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Theorem gzrngunit 19577
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 19565 . . . . 5 ℤ[i] ∈ (SubRing‘ℂfld)
2 gzrng.1 . . . . . 6 𝑍 = (ℂflds ℤ[i])
32subrgbas 18558 . . . . 5 (ℤ[i] ∈ (SubRing‘ℂfld) → ℤ[i] = (Base‘𝑍))
41, 3ax-mp 5 . . . 4 ℤ[i] = (Base‘𝑍)
5 eqid 2609 . . . 4 (Unit‘𝑍) = (Unit‘𝑍)
64, 5unitcl 18428 . . 3 (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℤ[i])
7 eqid 2609 . . . . . . . . . . . 12 (invr‘ℂfld) = (invr‘ℂfld)
8 eqid 2609 . . . . . . . . . . . 12 (invr𝑍) = (invr𝑍)
92, 7, 5, 8subrginv 18565 . . . . . . . . . . 11 ((ℤ[i] ∈ (SubRing‘ℂfld) ∧ 𝐴 ∈ (Unit‘𝑍)) → ((invr‘ℂfld)‘𝐴) = ((invr𝑍)‘𝐴))
101, 9mpan 701 . . . . . . . . . 10 (𝐴 ∈ (Unit‘𝑍) → ((invr‘ℂfld)‘𝐴) = ((invr𝑍)‘𝐴))
11 gzcn 15420 . . . . . . . . . . . 12 (𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)
126, 11syl 17 . . . . . . . . . . 11 (𝐴 ∈ (Unit‘𝑍) → 𝐴 ∈ ℂ)
13 0red 9897 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 0 ∈ ℝ)
14 1re 9895 . . . . . . . . . . . . . . 15 1 ∈ ℝ
1514a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℝ)
1612abscld 13969 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ∈ ℝ)
17 0lt1 10399 . . . . . . . . . . . . . . 15 0 < 1
1817a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 0 < 1)
192gzrngunitlem 19576 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
2013, 15, 16, 18, 19ltletrd 10048 . . . . . . . . . . . . 13 (𝐴 ∈ (Unit‘𝑍) → 0 < (abs‘𝐴))
2120gt0ne0d 10441 . . . . . . . . . . . 12 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≠ 0)
2212abs00ad 13824 . . . . . . . . . . . . 13 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
2322necon3bid 2825 . . . . . . . . . . . 12 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
2421, 23mpbid 220 . . . . . . . . . . 11 (𝐴 ∈ (Unit‘𝑍) → 𝐴 ≠ 0)
25 cnfldinv 19542 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
2612, 24, 25syl2anc 690 . . . . . . . . . 10 (𝐴 ∈ (Unit‘𝑍) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
2710, 26eqtr3d 2645 . . . . . . . . 9 (𝐴 ∈ (Unit‘𝑍) → ((invr𝑍)‘𝐴) = (1 / 𝐴))
282subrgring 18552 . . . . . . . . . . 11 (ℤ[i] ∈ (SubRing‘ℂfld) → 𝑍 ∈ Ring)
291, 28ax-mp 5 . . . . . . . . . 10 𝑍 ∈ Ring
305, 8unitinvcl 18443 . . . . . . . . . 10 ((𝑍 ∈ Ring ∧ 𝐴 ∈ (Unit‘𝑍)) → ((invr𝑍)‘𝐴) ∈ (Unit‘𝑍))
3129, 30mpan 701 . . . . . . . . 9 (𝐴 ∈ (Unit‘𝑍) → ((invr𝑍)‘𝐴) ∈ (Unit‘𝑍))
3227, 31eqeltrrd 2688 . . . . . . . 8 (𝐴 ∈ (Unit‘𝑍) → (1 / 𝐴) ∈ (Unit‘𝑍))
332gzrngunitlem 19576 . . . . . . . 8 ((1 / 𝐴) ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
3432, 33syl 17 . . . . . . 7 (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘(1 / 𝐴)))
35 1cnd 9912 . . . . . . . 8 (𝐴 ∈ (Unit‘𝑍) → 1 ∈ ℂ)
3635, 12, 24absdivd 13988 . . . . . . 7 (𝐴 ∈ (Unit‘𝑍) → (abs‘(1 / 𝐴)) = ((abs‘1) / (abs‘𝐴)))
3734, 36breqtrd 4603 . . . . . 6 (𝐴 ∈ (Unit‘𝑍) → 1 ≤ ((abs‘1) / (abs‘𝐴)))
38 1div1e1 10566 . . . . . 6 (1 / 1) = 1
39 abs1 13831 . . . . . . . 8 (abs‘1) = 1
4039eqcomi 2618 . . . . . . 7 1 = (abs‘1)
4140oveq1i 6537 . . . . . 6 (1 / (abs‘𝐴)) = ((abs‘1) / (abs‘𝐴))
4237, 38, 413brtr4g 4611 . . . . 5 (𝐴 ∈ (Unit‘𝑍) → (1 / 1) ≤ (1 / (abs‘𝐴)))
43 lerec 10755 . . . . . 6 ((((abs‘𝐴) ∈ ℝ ∧ 0 < (abs‘𝐴)) ∧ (1 ∈ ℝ ∧ 0 < 1)) → ((abs‘𝐴) ≤ 1 ↔ (1 / 1) ≤ (1 / (abs‘𝐴))))
4416, 20, 15, 18, 43syl22anc 1318 . . . . 5 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) ≤ 1 ↔ (1 / 1) ≤ (1 / (abs‘𝐴))))
4542, 44mpbird 245 . . . 4 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) ≤ 1)
46 letri3 9974 . . . . 5 (((abs‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
4716, 14, 46sylancl 692 . . . 4 (𝐴 ∈ (Unit‘𝑍) → ((abs‘𝐴) = 1 ↔ ((abs‘𝐴) ≤ 1 ∧ 1 ≤ (abs‘𝐴))))
4845, 19, 47mpbir2and 958 . . 3 (𝐴 ∈ (Unit‘𝑍) → (abs‘𝐴) = 1)
