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| Mirrors > Home > MPE Home > Th. List > gzcn | Structured version Visualization version GIF version | ||
| Description: A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.) |
| Ref | Expression |
|---|---|
| gzcn | ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elgz 16969 | . 2 ⊢ (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)) | |
| 2 | 1 | simp1bi 1146 | 1 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ‘cfv 6561 ℂcc 11153 ℤcz 12613 ℜcre 15136 ℑcim 15137 ℤ[i]cgz 16967 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-gz 16968 |
| This theorem is referenced by: gznegcl 16973 gzcjcl 16974 gzaddcl 16975 gzmulcl 16976 gzsubcl 16978 gzabssqcl 16979 4sqlem4a 16989 4sqlem4 16990 mul4sqlem 16991 mul4sq 16992 4sqlem12 16994 4sqlem17 16999 gzsubrg 21439 gzrngunitlem 21450 gzrngunit 21451 2sqlem2 27462 mul2sq 27463 2sqlem3 27464 cntotbnd 37803 |
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