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Theorem elgz 12994
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 5541 . . . . 5  |-  ( x  =  A  ->  (
Re `  x )  =  ( Re `  A ) )
21eleq1d 2362 . . . 4  |-  ( x  =  A  ->  (
( Re `  x
)  e.  ZZ  <->  ( Re `  A )  e.  ZZ ) )
3 fveq2 5541 . . . . 5  |-  ( x  =  A  ->  (
Im `  x )  =  ( Im `  A ) )
43eleq1d 2362 . . . 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 12993 . . 3  |-  ZZ [
_i ]  =  {
x  e.  CC  | 
( ( Re `  x )  e.  ZZ  /\  ( Im `  x
)  e.  ZZ ) }
75, 6elrab2 2938 . 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 1632    e. wcel 1696   ` cfv 5271   CCcc 8751   ZZcz 10040   Recre 11598   Imcim 11599   ZZ [ _i ]cgz 12992
This theorem is referenced by:  gzcn  12995  zgz  12996  igz  12997  gznegcl  12998  gzcjcl  12999  gzaddcl  13000  gzmulcl  13001  gzabssqcl  13004  4sqlem4a  13014  2sqlem2  20619  2sqlem3  20621  cntotbnd  26623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-gz 12993
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