Theorem elgz 12512

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 5555	. . . . 5 |- ( x = A -> ( Re ` x ) = ( Re ` A ) )
2	1	eleq1d 2262	. . . 4 |- ( x = A -> ( ( Re ` x ) e. ZZ <-> ( Re ` A ) e. ZZ ) )
3		fveq2 5555	. . . . 5 |- ( x = A -> ( Im ` x ) = ( Im ` A ) )
4	3	eleq1d 2262	. . . 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 12511	. . 3 |- ZZ[_i] = { x e. CC | ( ( Re ` x ) e. ZZ /\ ( Im ` x ) e. ZZ ) }
7	5, 6	elrab2 2920	. 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 2164   ` cfv 5255   CCcc 7872   ZZcz 9320   Recre 10987   Imcim 10988   ZZ[_i]cgz 12510

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 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-rab 2481  df-v 2762  df-un 3158  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-iota 5216  df-fv 5263  df-gz 12511

This theorem is referenced by:  gzcn  12513  zgz  12514  igz  12515  gznegcl  12516  gzcjcl  12517  gzaddcl  12518  gzmulcl  12519  gzabssqcl  12522  4sqlem4a  12532  2sqlem2  15272  2sqlem3  15274