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Theorem gzcn 16262
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 16261 . 2 (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))
21simp1bi 1141 1 (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  cfv 6350  cc 10529  cz 11975  cre 14450  cim 14451  ℤ[i]cgz 16259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-iota 6309  df-fv 6358  df-gz 16260
This theorem is referenced by:  gznegcl  16265  gzcjcl  16266  gzaddcl  16267  gzmulcl  16268  gzsubcl  16270  gzabssqcl  16271  4sqlem4a  16281  4sqlem4  16282  mul4sqlem  16283  mul4sq  16284  4sqlem12  16286  4sqlem17  16291  gzsubrg  20593  gzrngunitlem  20604  gzrngunit  20605  2sqlem2  25988  mul2sq  25989  2sqlem3  25990  cntotbnd  35068
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