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Theorem gzcn 15560
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 15559 . 2 (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))
21simp1bi 1074 1 (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  cfv 5847  cc 9878  cz 11321  cre 13771  cim 13772  ℤ[i]cgz 15557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-gz 15558
This theorem is referenced by:  gznegcl  15563  gzcjcl  15564  gzaddcl  15565  gzmulcl  15566  gzsubcl  15568  gzabssqcl  15569  4sqlem4a  15579  4sqlem4  15580  mul4sqlem  15581  mul4sq  15582  4sqlem12  15584  4sqlem17  15589  gzsubrg  19719  gzrngunitlem  19730  gzrngunit  19731  2sqlem2  25043  mul2sq  25044  2sqlem3  25045  cntotbnd  33227
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