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Theorem gzrngunit 20020
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 20008 . . . . 5 ℤ[i] ∈ (SubRing‘ℂfld)
2 gzrng.1 . . . . . 6 𝑍 = (ℂflds ℤ[i])
32subrgbas 18993 . . . . 5 (ℤ[i] ∈ (SubRing‘ℂfld) → ℤ[i] = (Base‘𝑍))
41, 3ax-mp 5 . . . 4 ℤ[i] = (Base‘𝑍)
5 eqid 2806 . . . 4 (Unit‘𝑍) = (Unit‘𝑍)
64, 5unitcl 18861 . . 3 (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
7 eqid 2806 . . . . . . . . . . . 12 (invr‘ℂfld) = (invr‘ℂfld)
8 eqid 2806 . . . . . . . . . . . 12 (invr𝑍) = (invr𝑍)
92, 7, 5, 8subrginv 19000 . . . . . . . . . . 11 ((ℤ[i] ∈ (SubRing‘ℂfld) ∧ 𝐴 ∈ (Unit‘𝑍)) → ((invr‘ℂfld)‘𝐴) = ((invr𝑍)‘𝐴))
101, 9mpan 673 . . . . . . . . . 10 (𝐴 ∈ (Unit‘𝑍) → ((invr‘ℂfld)‘𝐴) = ((invr𝑍)‘𝐴))
11 gzcn 15853 . . . . . . . . . . . 12 (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
126, 11syl 17 . . . . . . . . . . 11 (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ)
13 0red 10328 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 0 ∈ ℝ)
14 1re 10325 . . . . . . . . . . . . . . 15 1 ∈ ℝ
1514a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℝ)
1612abscld 14398 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
17 0lt1 10835 . . . . . . . . . . . . . . 15 0 < 1
1817a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 0 < 1)
192gzrngunitlem 20019 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
2013, 15, 16, 18, 19ltletrd 10482 . . . . . . . . . . . . 13 (𝐴 ∈ (Unit‘𝑍) → 0 < (abs‘𝐴))
2120gt0ne0d 10877 . . . . . . . . . . . 12 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≠ 0)
2212abs00ad 14253 . . . . . . . . . . . . 13 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
2322necon3bid 3022 . . . . . . . . . . . 12 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
2421, 23mpbid 223 . . . . . . . . . . 11 (𝐴 ∈ (Unit‘𝑍) → 𝐴 ≠ 0)
25 cnfldinv 19985 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
2612, 24, 25syl2anc 575 . . . . . . . . . 10 (𝐴 ∈ (Unit‘𝑍) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
2710, 26eqtr3d 2842 . . . . . . . . 9 (𝐴 ∈ (Unit‘𝑍) → ((invr𝑍)‘𝐴) = (1 / 𝐴))
282subrgring 18987 . . . . . . . . . . 11 (ℤ[i] ∈ (SubRing‘ℂfld) → 𝑍 ∈ Ring)
291, 28ax-mp 5 . . . . . . . . . 10 𝑍 ∈ Ring
305, 8unitinvcl 18876 . . . . . . . . . 10 ((𝑍 ∈ Ring ∧ 𝐴 ∈ (Unit‘𝑍)) → ((invr𝑍)‘𝐴) ∈ (Unit‘𝑍))
3129, 30mpan 673 . . . . . . . . 9 (𝐴 ∈ (Unit‘𝑍) → ((invr𝑍)‘𝐴) ∈ (Unit‘𝑍))
3227, 31eqeltrrd 2886 . . . . . . . 8 (𝐴 ∈ (Unit‘𝑍) → (1 / 𝐴) ∈ (Unit‘𝑍))
332gzrngunitlem 20019 . . . . . . . 8 ((1 / 𝐴) ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
3432, 33syl 17 . . . . . . 7 (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
35 1cnd 10320 . . . . . . . 8 (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℂ)
3635, 12, 24absdivd 14417 . . . . . . 7 (𝐴 ∈ (Unit‘𝑍) → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴)))
3734, 36breqtrd 4870 . . . . . 6 (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘1) / (abs‘𝐴)))
38 1div1e1 11002 . . . . . 6 (1 / 1) = 1
39 abs1 14260 . . . . . . . 8 (abs‘1) = 1
4039eqcomi 2815 . . . . . . 7 1 = (abs‘1)
4140oveq1i 6884 . . . . . 6 (1 / (abs‘𝐴)) = ((abs‘1) / (abs‘𝐴))
4237, 38, 413brtr4g 4878 . . . . 5 (𝐴 ∈ (Unit‘𝑍) → (1 / 1) ≤ (1 / (abs‘𝐴)))
43 lerec 11191 . . . . . 6 ((((abs‘𝐴) ∈ ℝ ∧ 0 < (abs‘𝐴)) ∧ (1 ∈ ℝ ∧ 0 < 1)) → ((abs‘𝐴) ≤ 1 ↔ (1 / 1) ≤ (1 / (abs‘𝐴))))
4416, 20, 15, 18, 43syl22anc 858 . . . . 5 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≤ 1 ↔ (1 / 1) ≤ (1 / (abs‘𝐴))))
4542, 44mpbird 248 . . . 4 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≤ 1)
46 letri3 10408 . . . . 5 (((abs‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
4716, 14, 46sylancl 576 . . . 4 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
4845, 19, 47mpbir2and 695 . . 3 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) = 1)
496, 48jca 503 . 2 (𝐴 ∈ (Unit‘𝑍) → (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