496, 48jca 552 . 2 (𝐴 ∈ (Unit‘𝑍) → (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
5011adantr 479 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℂ)
51 simpr 475 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1)
52 ax-1ne0 9861 . . . . . . 7 1 ≠ 0
5352a1i 11 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 1 ≠ 0)
5451, 53eqnetrd 2848 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (abs‘𝐴) ≠ 0)
55 fveq2 6088 . . . . . . 7 (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
56 abs0 13819 . . . . . . 7 (abs‘0) = 0
5755, 56syl6eq 2659 . . . . . 6 (𝐴 = 0 → (abs‘𝐴) = 0)
5857necon3i 2813 . . . . 5 ((abs‘𝐴) ≠ 0 → 𝐴 ≠ 0)
5954, 58syl 17 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ≠ 0)
60 eldifsn 4259 . . . 4 (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
6150, 59, 60sylanbrc 694 . . 3 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (ℂ ∖ {0}))
62 simpl 471 . . 3 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ ℤ[i])
6350, 59, 25syl2anc 690 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) = (1 / 𝐴))
6450absvalsqd 13975 . . . . . . 7 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
6551oveq1d 6542 . . . . . . . 8 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = (1↑2))
66 sq1 12775 . . . . . . . 8 (1↑2) = 1
6765, 66syl6eq 2659 . . . . . . 7 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((abs‘𝐴)↑2) = 1)
6864, 67eqtr3d 2645 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (𝐴 · (∗‘𝐴)) = 1)
6968oveq1d 6542 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (1 / 𝐴))
7050cjcld 13730 . . . . . 6 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℂ)
7170, 50, 59divcan3d 10655 . . . . 5 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((𝐴 · (∗‘𝐴)) / 𝐴) = (∗‘𝐴))
7263, 69, 713eqtr2d 2649 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) = (∗‘𝐴))
73 gzcjcl 15424 . . . . 5 (𝐴 ∈ ℤ[i] → (∗‘𝐴) ∈ ℤ[i])
7473adantr 479 . . . 4 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → (∗‘𝐴) ∈ ℤ[i])
7572, 74eqeltrd 2687 . . 3 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → ((invr‘ℂfld)‘𝐴) ∈ ℤ[i])
76 cnfldbas 19517 . . . . . 6 ℂ = (Base‘ℂfld)
77 cnfld0 19535 . . . . . 6 0 = (0g‘ℂfld)
78 cndrng 19540 . . . . . 6 fld ∈ DivRing
7976, 77, 78drngui 18522 . . . . 5 (ℂ ∖ {0}) = (Unit‘ℂfld)
802, 79, 5, 7subrgunit 18567 . . . 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 1238 . 2 ((𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1) → 𝐴 ∈ (Unit‘𝑍))
8349, 82impbii 197 1 (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  cdif 3536  {csn 4124   class class class wbr 4577  cfv 5790  (class class class)co 6527  cc 9790  cr 9791  0cc0 9792  1c1 9793   · cmul 9797   < clt 9930  cle 9931   / cdiv 10533  2c2 10917  cexp 12677  ccj 13630  abscabs 13768  ℤ[i]cgz 15417  Basecbs 15641  s cress 15642  Ringcrg 18316  Unitcui 18408  invrcinvr 18440  SubRingcsubrg 18545  fldccnfld 19513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870  ax-addf 9871  ax-mulf 9872
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-tpos 7216  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-rp 11665  df-fz 12153  df-seq 12619  df-exp 12678  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-gz 15418  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-mulr 15728  df-starv 15729  df-tset 15733  df-ple 15734  df-ds 15737  df-unif 15738  df-0g 15871  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-grp 17194  df-minusg 17195  df-subg 17360  df-cmn 17964  df-mgp 18259  df-ur 18271  df-ring 18318  df-cring 18319  df-oppr 18392  df-dvdsr 18410  df-unit 18411  df-invr 18441  df-dvr 18452  df-drng 18518  df-subrg 18547  df-cnfld 19514
This theorem is referenced by:  zringunit  19603
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