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

Theorem elgz 15554
Description: Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.)
Assertion
Ref Expression
elgz (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))

Proof of Theorem elgz
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6150 . . . . 5 (𝑥 = 𝐴 → (ℜ‘𝑥) = (ℜ‘𝐴))
21eleq1d 2688 . . . 4 (𝑥 = 𝐴 → ((ℜ‘𝑥) ∈ ℤ ↔ (ℜ‘𝐴) ∈ ℤ))
3 fveq2 6150 . . . . 5 (𝑥 = 𝐴 → (ℑ‘𝑥) = (ℑ‘𝐴))
43eleq1d 2688 . . . 4 (𝑥 = 𝐴 → ((ℑ‘𝑥) ∈ ℤ ↔ (ℑ‘𝐴) ∈ ℤ))
52, 4anbi12d 746 . . 3 (𝑥 = 𝐴 → (((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ) ↔ ((ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)))
6 df-gz 15553 . . 3 ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)}
75, 6elrab2 3353 . 2 (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ ((ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)))
8 3anass 1040 . 2 ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ) ↔ (𝐴 ∈ ℂ ∧ ((ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)))
97, 8bitr4i 267 1 (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  cfv 5850  cc 9879  cz 11322  cre 13766  cim 13767  ℤ[i]cgz 15552
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
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 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5813  df-fv 5858  df-gz 15553
This theorem is referenced by:  gzcn  15555  zgz  15556  igz  15557  gznegcl  15558  gzcjcl  15559  gzaddcl  15560  gzmulcl  15561  gzabssqcl  15564  4sqlem4a  15574  2sqlem2  25038  2sqlem3  25040  cntotbnd  33194
  Copyright terms: Public domain W3C validator