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Theorem gzcn 12995
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 12994 . 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 1696   ` cfv 5271   CCcc 8751   ZZcz 10040   Recre 11598   Imcim 11599   ZZ [ _i ]cgz 12992
This theorem is referenced by:  gznegcl  12998  gzcjcl  12999  gzaddcl  13000  gzmulcl  13001  gzsubcl  13003  gzabssqcl  13004  4sqlem4a  13014  4sqlem4  13015  mul4sqlem  13016  mul4sq  13017  4sqlem12  13019  4sqlem17  13024  gzsubrg  16442  gzrngunitlem  16452  gzrngunit  16453  2sqlem2  20619  mul2sq  20620  2sqlem3  20621  cntotbnd  26623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-gz 12993
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