Theorem elgz 12943

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 5639	. . . . 5 |- ( x = A -> ( Re ` x ) = ( Re ` A ) )
2	1	eleq1d 2300	. . . 4 |- ( x = A -> ( ( Re ` x ) e. ZZ <-> ( Re ` A ) e. ZZ ) )
3		fveq2 5639	. . . . 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 12942	. . 3 |- ZZ[_i] = { x e. CC | ( ( Re ` x ) e. ZZ /\ ( Im ` x ) e. ZZ ) }
7	5, 6	elrab2 2965	. 2 |- ( A e. ZZ[_i] <-> ( A e. CC /\ ( ( Re ` A ) e. ZZ /\ ( Im ` A ) e. ZZ ) ) )
8		3anass 1008	. 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 1004   = wceq 1397   e. wcel 2202   ` cfv 5326   CCcc 8029   ZZcz 9478   Recre 11400   Imcim 11401   ZZ[_i]cgz 12941

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-gz 12942

This theorem is referenced by:  gzcn  12944  zgz  12945  igz  12946  gznegcl  12947  gzcjcl  12948  gzaddcl  12949  gzmulcl  12950  gzabssqcl  12953  4sqlem4a  12963  2sqlem2  15843  2sqlem3  15845