MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elgz Unicode version

Theorem elgz 12978
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 5525 . . . . 5  |-  ( x  =  A  ->  (
Re `  x )  =  ( Re `  A ) )
21eleq1d 2349 . . . 4  |-  ( x  =  A  ->  (
( Re `  x
)  e.  ZZ  <->  ( Re `  A )  e.  ZZ ) )
3 fveq2 5525 . . . . 5  |-  ( x  =  A  ->  (
Im `  x )  =  ( Im `  A ) )
43eleq1d 2349 . . . 4  |-  ( x  =  A  ->  (
( Im `  x
)  e.  ZZ  <->  ( Im `  A )  e.  ZZ ) )
52, 4anbi12d 691 . . 3  |-  ( x  =  A  ->  (
( ( Re `  x )  e.  ZZ  /\  ( Im `  x
)  e.  ZZ )  <-> 
( ( Re `  A )  e.  ZZ  /\  ( Im `  A
)  e.  ZZ ) ) )
6 df-gz 12977 . . 3  |-  ZZ [
_i ]  =  {
x  e.  CC  | 
( ( Re `  x )  e.  ZZ  /\  ( Im `  x
)  e.  ZZ ) }
75, 6elrab2 2925 . 2  |-  ( A  e.  ZZ [ _i ] 
<->  ( A  e.  CC  /\  ( ( Re `  A )  e.  ZZ  /\  ( Im `  A
)  e.  ZZ ) ) )
8 3anass 938 . 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 243 1  |-  ( A  e.  ZZ [ _i ] 
<->  ( A  e.  CC  /\  ( Re `  A
)  e.  ZZ  /\  ( Im `  A )  e.  ZZ ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255   CCcc 8735   ZZcz 10024   Recre 11582   Imcim 11583   ZZ [ _i ]cgz 12976
This theorem is referenced by:  gzcn  12979  zgz  12980  igz  12981  gznegcl  12982  gzcjcl  12983  gzaddcl  12984  gzmulcl  12985  gzabssqcl  12988  4sqlem4a  12998  2sqlem2  20603  2sqlem3  20605  cntotbnd  26520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-gz 12977
  Copyright terms: Public domain W3C validator