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Theorem gzrngunit 20546
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 20534 . . . . 5 ℤ[i] ∈ (SubRing‘ℂfld)
2 gzrng.1 . . . . . 6 𝑍 = (ℂflds ℤ[i])
32subrgbas 19480 . . . . 5 (ℤ[i] ∈ (SubRing‘ℂfld) → ℤ[i] = (Base‘𝑍))
41, 3ax-mp 5 . . . 4 ℤ[i] = (Base‘𝑍)
5 eqid 2826 . . . 4 (Unit‘𝑍) = (Unit‘𝑍)
64, 5unitcl 19345 . . 3 (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
7 eqid 2826 . . . . . . . . . . . 12 (invr‘ℂfld) = (invr‘ℂfld)
8 eqid 2826 . . . . . . . . . . . 12 (invr𝑍) = (invr𝑍)
92, 7, 5, 8subrginv 19487 . . . . . . . . . . 11 ((ℤ[i] ∈ (SubRing‘ℂfld) ∧ 𝐴 ∈ (Unit‘𝑍)) → ((invr‘ℂfld)‘𝐴) = ((invr𝑍)‘𝐴))
101, 9mpan 686 . . . . . . . . . 10 (𝐴 ∈ (Unit‘𝑍) → ((invr‘ℂfld)‘𝐴) = ((invr𝑍)‘𝐴))
11 gzcn 16263 . . . . . . . . . . . 12 (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
126, 11syl 17 . . . . . . . . . . 11 (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ)
13 0red 10638 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 0 ∈ ℝ)
14 1re 10635 . . . . . . . . . . . . . . 15 1 ∈ ℝ
1514a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℝ)
1612abscld 14791 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
17 0lt1 11156 . . . . . . . . . . . . . . 15 0 < 1
1817a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 0 < 1)
192gzrngunitlem 20545 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
2013, 15, 16, 18, 19ltletrd 10794 . . . . . . . . . . . . 13 (𝐴 ∈ (Unit‘𝑍) → 0 < (abs‘𝐴))
2120gt0ne0d 11198 . . . . . . . . . . . 12 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≠ 0)
2212abs00ad 14645 . . . . . . . . . . . . 13 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
2322necon3bid 3065 . . . . . . . . . . . 12 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
2421, 23mpbid 233 . . . . . . . . . . 11 (𝐴 ∈ (Unit‘𝑍) → 𝐴 ≠ 0)
25 cnfldinv 20511 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
2612, 24, 25syl2anc 584 . . . . . . . . . 10 (𝐴 ∈ (Unit‘𝑍) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
2710, 26eqtr3d 2863 . . . . . . . . 9 (𝐴 ∈ (Unit‘𝑍) → ((invr𝑍)‘𝐴) = (1 / 𝐴))
282subrgring 19474 . . . . . . . . . . 11 (ℤ[i] ∈ (SubRing‘ℂfld) → 𝑍 ∈ Ring)
291, 28ax-mp 5 . . . . . . . . . 10 𝑍 ∈ Ring
305, 8unitinvcl 19360 . . . . . . . . . 10 ((𝑍 ∈ Ring ∧ 𝐴 ∈ (Unit‘𝑍)) → ((invr𝑍)‘𝐴) ∈ (Unit‘𝑍))
3129, 30mpan 686 . . . . . . . . 9 (𝐴 ∈ (Unit‘𝑍) → ((invr𝑍)‘𝐴) ∈ (Unit‘𝑍))
3227, 31eqeltrrd 2919 . . . . . . . 8 (𝐴 ∈ (Unit‘𝑍) → (1 / 𝐴) ∈ (Unit‘𝑍))
332gzrngunitlem 20545 . . . . . . . 8 ((1 / 𝐴) ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
3432, 33syl 17 . . . . . . 7 (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
35 1cnd 10630 . . . . . . . 8 (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℂ)
3635, 12, 24absdivd 14810 . . . . . . 7 (𝐴 ∈ (Unit‘𝑍) → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴)))
3734, 36breqtrd 5089 . . . . . 6 (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘1) / (abs‘𝐴)))
38 1div1e1 11324 . . . . . 6 (1 / 1) = 1
39 abs1 14652 . . . . . . . 8 (abs‘1) = 1
4039eqcomi 2835 . . . . . . 7 1 = (abs‘1)
4140oveq1i 7160 . . . . . 6 (1 / (abs‘𝐴)) = ((abs‘1) / (abs‘𝐴))
4237, 38, 413brtr4g 5097 . . . . 5 (𝐴 ∈ (Unit‘𝑍) → (1 / 1) ≤ (1 / (abs‘𝐴)))
43 lerec 11517 . . . . . 6 ((((abs‘𝐴) ∈ ℝ ∧ 0 < (abs‘𝐴)) ∧ (1 ∈ ℝ ∧ 0 < 1)) → ((abs‘𝐴) ≤ 1 ↔ (1 / 1) ≤ (1 / (abs‘𝐴))))
4416, 20, 15, 18, 43syl22anc 836 . . . . 5 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≤ 1 ↔ (1 / 1) ≤ (1 / (abs‘𝐴))))
4542, 44mpbird 258 . . . 4 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≤ 1)
46 letri3 10720 . . . . 5 (((abs‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
4716, 14, 46sylancl 586 . . . 4 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
