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Theorem elgz 16866
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 6891 . . . . 5 (𝑥 = 𝐴 → (ℜ‘𝑥) = (ℜ‘𝐴))
21eleq1d 2818 . . . 4 (𝑥 = 𝐴 → ((ℜ‘𝑥) ∈ ℤ ↔ (ℜ‘𝐴) ∈ ℤ))
3 fveq2 6891 . . . . 5 (𝑥 = 𝐴 → (ℑ‘𝑥) = (ℑ‘𝐴))
43eleq1d 2818 . . . 4 (𝑥 = 𝐴 → ((ℑ‘𝑥) ∈ ℤ ↔ (ℑ‘𝐴) ∈ ℤ))
52, 4anbi12d 631 . . 3 (𝑥 = 𝐴 → (((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ) ↔ ((ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)))
6 df-gz 16865 . . 3 ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)}
75, 6elrab2 3686 . 2 (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ ((ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)))
8 3anass 1095 . 2 ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ) ↔ (𝐴 ∈ ℂ ∧ ((ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)))
97, 8bitr4i 277 1 (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  cfv 6543  cc 11110  cz 12560  cre 15046  cim 15047  ℤ[i]cgz 16864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-gz 16865
This theorem is referenced by:  gzcn  16867  zgz  16868  igz  16869  gznegcl  16870  gzcjcl  16871  gzaddcl  16872  gzmulcl  16873  gzabssqcl  16876  4sqlem4a  16886  2sqlem2  26928  2sqlem3  26930  cntotbnd  36756
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