Theorem gcdid0

Description: The gcd of an integer and 0 is the integer's absolute value. Theorem 1.4(d)2 in ApostolNT p. 16. (Contributed by Paul Chapman, 31-Mar-2011)

	Ref	Expression
Assertion	gcdid0	|- ( N e. ZZ -> ( N gcd 0 ) = ( abs ` N ) )

Proof

Step	Hyp	Ref	Expression
1		0z	|- 0 e. ZZ
2		gcdcom	|- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 gcd N ) = ( N gcd 0 ) )
3	1 2	mpan	|- ( N e. ZZ -> ( 0 gcd N ) = ( N gcd 0 ) )
4		gcd0id	|- ( N e. ZZ -> ( 0 gcd N ) = ( abs ` N ) )
5	3 4	eqtr3d	|- ( N e. ZZ -> ( N gcd 0 ) = ( abs ` N ) )