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Theorem gzcn 12979
Description: A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)
Assertion
Ref Expression
gzcn  |-  ( A  e.  ZZ [ _i ]  ->  A  e.  CC )

Proof of Theorem gzcn
StepHypRef Expression
1 elgz 12978 . 2  |-  ( A  e.  ZZ [ _i ] 
<->  ( A  e.  CC  /\  ( Re `  A
)  e.  ZZ  /\  ( Im `  A )  e.  ZZ ) )
21simp1bi 970 1  |-  ( A  e.  ZZ [ _i ]  ->  A  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   ` cfv 5255   CCcc 8735   ZZcz 10024   Recre 11582   Imcim 11583   ZZ [ _i ]cgz 12976
This theorem is referenced by:  gznegcl  12982  gzcjcl  12983  gzaddcl  12984  gzmulcl  12985  gzsubcl  12987  gzabssqcl  12988  4sqlem4a  12998  4sqlem4  12999  mul4sqlem  13000  mul4sq  13001  4sqlem12  13003  4sqlem17  13008  gzsubrg  16426  gzrngunitlem  16436  gzrngunit  16437  2sqlem2  20603  mul2sq  20604  2sqlem3  20605  cntotbnd  26520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-gz 12977
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