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Mirrors > Home > MPE Home > Th. List > zgz | Structured version Visualization version GIF version |
Description: An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.) |
Ref | Expression |
---|---|
zgz | ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℤ[i]) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 12438 | . 2 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
2 | zre 12437 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
3 | 2 | rered 15044 | . . 3 ⊢ (𝐴 ∈ ℤ → (ℜ‘𝐴) = 𝐴) |
4 | id 22 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℤ) | |
5 | 3, 4 | eqeltrd 2839 | . 2 ⊢ (𝐴 ∈ ℤ → (ℜ‘𝐴) ∈ ℤ) |
6 | 2 | reim0d 15045 | . . 3 ⊢ (𝐴 ∈ ℤ → (ℑ‘𝐴) = 0) |
7 | 0z 12444 | . . 3 ⊢ 0 ∈ ℤ | |
8 | 6, 7 | eqeltrdi 2847 | . 2 ⊢ (𝐴 ∈ ℤ → (ℑ‘𝐴) ∈ ℤ) |
9 | elgz 16739 | . 2 ⊢ (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)) | |
10 | 1, 5, 8, 9 | syl3anbrc 1344 | 1 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℤ[i]) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ‘cfv 6492 ℂcc 10983 0cc0 10985 ℤcz 12433 ℜcre 14917 ℑcim 14918 ℤ[i]cgz 16737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-2 12150 df-z 12434 df-cj 14919 df-re 14920 df-im 14921 df-gz 16738 |
This theorem is referenced by: gzreim 16747 mul4sqlem 16761 4sqlem13 16765 4sqlem19 16771 gzsubrg 20780 zringunit 20816 2sqlem9 26703 2sqlem10 26704 |
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