4845, 19, 47mpbir2and 709 . . 3 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) = 1)
496, 48jca 512 . 2 (𝐴 ∈ (Unit‘𝑍) → (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
5011adantr 481 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℂ)
51 simpr 485 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1)
52 ax-1ne0 10600 . . . . . . 7 1 ≠ 0
5352a1i 11 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 1 ≠ 0)
5451, 53eqnetrd 3088 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) ≠ 0)
55 fveq2 6669 . . . . . . 7 (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
56 abs0 14640 . . . . . . 7 (abs‘0) = 0
5755, 56syl6eq 2877 . . . . . 6 (𝐴 = 0 → (abs‘𝐴) = 0)
5857necon3i 3053 . . . . 5 ((abs‘𝐴) ≠ 0 → 𝐴 ≠ 0)
5954, 58syl 17 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ≠ 0)
60 eldifsn 4718 . . . 4 (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
6150, 59, 60sylanbrc 583 . . 3 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (ℂ ∖ {0}))
62 simpl 483 . . 3 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℤ[i])
6350, 59, 25syl2anc 584 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
6450absvalsqd 14797 . . . . . . 7 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
6551oveq1d 7165 . . . . . . . 8 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (1↑2))
66 sq1 13553 . . . . . . . 8 (1↑2) = 1
6765, 66syl6eq 2877 . . . . . . 7 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
6864, 67eqtr3d 2863 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (𝐴 · (∗‘𝐴)) = 1)
6968oveq1d 7165 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (1 / 𝐴))
7050cjcld 14550 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℂ)
7170, 50, 59divcan3d 11415 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
7263, 69, 713eqtr2d 2867 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) = (∗‘𝐴))
73 gzcjcl 16267 . . . . 5 (𝐴 ∈ ℤ[i] → (∗‘𝐴) ∈ ℤ[i])
7473adantr 481 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℤ[i])
7572, 74eqeltrd 2918 . . 3 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) ∈ ℤ[i])
76 cnfldbas 20484 . . . . . 6 ℂ = (Base‘ℂfld)
77 cnfld0 20504 . . . . . 6 0 = (0g‘ℂfld)
78 cndrng 20509 . . . . . 6 fld ∈ DivRing
7976, 77, 78drngui 19444 . . . . 5 (ℂ ∖ {0}) = (Unit‘ℂfld)
802, 79, 5, 7subrgunit 19489 . . . 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 1337 . 2 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (Unit‘𝑍))
8349, 82impbii 210 1 (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  cdif 3937  {csn 4564   class class class wbr 5063  cfv 6354  (class class class)co 7150  cc 10529  cr 10530  0cc0 10531  1c1 10532   · cmul 10536   < clt 10669  cle 10670   / cdiv 11291  2c2 11686  cexp 13424  ccj 14450  abscabs 14588  ℤ[i]cgz 16260  Basecbs 16478  s cress 16479  Ringcrg 19233  Unitcui 19325  invrcinvr 19357  SubRingcsubrg 19467  fldccnfld 20480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609  ax-addf 10610  ax-mulf 10611
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-tpos 7888  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-rp 12385  df-fz 12888  df-seq 13365  df-exp 13425  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-gz 16261  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-starv 16575  df-tset 16579  df-ple 16580  df-ds 16582  df-unif 16583  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18051  df-minusg 18052  df-subg 18221  df-cmn 18844  df-mgp 19176  df-ur 19188  df-ring 19235  df-cring 19236  df-oppr 19309  df-dvdsr 19327  df-unit 19328  df-invr 19358  df-dvr 19369  df-drng 19440  df-subrg 19469  df-cnfld 20481
This theorem is referenced by:  zringunit  20570
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