5011adantr 468 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℂ)
51 simpr 473 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1)
52 ax-1ne0 10290 . . . . . . 7 1 ≠ 0
5352a1i 11 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 1 ≠ 0)
5451, 53eqnetrd 3045 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) ≠ 0)
55 fveq2 6408 . . . . . . 7 (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
56 abs0 14248 . . . . . . 7 (abs‘0) = 0
5755, 56syl6eq 2856 . . . . . 6 (𝐴 = 0 → (abs‘𝐴) = 0)
5857necon3i 3010 . . . . 5 ((abs‘𝐴) ≠ 0 → 𝐴 ≠ 0)
5954, 58syl 17 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ≠ 0)
60 eldifsn 4508 . . . 4 (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
6150, 59, 60sylanbrc 574 . . 3 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (ℂ ∖ {0}))
62 simpl 470 . . 3 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℤ[i])
6350, 59, 25syl2anc 575 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
6450absvalsqd 14404 . . . . . . 7 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
6551oveq1d 6889 . . . . . . . 8 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (1↑2))
66 sq1 13181 . . . . . . . 8 (1↑2) = 1
6765, 66syl6eq 2856 . . . . . . 7 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
6864, 67eqtr3d 2842 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (𝐴 · (∗‘𝐴)) = 1)
6968oveq1d 6889 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (1 / 𝐴))
7050cjcld 14159 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℂ)
7170, 50, 59divcan3d 11091 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
7263, 69, 713eqtr2d 2846 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) = (∗‘𝐴))
73 gzcjcl 15857 . . . . 5 (𝐴 ∈ ℤ[i] → (∗‘𝐴) ∈ ℤ[i])
7473adantr 468 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℤ[i])
7572, 74eqeltrd 2885 . . 3 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) ∈ ℤ[i])
76 cnfldbas 19958 . . . . . 6 ℂ = (Base‘ℂfld)
77 cnfld0 19978 . . . . . 6 0 = (0g‘ℂfld)
78 cndrng 19983 . . . . . 6 fld ∈ DivRing
7976, 77, 78drngui 18957 . . . . 5 (ℂ ∖ {0}) = (Unit‘ℂfld)
802, 79, 5, 7subrgunit 19002 . . . 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 1436 . 2 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (Unit‘𝑍))
8349, 82impbii 200 1 (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wne 2978  cdif 3766  {csn 4370   class class class wbr 4844  cfv 6101  (class class class)co 6874  cc 10219  cr 10220  0cc0 10221  1c1 10222   · cmul 10226   < clt 10359  cle 10360   / cdiv 10969  2c2 11356  cexp 13083  ccj 14059  abscabs 14197  ℤ[i]cgz 15850  Basecbs 16068  s cress 16069  Ringcrg 18749  Unitcui 18841  invrcinvr 18873  SubRingcsubrg 18980  fldccnfld 19954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298  ax-pre-sup 10299  ax-addf 10300  ax-mulf 10301
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-tpos 7587  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-oadd 7800  df-er 7979  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-sup 8587  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-div 10970  df-nn 11306  df-2 11364  df-3 11365  df-4 11366  df-5 11367  df-6 11368  df-7 11369  df-8 11370  df-9 11371  df-n0 11560  df-z 11644  df-dec 11760  df-uz 11905  df-rp 12047  df-fz 12550  df-seq 13025  df-exp 13084  df-cj 14062  df-re 14063  df-im 14064  df-sqrt 14198  df-abs 14199  df-gz 15851  df-struct 16070  df-ndx 16071  df-slot 16072  df-base 16074  df-sets 16075  df-ress 16076  df-plusg 16166  df-mulr 16167  df-starv 16168  df-tset 16172  df-ple 16173  df-ds 16175  df-unif 16176  df-0g 16307  df-mgm 17447  df-sgrp 17489  df-mnd 17500  df-grp 17630  df-minusg 17631  df-subg 17793  df-cmn 18396  df-mgp 18692  df-ur 18704  df-ring 18751  df-cring 18752  df-oppr 18825  df-dvdsr 18843  df-unit 18844  df-invr 18874  df-dvr 18885  df-drng 18953  df-subrg 18982  df-cnfld 19955
This theorem is referenced by:  zringunit  20044
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