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Theorem gzcn 14302
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 14301 . 2  |-  ( A  e.  ZZ[_i]  <->  ( A  e.  CC  /\  ( Re
`  A )  e.  ZZ  /\  ( Im
`  A )  e.  ZZ ) )
21simp1bi 1011 1  |-  ( A  e.  ZZ[_i]  ->  A  e.  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   ` cfv 5586   CCcc 9486   ZZcz 10860   Recre 12887   Imcim 12888   ZZ[_i]cgz 14299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-gz 14300
This theorem is referenced by:  gznegcl  14305  gzcjcl  14306  gzaddcl  14307  gzmulcl  14308  gzsubcl  14310  gzabssqcl  14311  4sqlem4a  14321  4sqlem4  14322  mul4sqlem  14323  mul4sq  14324  4sqlem12  14326  4sqlem17  14331  gzsubrg  18237  gzrngunitlem  18247  gzrngunit  18248  2sqlem2  23364  mul2sq  23365  2sqlem3  23366  cntotbnd  29893
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