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Theorem gzcn 14551
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 14550 . 2  |-  ( A  e.  ZZ[_i]  <->  ( A  e.  CC  /\  ( Re
`  A )  e.  ZZ  /\  ( Im
`  A )  e.  ZZ ) )
21simp1bi 1012 1  |-  ( A  e.  ZZ[_i]  ->  A  e.  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842   ` cfv 5525   CCcc 9440   ZZcz 10825   Recre 12986   Imcim 12987   ZZ[_i]cgz 14548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5489  df-fv 5533  df-gz 14549
This theorem is referenced by:  gznegcl  14554  gzcjcl  14555  gzaddcl  14556  gzmulcl  14557  gzsubcl  14559  gzabssqcl  14560  4sqlem4a  14570  4sqlem4  14571  mul4sqlem  14572  mul4sq  14573  4sqlem12  14575  4sqlem17  14580  gzsubrg  18684  gzrngunitlem  18694  gzrngunit  18695  2sqlem2  23912  mul2sq  23913  2sqlem3  23914  cntotbnd  31555
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