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Theorem gzcn 15843
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 15842 . 2 (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))
21simp1bi 1139 1 (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  cfv 6031  cc 10136  cz 11579  cre 14045  cim 14046  ℤ[i]cgz 15840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-gz 15841
This theorem is referenced by:  gznegcl  15846  gzcjcl  15847  gzaddcl  15848  gzmulcl  15849  gzsubcl  15851  gzabssqcl  15852  4sqlem4a  15862  4sqlem4  15863  mul4sqlem  15864  mul4sq  15865  4sqlem12  15867  4sqlem17  15872  gzsubrg  20015  gzrngunitlem  20026  gzrngunit  20027  2sqlem2  25364  mul2sq  25365  2sqlem3  25366  cntotbnd  33927
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