Metamath Proof Explorer

Theorem gcdnn0id

Description: The gcd of a nonnegative integer and itself is the integer. (Contributed by SN, 25-Aug-2024)

Ref Expression
Assertion gcdnn0id 
|- ( N e. NN0 -> ( N gcd N ) = N )

Proof

Step Hyp Ref Expression
1 nn0z 
 |-  ( N e. NN0 -> N e. ZZ )
2 gcdid 
 |-  ( N e. ZZ -> ( N gcd N ) = ( abs ` N ) )
3 1 2 syl 
 |-  ( N e. NN0 -> ( N gcd N ) = ( abs ` N ) )
4 nn0re 
 |-  ( N e. NN0 -> N e. RR )
5 nn0ge0 
 |-  ( N e. NN0 -> 0 <_ N )
6 4 5 absidd 
 |-  ( N e. NN0 -> ( abs ` N ) = N )
7 3 6 eqtrd 
 |-  ( N e. NN0 -> ( N gcd N ) = N )