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Definition df-gz 13300
Description: Define the set of gaussian integers, which are complex numbers whose real and imaginary parts are integers. (Note that the  [
_i ] is actually part of the symbol token and has no independent meaning.) (Contributed by Mario Carneiro, 14-Jul-2014.)
Assertion
Ref Expression
df-gz  |-  ZZ [
_i ]  =  {
x  e.  CC  | 
( ( Re `  x )  e.  ZZ  /\  ( Im `  x
)  e.  ZZ ) }

Detailed syntax breakdown of Definition df-gz
StepHypRef Expression
1 cgz 13299 . 2  class  ZZ [
_i ]
2 vx . . . . . . 7  set  x
32cv 1652 . . . . . 6  class  x
4 cre 11904 . . . . . 6  class  Re
53, 4cfv 5456 . . . . 5  class  ( Re
`  x )
6 cz 10284 . . . . 5  class  ZZ
75, 6wcel 1726 . . . 4  wff  ( Re
`  x )  e.  ZZ
8 cim 11905 . . . . . 6  class  Im
93, 8cfv 5456 . . . . 5  class  ( Im
`  x )
109, 6wcel 1726 . . . 4  wff  ( Im
`  x )  e.  ZZ
117, 10wa 360 . . 3  wff  ( ( Re `  x )  e.  ZZ  /\  (
Im `  x )  e.  ZZ )
12 cc 8990 . . 3  class  CC
1311, 2, 12crab 2711 . 2  class  { x  e.  CC  |  ( ( Re `  x )  e.  ZZ  /\  (
Im `  x )  e.  ZZ ) }
141, 13wceq 1653 1  wff  ZZ [
_i ]  =  {
x  e.  CC  | 
( ( Re `  x )  e.  ZZ  /\  ( Im `  x
)  e.  ZZ ) }
Colors of variables: wff set class
This definition is referenced by:  elgz  13301
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