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Mirrors > Home > MPE Home > Th. List > elgz | Structured version Visualization version GIF version |
Description: Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.) |
Ref | Expression |
---|---|
elgz | ⊢ (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6887 | . . . . 5 ⊢ (𝑥 = 𝐴 → (ℜ‘𝑥) = (ℜ‘𝐴)) | |
2 | 1 | eleq1d 2819 | . . . 4 ⊢ (𝑥 = 𝐴 → ((ℜ‘𝑥) ∈ ℤ ↔ (ℜ‘𝐴) ∈ ℤ)) |
3 | fveq2 6887 | . . . . 5 ⊢ (𝑥 = 𝐴 → (ℑ‘𝑥) = (ℑ‘𝐴)) | |
4 | 3 | eleq1d 2819 | . . . 4 ⊢ (𝑥 = 𝐴 → ((ℑ‘𝑥) ∈ ℤ ↔ (ℑ‘𝐴) ∈ ℤ)) |
5 | 2, 4 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝐴 → (((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ) ↔ ((ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))) |
6 | df-gz 16858 | . . 3 ⊢ ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)} | |
7 | 5, 6 | elrab2 3684 | . 2 ⊢ (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ ((ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))) |
8 | 3anass 1096 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ) ↔ (𝐴 ∈ ℂ ∧ ((ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))) | |
9 | 7, 8 | bitr4i 278 | 1 ⊢ (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6539 ℂcc 11103 ℤcz 12553 ℜcre 15039 ℑcim 15040 ℤ[i]cgz 16857 |
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 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-iota 6491 df-fv 6547 df-gz 16858 |
This theorem is referenced by: gzcn 16860 zgz 16861 igz 16862 gznegcl 16863 gzcjcl 16864 gzaddcl 16865 gzmulcl 16866 gzabssqcl 16869 4sqlem4a 16879 2sqlem2 26900 2sqlem3 26902 cntotbnd 36601 |
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