Theorem elgz 12406

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 5534	. . . . 5 |- ( x = A -> ( Re ` x ) = ( Re ` A ) )
2	1	eleq1d 2258	. . . 4 |- ( x = A -> ( ( Re ` x ) e. ZZ <-> ( Re ` A ) e. ZZ ) )
3		fveq2 5534	. . . . 5 |- ( x = A -> ( Im ` x ) = ( Im ` A ) )
4	3	eleq1d 2258	. . . 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 12405	. . 3 |- ZZ[_i] = { x e. CC | ( ( Re ` x ) e. ZZ /\ ( Im ` x ) e. ZZ ) }
7	5, 6	elrab2 2911	. 2 |- ( A e. ZZ[_i] <-> ( A e. CC /\ ( ( Re ` A ) e. ZZ /\ ( Im ` A ) e. ZZ ) ) )
8		3anass 984	. 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 980   = wceq 1364   e. wcel 2160   ` cfv 5235   CCcc 7840   ZZcz 9284   Recre 10884   Imcim 10885   ZZ[_i]cgz 12404

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-gz 12405

This theorem is referenced by:  gzcn  12407  zgz  12408  igz  12409  gznegcl  12410  gzcjcl  12411  gzaddcl  12412  gzmulcl  12413  gzabssqcl  12416  4sqlem4a  12426  2sqlem2  14940  2sqlem3  14942