Prove the absolute value of ab is equal to absolute value of a multiplied by absolute value of b

If either a or b or both are 0, we have |ab|= 0 and |a|*|b|=0

For any real number a and b non equal to 0, one of the 3 following cases are true:

i) both a and b are positive:

then  |ab|=ab, |a|=a, |b|=b

ab=a*b
|ab|=|a|*|b|

ii) both a and b are negative:

then |a|=-a, |b|=-b

|ab|=ab, for example if a=-3, b=-7: |(-3)(-7)|=|21|=21=(-3)(-7)=a*b

so

ab=(-a)*(-b)

|ab|=|a|*|b|

iii) one of them is positive and the other negative. 

In our case let a be positive, b negative:

|a|=a, |b|=-b, 

and |ab|=-ab, for example if a=3, b=-4; |3*(-4)|=|-12|=12=3*(4)=a*(-b)

thus:

-ab= a*(-b)
|ab|= |a||b|.

In each possible case of the signs of a and b we get: |ab|= |a||b|.