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Theorem elgz 14460
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 5872 . . . . 5  |-  ( x  =  A  ->  (
Re `  x )  =  ( Re `  A ) )
21eleq1d 2526 . . . 4  |-  ( x  =  A  ->  (
( Re `  x
)  e.  ZZ  <->  ( Re `  A )  e.  ZZ ) )
3 fveq2 5872 . . . . 5  |-  ( x  =  A  ->  (
Im `  x )  =  ( Im `  A ) )
43eleq1d 2526 . . . 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 14459 . . 3  |-  ZZ[_i]  =  {
x  e.  CC  | 
( ( Re `  x )  e.  ZZ  /\  ( Im `  x
)  e.  ZZ ) }
75, 6elrab2 3259 . 2  |-  ( A  e.  ZZ[_i]  <->  ( A  e.  CC  /\  ( ( Re `  A )  e.  ZZ  /\  (
Im `  A )  e.  ZZ ) ) )
8 3anass 977 . 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 973    = wceq 1395    e. wcel 1819   ` cfv 5594   CCcc 9507   ZZcz 10885   Recre 12941   Imcim 12942   ZZ[_i]cgz 14458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-gz 14459
This theorem is referenced by:  gzcn  14461  zgz  14462  igz  14463  gznegcl  14464  gzcjcl  14465  gzaddcl  14466  gzmulcl  14467  gzabssqcl  14470  4sqlem4a  14480  2sqlem2  23764  2sqlem3  23766  cntotbnd  30454
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