![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 16257 | . 2 ⊢ (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)) | |
2 | 1 | simp1bi 1142 | 1 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ‘cfv 6324 ℂcc 10524 ℤcz 11969 ℜcre 14448 ℑcim 14449 ℤ[i]cgz 16255 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-gz 16256 |
This theorem is referenced by: gznegcl 16261 gzcjcl 16262 gzaddcl 16263 gzmulcl 16264 gzsubcl 16266 gzabssqcl 16267 4sqlem4a 16277 4sqlem4 16278 mul4sqlem 16279 mul4sq 16280 4sqlem12 16282 4sqlem17 16287 gzsubrg 20145 gzrngunitlem 20156 gzrngunit 20157 2sqlem2 26002 mul2sq 26003 2sqlem3 26004 cntotbnd 35234 |
Copyright terms: Public domain | W3C validator |