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Theorem gzrngunit 20034
Description: The units on ℤ[i] are the gaussian integers with norm 1. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypothesis
Ref Expression
gzrng.1 𝑍 = (ℂflds ℤ[i])
Assertion
Ref Expression
gzrngunit (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))

Proof of Theorem gzrngunit
StepHypRef Expression
1 gzsubrg 20022 . . . . 5 ℤ[i] ∈ (SubRing‘ℂfld)
2 gzrng.1 . . . . . 6 𝑍 = (ℂflds ℤ[i])
32subrgbas 19011 . . . . 5 (ℤ[i] ∈ (SubRing‘ℂfld) → ℤ[i] = (Base‘𝑍))
41, 3ax-mp 5 . . . 4 ℤ[i] = (Base‘𝑍)
5 eqid 2760 . . . 4 (Unit‘𝑍) = (Unit‘𝑍)
64, 5unitcl 18879 . . 3 (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
7 eqid 2760 . . . . . . . . . . . 12 (invr‘ℂfld) = (invr‘ℂfld)
8 eqid 2760 . . . . . . . . . . . 12 (invr𝑍) = (invr𝑍)
92, 7, 5, 8subrginv 19018 . . . . . . . . . . 11 ((ℤ[i] ∈ (SubRing‘ℂfld) ∧ 𝐴 ∈ (Unit‘𝑍)) → ((invr‘ℂfld)‘𝐴) = ((invr𝑍)‘𝐴))
101, 9mpan 708 . . . . . . . . . 10 (𝐴 ∈ (Unit‘𝑍) → ((invr‘ℂfld)‘𝐴) = ((invr𝑍)‘𝐴))
11 gzcn 15858 . . . . . . . . . . . 12 (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
126, 11syl 17 . . . . . . . . . . 11 (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ)
13 0red 10253 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 0 ∈ ℝ)
14 1re 10251 . . . . . . . . . . . . . . 15 1 ∈ ℝ
1514a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℝ)
1612abscld 14394 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
17 0lt1 10762 . . . . . . . . . . . . . . 15 0 < 1
1817a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 0 < 1)
192gzrngunitlem 20033 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
2013, 15, 16, 18, 19ltletrd 10409 . . . . . . . . . . . . 13 (𝐴 ∈ (Unit‘𝑍) → 0 < (abs‘𝐴))
2120gt0ne0d 10804 . . . . . . . . . . . 12 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≠ 0)
2212abs00ad 14249 . . . . . . . . . . . . 13 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
2322necon3bid 2976 . . . . . . . . . . . 12 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
2421, 23mpbid 222 . . . . . . . . . . 11 (𝐴 ∈ (Unit‘𝑍) → 𝐴 ≠ 0)
25 cnfldinv 19999 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
2612, 24, 25syl2anc 696 . . . . . . . . . 10 (𝐴 ∈ (Unit‘𝑍) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
2710, 26eqtr3d 2796 . . . . . . . . 9 (𝐴 ∈ (Unit‘𝑍) → ((invr𝑍)‘𝐴) = (1 / 𝐴))
282subrgring 19005 . . . . . . . . . . 11 (ℤ[i] ∈ (SubRing‘ℂfld) → 𝑍 ∈ Ring)
291, 28ax-mp 5 . . . . . . . . . 10 𝑍 ∈ Ring
305, 8unitinvcl 18894 . . . . . . . . . 10 ((𝑍 ∈ Ring ∧ 𝐴 ∈ (Unit‘𝑍)) → ((invr𝑍)‘𝐴) ∈ (Unit‘𝑍))
3129, 30mpan 708 . . . . . . . . 9 (𝐴 ∈ (Unit‘𝑍) → ((invr𝑍)‘𝐴) ∈ (Unit‘𝑍))
3227, 31eqeltrrd 2840 . . . . . . . 8 (𝐴 ∈ (Unit‘𝑍) → (1 / 𝐴) ∈ (Unit‘𝑍))
332gzrngunitlem 20033 . . . . . . . 8 ((1 / 𝐴) ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
3432, 33syl 17 . . . . . . 7 (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
35 1cnd 10268 . . . . . . . 8 (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℂ)
3635, 12, 24absdivd 14413 . . . . . . 7 (𝐴 ∈ (Unit‘𝑍) → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴)))
3734, 36breqtrd 4830 . . . . . 6 (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘1) / (abs‘𝐴)))
38 1div1e1 10929 . . . . . 6 (1 / 1) = 1
39 abs1 14256 . . . . . . . 8 (abs‘1) = 1
4039eqcomi 2769 . . . . . . 7 1 = (abs‘1)
4140oveq1i 6824 . . . . . 6 (1 / (abs‘𝐴)) = ((abs‘1) / (abs‘𝐴))
4237, 38, 413brtr4g 4838 . . . . 5 (𝐴 ∈ (Unit‘𝑍) → (1 / 1) ≤ (1 / (abs‘𝐴)))
43 lerec 11118 . . . . . 6 ((((abs‘𝐴) ∈ ℝ ∧ 0 < (abs‘𝐴)) ∧ (1 ∈ ℝ ∧ 0 < 1)) → ((abs‘𝐴) ≤ 1 ↔ (1 / 1) ≤ (1 / (abs‘𝐴))))
4416, 20, 15, 18, 43syl22anc 1478 . . . . 5 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≤ 1 ↔ (1 / 1) ≤ (1 / (abs‘𝐴))))
4542, 44mpbird 247 . . . 4 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≤ 1)
46 letri3 10335 . . . . 5 (((abs‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
4716, 14, 46sylancl 697 . . . 4 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
