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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 15473 | . 2 ⊢ (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)) | |
2 | 1 | simp1bi 1069 | 1 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ‘cfv 5804 ℂcc 9813 ℤcz 11254 ℜcre 13685 ℑcim 13686 ℤ[i]cgz 15471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-gz 15472 |
This theorem is referenced by: gznegcl 15477 gzcjcl 15478 gzaddcl 15479 gzmulcl 15480 gzsubcl 15482 gzabssqcl 15483 4sqlem4a 15493 4sqlem4 15494 mul4sqlem 15495 mul4sq 15496 4sqlem12 15498 4sqlem17 15503 gzsubrg 19619 gzrngunitlem 19630 gzrngunit 19631 2sqlem2 24943 mul2sq 24944 2sqlem3 24945 cntotbnd 32765 |
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