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Theorem elgz 13991
Description: Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.)
Assertion
Ref Expression
elgz  |-  ( A  e.  ZZ[_i]  <->  ( A  e.  CC  /\  ( Re
`  A )  e.  ZZ  /\  ( Im
`  A )  e.  ZZ ) )

Proof of Theorem elgz
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5690 . . . . 5  |-  ( x  =  A  ->  (
Re `  x )  =  ( Re `  A ) )
21eleq1d 2508 . . . 4  |-  ( x  =  A  ->  (
( Re `  x
)  e.  ZZ  <->  ( Re `  A )  e.  ZZ ) )
3 fveq2 5690 . . . . 5  |-  ( x  =  A  ->  (
Im `  x )  =  ( Im `  A ) )
43eleq1d 2508 . . . 4  |-  ( x  =  A  ->  (
( Im `  x
)  e.  ZZ  <->  ( Im `  A )  e.  ZZ ) )
52, 4anbi12d 710 . . 3  |-  ( x  =  A  ->  (
( ( Re `  x )  e.  ZZ  /\  ( Im `  x
)  e.  ZZ )  <-> 
( ( Re `  A )  e.  ZZ  /\  ( Im `  A
)  e.  ZZ ) ) )
6 df-gz 13990 . . 3  |-  ZZ[_i]  =  {
x  e.  CC  | 
( ( Re `  x )  e.  ZZ  /\  ( Im `  x
)  e.  ZZ ) }
75, 6elrab2 3118 . 2  |-  ( A  e.  ZZ[_i]  <->  ( A  e.  CC  /\  ( ( Re `  A )  e.  ZZ  /\  (
Im `  A )  e.  ZZ ) ) )
8 3anass 969 . 2  |-  ( ( A  e.  CC  /\  ( Re `  A )  e.  ZZ  /\  (
Im `  A )  e.  ZZ )  <->  ( A  e.  CC  /\  ( ( Re `  A )  e.  ZZ  /\  (
Im `  A )  e.  ZZ ) ) )
97, 8bitr4i 252 1  |-  ( A  e.  ZZ[_i]  <->  ( A  e.  CC  /\  ( Re
`  A )  e.  ZZ  /\  ( Im
`  A )  e.  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5417   CCcc 9279   ZZcz 10645   Recre 12585   Imcim 12586   ZZ[_i]cgz 13989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-iota 5380  df-fv 5425  df-gz 13990
This theorem is referenced by:  gzcn  13992  zgz  13993  igz  13994  gznegcl  13995  gzcjcl  13996  gzaddcl  13997  gzmulcl  13998  gzabssqcl  14001  4sqlem4a  14011  2sqlem2  22702  2sqlem3  22704  cntotbnd  28693
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