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Theorem gzcn 16970
Description: A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)
Assertion
Ref Expression
gzcn (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)

Proof of Theorem gzcn
StepHypRef Expression
1 elgz 16969 . 2 (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))
21simp1bi 1146 1 (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  cfv 6561  cc 11153  cz 12613  cre 15136  cim 15137  ℤ[i]cgz 16967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-gz 16968
This theorem is referenced by:  gznegcl  16973  gzcjcl  16974  gzaddcl  16975  gzmulcl  16976  gzsubcl  16978  gzabssqcl  16979  4sqlem4a  16989  4sqlem4  16990  mul4sqlem  16991  mul4sq  16992  4sqlem12  16994  4sqlem17  16999  gzsubrg  21439  gzrngunitlem  21450  gzrngunit  21451  2sqlem2  27462  mul2sq  27463  2sqlem3  27464  cntotbnd  37803
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