Theorem elgz 12371

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 5517	. . . . 5 |- ( x = A -> ( Re ` x ) = ( Re ` A ) )
2	1	eleq1d 2246	. . . 4 |- ( x = A -> ( ( Re ` x ) e. ZZ <-> ( Re ` A ) e. ZZ ) )
3		fveq2 5517	. . . . 5 |- ( x = A -> ( Im ` x ) = ( Im ` A ) )
4	3	eleq1d 2246	. . . 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 12370	. . 3 |- ZZ[_i] = { x e. CC | ( ( Re ` x ) e. ZZ /\ ( Im ` x ) e. ZZ ) }
7	5, 6	elrab2 2898	. 2 |- ( A e. ZZ[_i] <-> ( A e. CC /\ ( ( Re ` A ) e. ZZ /\ ( Im ` A ) e. ZZ ) ) )
8		3anass 982	. 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 978   = wceq 1353   e. wcel 2148   ` cfv 5218   CCcc 7811   ZZcz 9255   Recre 10851   Imcim 10852   ZZ[_i]cgz 12369

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-rab 2464  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-gz 12370

This theorem is referenced by:  gzcn  12372  zgz  12373  igz  12374  gznegcl  12375  gzcjcl  12376  gzaddcl  12377  gzmulcl  12378  gzabssqcl  12381  4sqlem4a  12391  2sqlem2  14547  2sqlem3  14549