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Theorem gzrngunit 21211
Description: The units on β„€[i] are the gaussian integers with norm 1. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypothesis
Ref Expression
gzrng.1 𝑍 = (β„‚fld β†Ύs β„€[i])
Assertion
Ref Expression
gzrngunit (𝐴 ∈ (Unitβ€˜π‘) ↔ (𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1))

Proof of Theorem gzrngunit
StepHypRef Expression
1 gzsubrg 21199 . . . . 5 β„€[i] ∈ (SubRingβ€˜β„‚fld)
2 gzrng.1 . . . . . 6 𝑍 = (β„‚fld β†Ύs β„€[i])
32subrgbas 20471 . . . . 5 (β„€[i] ∈ (SubRingβ€˜β„‚fld) β†’ β„€[i] = (Baseβ€˜π‘))
41, 3ax-mp 5 . . . 4 β„€[i] = (Baseβ€˜π‘)
5 eqid 2730 . . . 4 (Unitβ€˜π‘) = (Unitβ€˜π‘)
64, 5unitcl 20266 . . 3 (𝐴 ∈ (Unitβ€˜π‘) β†’ 𝐴 ∈ β„€[i])
7 eqid 2730 . . . . . . . . . . . 12 (invrβ€˜β„‚fld) = (invrβ€˜β„‚fld)
8 eqid 2730 . . . . . . . . . . . 12 (invrβ€˜π‘) = (invrβ€˜π‘)
92, 7, 5, 8subrginv 20478 . . . . . . . . . . 11 ((β„€[i] ∈ (SubRingβ€˜β„‚fld) ∧ 𝐴 ∈ (Unitβ€˜π‘)) β†’ ((invrβ€˜β„‚fld)β€˜π΄) = ((invrβ€˜π‘)β€˜π΄))
101, 9mpan 686 . . . . . . . . . 10 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((invrβ€˜β„‚fld)β€˜π΄) = ((invrβ€˜π‘)β€˜π΄))
11 gzcn 16869 . . . . . . . . . . . 12 (𝐴 ∈ β„€[i] β†’ 𝐴 ∈ β„‚)
126, 11syl 17 . . . . . . . . . . 11 (𝐴 ∈ (Unitβ€˜π‘) β†’ 𝐴 ∈ β„‚)
13 0red 11221 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unitβ€˜π‘) β†’ 0 ∈ ℝ)
14 1re 11218 . . . . . . . . . . . . . . 15 1 ∈ ℝ
1514a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unitβ€˜π‘) β†’ 1 ∈ ℝ)
1612abscld 15387 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unitβ€˜π‘) β†’ (absβ€˜π΄) ∈ ℝ)
17 0lt1 11740 . . . . . . . . . . . . . . 15 0 < 1
1817a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unitβ€˜π‘) β†’ 0 < 1)
192gzrngunitlem 21210 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unitβ€˜π‘) β†’ 1 ≀ (absβ€˜π΄))
2013, 15, 16, 18, 19ltletrd 11378 . . . . . . . . . . . . 13 (𝐴 ∈ (Unitβ€˜π‘) β†’ 0 < (absβ€˜π΄))
2120gt0ne0d 11782 . . . . . . . . . . . 12 (𝐴 ∈ (Unitβ€˜π‘) β†’ (absβ€˜π΄) β‰  0)
2212abs00ad 15241 . . . . . . . . . . . . 13 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((absβ€˜π΄) = 0 ↔ 𝐴 = 0))
2322necon3bid 2983 . . . . . . . . . . . 12 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((absβ€˜π΄) β‰  0 ↔ 𝐴 β‰  0))
2421, 23mpbid 231 . . . . . . . . . . 11 (𝐴 ∈ (Unitβ€˜π‘) β†’ 𝐴 β‰  0)
25 cnfldinv 21176 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) β†’ ((invrβ€˜β„‚fld)β€˜π΄) = (1 / 𝐴))
2612, 24, 25syl2anc 582 . . . . . . . . . 10 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((invrβ€˜β„‚fld)β€˜π΄) = (1 / 𝐴))
2710, 26eqtr3d 2772 . . . . . . . . 9 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((invrβ€˜π‘)β€˜π΄) = (1 / 𝐴))
282subrgring 20464 . . . . . . . . . . 11 (β„€[i] ∈ (SubRingβ€˜β„‚fld) β†’ 𝑍 ∈ Ring)
291, 28ax-mp 5 . . . . . . . . . 10 𝑍 ∈ Ring
305, 8unitinvcl 20281 . . . . . . . . . 10 ((𝑍 ∈ Ring ∧ 𝐴 ∈ (Unitβ€˜π‘)) β†’ ((invrβ€˜π‘)β€˜π΄) ∈ (Unitβ€˜π‘))
3129, 30mpan 686 . . . . . . . . 9 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((invrβ€˜π‘)β€˜π΄) ∈ (Unitβ€˜π‘))
3227, 31eqeltrrd 2832 . . . . . . . 8 (𝐴 ∈ (Unitβ€˜π‘) β†’ (1 / 𝐴) ∈ (Unitβ€˜π‘))
332gzrngunitlem 21210 . . . . . . . 8 ((1 / 𝐴) ∈ (Unitβ€˜π‘) β†’ 1 ≀ (absβ€˜(1 / 𝐴)))
3432, 33syl 17 . . . . . . 7 (𝐴 ∈ (Unitβ€˜π‘) β†’ 1 ≀ (absβ€˜(1 / 𝐴)))
35 1cnd 11213 . . . . . . . 8 (𝐴 ∈ (Unitβ€˜π‘) β†’ 1 ∈ β„‚)
3635, 12, 24absdivd 15406 . . . . . . 7 (𝐴 ∈ (Unitβ€˜π‘) β†’ (absβ€˜(1 / 𝐴)) = ((absβ€˜1) / (absβ€˜π΄)))
