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Theorem elgz 13219
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 5661 . . . . 5  |-  ( x  =  A  ->  (
Re `  x )  =  ( Re `  A ) )
21eleq1d 2446 . . . 4  |-  ( x  =  A  ->  (
( Re `  x
)  e.  ZZ  <->  ( Re `  A )  e.  ZZ ) )
3 fveq2 5661 . . . . 5  |-  ( x  =  A  ->  (
Im `  x )  =  ( Im `  A ) )
43eleq1d 2446 . . . 4  |-  ( x  =  A  ->  (
( Im `  x
)  e.  ZZ  <->  ( Im `  A )  e.  ZZ ) )
52, 4anbi12d 692 . . 3  |-  ( x  =  A  ->  (
( ( Re `  x )  e.  ZZ  /\  ( Im `  x
)  e.  ZZ )  <-> 
( ( Re `  A )  e.  ZZ  /\  ( Im `  A
)  e.  ZZ ) ) )
6 df-gz 13218 . . 3  |-  ZZ [
_i ]  =  {
x  e.  CC  | 
( ( Re `  x )  e.  ZZ  /\  ( Im `  x
)  e.  ZZ ) }
75, 6elrab2 3030 . 2  |-  ( A  e.  ZZ [ _i ] 
<->  ( A  e.  CC  /\  ( ( Re `  A )  e.  ZZ  /\  ( Im `  A
)  e.  ZZ ) ) )
8 3anass 940 . 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 244 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5387   CCcc 8914   ZZcz 10207   Recre 11822   Imcim 11823   ZZ [ _i ]cgz 13217
This theorem is referenced by:  gzcn  13220  zgz  13221  igz  13222  gznegcl  13223  gzcjcl  13224  gzaddcl  13225  gzmulcl  13226  gzabssqcl  13229  4sqlem4a  13239  2sqlem2  21008  2sqlem3  21010  cntotbnd  26189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-gz 13218
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