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Mirrors > Home > MPE Home > Th. List > gzmulcl | Structured version Visualization version GIF version |
Description: The gaussian integers are closed under multiplication. (Contributed by Mario Carneiro, 14-Jul-2014.) |
Ref | Expression |
---|---|
gzmulcl | ⊢ ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (𝐴 · 𝐵) ∈ ℤ[i]) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gzcn 16621 | . . 3 ⊢ (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ) | |
2 | gzcn 16621 | . . 3 ⊢ (𝐵 ∈ ℤ[i] → 𝐵 ∈ ℂ) | |
3 | mulcl 10943 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) | |
4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (𝐴 · 𝐵) ∈ ℂ) |
5 | remul 14828 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵)))) | |
6 | 1, 2, 5 | syl2an 596 | . . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵)))) |
7 | elgz 16620 | . . . . . 6 ⊢ (𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ)) | |
8 | 7 | simp2bi 1145 | . . . . 5 ⊢ (𝐴 ∈ ℤ[i] → (ℜ‘𝐴) ∈ ℤ) |
9 | elgz 16620 | . . . . . 6 ⊢ (𝐵 ∈ ℤ[i] ↔ (𝐵 ∈ ℂ ∧ (ℜ‘𝐵) ∈ ℤ ∧ (ℑ‘𝐵) ∈ ℤ)) | |
10 | 9 | simp2bi 1145 | . . . . 5 ⊢ (𝐵 ∈ ℤ[i] → (ℜ‘𝐵) ∈ ℤ) |
11 | zmulcl 12357 | . . . . 5 ⊢ (((ℜ‘𝐴) ∈ ℤ ∧ (ℜ‘𝐵) ∈ ℤ) → ((ℜ‘𝐴) · (ℜ‘𝐵)) ∈ ℤ) | |
12 | 8, 10, 11 | syl2an 596 | . . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → ((ℜ‘𝐴) · (ℜ‘𝐵)) ∈ ℤ) |
13 | 7 | simp3bi 1146 | . . . . 5 ⊢ (𝐴 ∈ ℤ[i] → (ℑ‘𝐴) ∈ ℤ) |
14 | 9 | simp3bi 1146 | . . . . 5 ⊢ (𝐵 ∈ ℤ[i] → (ℑ‘𝐵) ∈ ℤ) |
15 | zmulcl 12357 | . . . . 5 ⊢ (((ℑ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐵) ∈ ℤ) → ((ℑ‘𝐴) · (ℑ‘𝐵)) ∈ ℤ) | |
16 | 13, 14, 15 | syl2an 596 | . . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → ((ℑ‘𝐴) · (ℑ‘𝐵)) ∈ ℤ) |
17 | 12, 16 | zsubcld 12419 | . . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵))) ∈ ℤ) |
18 | 6, 17 | eqeltrd 2839 | . 2 ⊢ ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (ℜ‘(𝐴 · 𝐵)) ∈ ℤ) |
19 | immul 14835 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵)))) | |
20 | 1, 2, 19 | syl2an 596 | . . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵)))) |
21 | zmulcl 12357 | . . . . 5 ⊢ (((ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐵) ∈ ℤ) → ((ℜ‘𝐴) · (ℑ‘𝐵)) ∈ ℤ) | |
22 | 8, 14, 21 | syl2an 596 | . . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → ((ℜ‘𝐴) · (ℑ‘𝐵)) ∈ ℤ) |
23 | zmulcl 12357 | . . . . 5 ⊢ (((ℑ‘𝐴) ∈ ℤ ∧ (ℜ‘𝐵) ∈ ℤ) → ((ℑ‘𝐴) · (ℜ‘𝐵)) ∈ ℤ) | |
24 | 13, 10, 23 | syl2an 596 | . . . 4 ⊢ ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → ((ℑ‘𝐴) · (ℜ‘𝐵)) ∈ ℤ) |
25 | 22, 24 | zaddcld 12418 | . . 3 ⊢ ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵))) ∈ ℤ) |
26 | 20, 25 | eqeltrd 2839 | . 2 ⊢ ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (ℑ‘(𝐴 · 𝐵)) ∈ ℤ) |
27 | elgz 16620 | . 2 ⊢ ((𝐴 · 𝐵) ∈ ℤ[i] ↔ ((𝐴 · 𝐵) ∈ ℂ ∧ (ℜ‘(𝐴 · 𝐵)) ∈ ℤ ∧ (ℑ‘(𝐴 · 𝐵)) ∈ ℤ)) | |
28 | 4, 18, 26, 27 | syl3anbrc 1342 | 1 ⊢ ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (𝐴 · 𝐵) ∈ ℤ[i]) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ‘cfv 6427 (class class class)co 7268 ℂcc 10857 + caddc 10862 · cmul 10864 − cmin 11193 ℤcz 12307 ℜcre 14796 ℑcim 14797 ℤ[i]cgz 16618 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-n0 12222 df-z 12308 df-cj 14798 df-re 14799 df-im 14800 df-gz 16619 |
This theorem is referenced by: gzreim 16628 mul4sqlem 16642 gzsubrg 20640 mul2sq 26555 2sqlem3 26556 |
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