MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gzcn Structured version   Visualization version   GIF version

Theorem gzcn 16865
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 16864 . 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 2107  cfv 6544  cc 11108  cz 12558  cre 15044  cim 15045  ℤ[i]cgz 16862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-gz 16863
This theorem is referenced by:  gznegcl  16868  gzcjcl  16869  gzaddcl  16870  gzmulcl  16871  gzsubcl  16873  gzabssqcl  16874  4sqlem4a  16884  4sqlem4  16885  mul4sqlem  16886  mul4sq  16887  4sqlem12  16889  4sqlem17  16894  gzsubrg  20999  gzrngunitlem  21010  gzrngunit  21011  2sqlem2  26921  mul2sq  26922  2sqlem3  26923  cntotbnd  36664
  Copyright terms: Public domain W3C validator