4845, 19, 47mpbir2and 995 . . 3 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) = 1)
496, 48jca 555 . 2 (𝐴 ∈ (Unit‘𝑍) → (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
5011adantr 472 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℂ)
51 simpr 479 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1)
52 ax-1ne0 10217 . . . . . . 7 1 ≠ 0
5352a1i 11 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 1 ≠ 0)
5451, 53eqnetrd 2999 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) ≠ 0)
55 fveq2 6353 . . . . . . 7 (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
56 abs0 14244 . . . . . . 7 (abs‘0) = 0
5755, 56syl6eq 2810 . . . . . 6 (𝐴 = 0 → (abs‘𝐴) = 0)
5857necon3i 2964 . . . . 5 ((abs‘𝐴) ≠ 0 → 𝐴 ≠ 0)
5954, 58syl 17 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ≠ 0)
60 eldifsn 4462 . . . 4 (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
6150, 59, 60sylanbrc 701 . . 3 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (ℂ ∖ {0}))
62 simpl 474 . . 3 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℤ[i])
6350, 59, 25syl2anc 696 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
6450absvalsqd 14400 . . . . . . 7 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
6551oveq1d 6829 . . . . . . . 8 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (1↑2))
66 sq1 13172 . . . . . . . 8 (1↑2) = 1
6765, 66syl6eq 2810 . . . . . . 7 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
6864, 67eqtr3d 2796 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (𝐴 · (∗‘𝐴)) = 1)
6968oveq1d 6829 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (1 / 𝐴))
7050cjcld 14155 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℂ)
7170, 50, 59divcan3d 11018 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
7263, 69, 713eqtr2d 2800 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) = (∗‘𝐴))
73 gzcjcl 15862 . . . . 5 (𝐴 ∈ ℤ[i] → (∗‘𝐴) ∈ ℤ[i])
7473adantr 472 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℤ[i])
7572, 74eqeltrd 2839 . . 3 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) ∈ ℤ[i])
76 cnfldbas 19972 . . . . . 6 ℂ = (Base‘ℂfld)
77 cnfld0 19992 . . . . . 6 0 = (0g‘ℂfld)
78 cndrng 19997 . . . . . 6 fld ∈ DivRing
7976, 77, 78drngui 18975 . . . . 5 (ℂ ∖ {0}) = (Unit‘ℂfld)
802, 79, 5, 7subrgunit 19020 . . . 4 (ℤ[i] ∈ (SubRing‘ℂfld) → (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐴 ∈ ℤ[i] ∧ ((invr‘ℂfld)‘𝐴) ∈ ℤ[i])))
811, 80ax-mp 5 . . 3 (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐴 ∈ ℤ[i] ∧ ((invr‘ℂfld)‘𝐴) ∈ ℤ[i]))
8261, 62, 75, 81syl3anbrc 1429 . 2 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (Unit‘𝑍))
8349, 82impbii 199 1 (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  cdif 3712  {csn 4321   class class class wbr 4804  cfv 6049  (class class class)co 6814  cc 10146  cr 10147  0cc0 10148  1c1 10149   · cmul 10153   < clt 10286  cle 10287   / cdiv 10896  2c2 11282  cexp 13074  ccj 14055  abscabs 14193  ℤ[i]cgz 15855  Basecbs 16079  s cress 16080  Ringcrg 18767  Unitcui 18859  invrcinvr 18891  SubRingcsubrg 18998  fldccnfld 19968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226  ax-addf 10227  ax-mulf 10228
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-tpos 7522  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-rp 12046  df-fz 12540  df-seq 13016  df-exp 13075  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-gz 15856  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-ress 16087  df-plusg 16176  df-mulr 16177  df-starv 16178  df-tset 16182  df-ple 16183  df-ds 16186  df-unif 16187  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-grp 17646  df-minusg 17647  df-subg 17812  df-cmn 18415  df-mgp 18710  df-ur 18722  df-ring 18769  df-cring 18770  df-oppr 18843  df-dvdsr 18861  df-unit 18862  df-invr 18892  df-dvr 18903  df-drng 18971  df-subrg 19000  df-cnfld 19969
This theorem is referenced by:  zringunit  20058
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