Theorem elgz 13024

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.

Step	Hyp	Ref	Expression
1		fveq2 5648	. . . . 5 |- ( x = A -> ( Re ` x ) = ( Re ` A ) )
2	1	eleq1d 2300	. . . 4 |- ( x = A -> ( ( Re ` x ) e. ZZ <-> ( Re ` A ) e. ZZ ) )
3		fveq2 5648	. . . . 5 |- ( x = A -> ( Im ` x ) = ( Im ` A ) )
4	3	eleq1d 2300	. . . 4 |- ( x = A -> ( ( Im ` x ) e. ZZ <-> ( Im ` A ) e. ZZ ) )
5	2, 4	anbi12d 473	. . 3 |- ( x = A -> ( ( ( Re ` x ) e. ZZ /\ ( Im ` x ) e. ZZ ) <-> ( ( Re ` A ) e. ZZ /\ ( Im ` A ) e. ZZ ) ) )
6		df-gz 13023	. . 3 |- ZZ[_i] = { x e. CC | ( ( Re ` x ) e. ZZ /\ ( Im ` x ) e. ZZ ) }
7	5, 6	elrab2 2966	. 2 |- ( A e. ZZ[_i] <-> ( A e. CC /\ ( ( Re ` A ) e. ZZ /\ ( Im ` A ) e. ZZ ) ) )
8		3anass 1009	. 2 |- ( ( A e. CC /\ ( Re ` A ) e. ZZ /\ ( Im ` A ) e. ZZ ) <-> ( A e. CC /\ ( ( Re ` A ) e. ZZ /\ ( Im ` A ) e. ZZ ) ) )
9	7, 8	bitr4i 187	1 |- ( A e. ZZ[_i] <-> ( A e. CC /\ ( Re ` A ) e. ZZ /\ ( Im ` A ) e. ZZ ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   ` cfv 5333   CCcc 8090   ZZcz 9540   Recre 11480   Imcim 11481   ZZ[_i]cgz 13022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-rab 2520  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-gz 13023

This theorem is referenced by:  gzcn  13025  zgz  13026  igz  13027  gznegcl  13028  gzcjcl  13029  gzaddcl  13030  gzmulcl  13031  gzabssqcl  13034  4sqlem4a  13044  2sqlem2  15934  2sqlem3  15936