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Given that $x\cdot(y+z)=(x\cdot y)+(x\cdot z)$ and $x+(y\cdot z)=(x+y)\cdot (x+z)$, what is the name for the opposite of those rules?

Say I'm trying to prove the opposite, and I need to simplify from the form $(x\cdot y)+(x\cdot z)$ to $x\cdot(y+z)$, what would I write for the rule I used to do it?


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up vote 5 down vote accepted

The distributive law applies to, and can be cited for, "moving" from either direction to the other; i.e., since it is an equality/identity, what you refer to as the "opposite" is really nothing other than the "equivalent", i.e., the distributive law you're invoking ensures $$x\cdot(y+z)=(x\cdot y)+(x\cdot z) \iff (x\cdot y)+(x\cdot z) = x\cdot(y+z)$$

Hence, you can cite the distributive law as your justification (multiplication distributes over addition in $\mathbb{R}$, assuming x, y, and z are real numbers. Ditto for x, y, z in $\mathbb{Z}$, or in $\mathbb{Q}$, etc.)

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