(related to Proposition: Distributivity Law For Integers)
By the definition of integers, the integers \(x,y,z \in \mathbb Z\) are identified by pairs \(x:=[a,b]\), \(y:=[c,d]\) and \(z:=[e,f]\) of natural numbers \(a,b,c,d,e,f\in\mathbb N\). Because multiplying integers is commutative, it is sufficient to show the left-distributivity law \[x\cdot(y+z)=(x\cdot y)+(x\cdot z).\]
The left-distributivity law can be proven using the following mathematical definitions and concepts: * definition of adding integers, * definition of multiplying integers, * distributivity law for natural numbers, and * commutativity law for adding natural numbers. The proof follows:
\[\begin{array}{ccll} x\cdot(y+z)&=&[a,b]\cdot([c,d] + [e,f])&\text{by definition of integers}\\ &=&[a,b]\cdot[c+e,d+f]&\text{by definition of adding integers}\\ &=&[a(c+e)+b(d+f),a(d+f)+b(c+e)]&\text{by definition of multiplying integers}\\ &=&[ac+ae+bd+bf,ad+af+bc+be]&\text{by distributivity law for natural numbers}\\ &=&[ac+bd+ae+bf,ad+bc+af+be]&\text{by commutativity of adding natural numbers}\\ &=&[ac+bd,ad+bc]+[ae+bf,af+be]&\text{by definition of adding integers}\\ &=&([a,b]\cdot[c,d])+([a,b]\cdot[e,f])&\text{by definition of multiplying integers}\\ &=&(x\cdot y)+(x\cdot z)&\text{by definition of integers} \end{array} \]
