm proving the distributive property for fuzzy sets and I'm having a bad time with the resolution of the following expression:
$$\max[\varphi_A(x),\min[\varphi_B(x),\varphi_C(x)]] = \\ \frac{1}{2}[\varphi_A(x) + \min[\varphi_B(x),\varphi_C(x)] + |\varphi_A(x) - \min[\varphi_B(x),\varphi_C(x)]|] = \\ \frac{1}{2}[\varphi_A(x) + \frac{1}{2}[\varphi_B(x) + \varphi_C(x) + |\varphi_B(x) - \varphi_C(x)|] + |\varphi_A(x) - \frac{1}{2}[\varphi_B(x) + \varphi_C(x) + |\varphi_B(x) - \varphi_C(x)|]|] = ?$$
from this point on I'm clueless. How can I manipulate those absolute values?
That’s doing it the hard way. Just prove that for any $x,y,z\in\Bbb R$,
$$\max\big\{x,\min\{y,z\}\big\}=\min\big\{\max\{x,y\},\max\{x,z\}\big\}\,.\tag{1}$$
Without loss of generality we may assume that $y\le z$. If $x<y$, both sides of $(1)$ are $y$, and if $x\ge y$, both sides are $x$: the righthand side is either $\min\{x,x\}=x$ or $\min\{x,z\}=x$.