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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 16871 | . 2 ⊢ (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)) | |
2 | 1 | simp1bi 1144 | 1 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ‘cfv 6543 ℂcc 11114 ℤcz 12565 ℜcre 15051 ℑcim 15052 ℤ[i]cgz 16869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 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 16870 |
This theorem is referenced by: gznegcl 16875 gzcjcl 16876 gzaddcl 16877 gzmulcl 16878 gzsubcl 16880 gzabssqcl 16881 4sqlem4a 16891 4sqlem4 16892 mul4sqlem 16893 mul4sq 16894 4sqlem12 16896 4sqlem17 16901 gzsubrg 21288 gzrngunitlem 21299 gzrngunit 21300 2sqlem2 27264 mul2sq 27265 2sqlem3 27266 cntotbnd 37128 |
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