Zero Product Property PROOF

If $ab=0$ then $a=0$ OR $b=0$.

Case 1: If $a-0$ then the statement is true. Case 2: If $a\ne 0$, then $a\ne 0$ $a\left(\frac{1}{a}\right)=1$ ; Existence of Additive Inverse

Now, by assumption, $ab=0$ $ab\left(\frac{1}{a}\right)=0\left(\frac{1}{a}\right)$ ; MPE $a\left(b\frac{1}{a}\right)=0$ ; Associativity $a(\left(\frac{1}{a}\right)b)$ ; Commutativity $(a\frac{1}{a})b=0$ ; Associativity $1b=0$ ; Existence of Multiplicative Inverse $b=0$ ; Existence of Multiplicative Identity

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