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Mirrors > Home > ILE Home > Th. List > elgz | 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 5510 | . . . . 5 ⊢ (𝑥 = 𝐴 → (ℜ‘𝑥) = (ℜ‘𝐴)) | |
2 | 1 | eleq1d 2246 | . . . 4 ⊢ (𝑥 = 𝐴 → ((ℜ‘𝑥) ∈ ℤ ↔ (ℜ‘𝐴) ∈ ℤ)) |
3 | fveq2 5510 | . . . . 5 ⊢ (𝑥 = 𝐴 → (ℑ‘𝑥) = (ℑ‘𝐴)) | |
4 | 3 | eleq1d 2246 | . . . 4 ⊢ (𝑥 = 𝐴 → ((ℑ‘𝑥) ∈ ℤ ↔ (ℑ‘𝐴) ∈ ℤ)) |
5 | 2, 4 | anbi12d 473 | . . 3 ⊢ (𝑥 = 𝐴 → (((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ) ↔ ((ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))) |
6 | df-gz 12338 | . . 3 ⊢ ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)} | |
7 | 5, 6 | elrab2 2896 | . 2 ⊢ (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ ((ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))) |
8 | 3anass 982 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ) ↔ (𝐴 ∈ ℂ ∧ ((ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))) | |
9 | 7, 8 | bitr4i 187 | 1 ⊢ (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ‘cfv 5211 ℂcc 7787 ℤcz 9229 ℜcre 10820 ℑcim 10821 ℤ[i]cgz 12337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-iota 5173 df-fv 5219 df-gz 12338 |
This theorem is referenced by: gzcn 12340 zgz 12341 igz 12342 gznegcl 12343 gzcjcl 12344 gzaddcl 12345 gzmulcl 12346 gzabssqcl 12349 4sqlem4a 12359 2sqlem2 14084 2sqlem3 14086 |
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