MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gzcn Structured version   Unicode version

Theorem gzcn 13290
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 13289 . 2  |-  ( A  e.  ZZ [ _i ] 
<->  ( A  e.  CC  /\  ( Re `  A
)  e.  ZZ  /\  ( Im `  A )  e.  ZZ ) )
21simp1bi 972 1  |-  ( A  e.  ZZ [ _i ]  ->  A  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   ` cfv 5446   CCcc 8978   ZZcz 10272   Recre 11892   Imcim 11893   ZZ [ _i ]cgz 13287
This theorem is referenced by:  gznegcl  13293  gzcjcl  13294  gzaddcl  13295  gzmulcl  13296  gzsubcl  13298  gzabssqcl  13299  4sqlem4a  13309  4sqlem4  13310  mul4sqlem  13311  mul4sq  13312  4sqlem12  13314  4sqlem17  13319  gzsubrg  16743  gzrngunitlem  16753  gzrngunit  16754  2sqlem2  21138  mul2sq  21139  2sqlem3  21140  cntotbnd  26459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-gz 13288
  Copyright terms: Public domain W3C validator