3734, 36breqtrd 5173 . . . . . 6 (𝐴 ∈ (Unitβ€˜π‘) β†’ 1 ≀ ((absβ€˜1) / (absβ€˜π΄)))
38 1div1e1 11908 . . . . . 6 (1 / 1) = 1
39 abs1 15248 . . . . . . . 8 (absβ€˜1) = 1
4039eqcomi 2739 . . . . . . 7 1 = (absβ€˜1)
4140oveq1i 7421 . . . . . 6 (1 / (absβ€˜π΄)) = ((absβ€˜1) / (absβ€˜π΄))
4237, 38, 413brtr4g 5181 . . . . 5 (𝐴 ∈ (Unitβ€˜π‘) β†’ (1 / 1) ≀ (1 / (absβ€˜π΄)))
43 lerec 12101 . . . . . 6 ((((absβ€˜π΄) ∈ ℝ ∧ 0 < (absβ€˜π΄)) ∧ (1 ∈ ℝ ∧ 0 < 1)) β†’ ((absβ€˜π΄) ≀ 1 ↔ (1 / 1) ≀ (1 / (absβ€˜π΄))))
4416, 20, 15, 18, 43syl22anc 835 . . . . 5 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((absβ€˜π΄) ≀ 1 ↔ (1 / 1) ≀ (1 / (absβ€˜π΄))))
4542, 44mpbird 256 . . . 4 (𝐴 ∈ (Unitβ€˜π‘) β†’ (absβ€˜π΄) ≀ 1)
46 letri3 11303 . . . . 5 (((absβ€˜π΄) ∈ ℝ ∧ 1 ∈ ℝ) β†’ ((absβ€˜π΄) = 1 ↔ ((absβ€˜π΄) ≀ 1 ∧ 1 ≀ (absβ€˜π΄))))
4716, 14, 46sylancl 584 . . . 4 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((absβ€˜π΄) = 1 ↔ ((absβ€˜π΄) ≀ 1 ∧ 1 ≀ (absβ€˜π΄))))
4845, 19, 47mpbir2and 709 . . 3 (𝐴 ∈ (Unitβ€˜π‘) β†’ (absβ€˜π΄) = 1)
496, 48jca 510 . 2 (𝐴 ∈ (Unitβ€˜π‘) β†’ (𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1))
5011adantr 479 . . . 4 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ 𝐴 ∈ β„‚)
51 simpr 483 . . . . . 6 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ (absβ€˜π΄) = 1)
52 ax-1ne0 11181 . . . . . . 7 1 β‰  0
5352a1i 11 . . . . . 6 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ 1 β‰  0)
5451, 53eqnetrd 3006 . . . . 5 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ (absβ€˜π΄) β‰  0)
55 fveq2 6890 . . . . . . 7 (𝐴 = 0 β†’ (absβ€˜π΄) = (absβ€˜0))
56 abs0 15236 . . . . . . 7 (absβ€˜0) = 0
5755, 56eqtrdi 2786 . . . . . 6 (𝐴 = 0 β†’ (absβ€˜π΄) = 0)
5857necon3i 2971 . . . . 5 ((absβ€˜π΄) β‰  0 β†’ 𝐴 β‰  0)
5954, 58syl 17 . . . 4 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ 𝐴 β‰  0)
60 eldifsn 4789 . . . 4 (𝐴 ∈ (β„‚ βˆ– {0}) ↔ (𝐴 ∈ β„‚ ∧ 𝐴 β‰  0))
6150, 59, 60sylanbrc 581 . . 3 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ 𝐴 ∈ (β„‚ βˆ– {0}))
62 simpl 481 . . 3 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ 𝐴 ∈ β„€[i])
6350, 59, 25syl2anc 582 . . . . 5 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((invrβ€˜β„‚fld)β€˜π΄) = (1 / 𝐴))
6450absvalsqd 15393 . . . . . . 7 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((absβ€˜π΄)↑2) = (𝐴 Β· (βˆ—β€˜π΄)))
6551oveq1d 7426 . . . . . . . 8 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((absβ€˜π΄)↑2) = (1↑2))
66 sq1 14163 . . . . . . . 8 (1↑2) = 1
6765, 66eqtrdi 2786 . . . . . . 7 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((absβ€˜π΄)↑2) = 1)
6864, 67eqtr3d 2772 . . . . . 6 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ (𝐴 Β· (βˆ—β€˜π΄)) = 1)
6968oveq1d 7426 . . . . 5 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((𝐴 Β· (βˆ—β€˜π΄)) / 𝐴) = (1 / 𝐴))
7050cjcld 15147 . . . . . 6 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ (βˆ—β€˜π΄) ∈ β„‚)
7170, 50, 59divcan3d 11999 . . . . 5 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((𝐴 Β· (βˆ—β€˜π΄)) / 𝐴) = (βˆ—β€˜π΄))
7263, 69, 713eqtr2d 2776 . . . 4 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((invrβ€˜β„‚fld)β€˜π΄) = (βˆ—β€˜π΄))
73 gzcjcl 16873 . . . . 5 (𝐴 ∈ β„€[i] β†’ (βˆ—β€˜π΄) ∈ β„€[i])
7473adantr 479 . . . 4 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ (βˆ—β€˜π΄) ∈ β„€[i])
7572, 74eqeltrd 2831 . . 3 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((invrβ€˜β„‚fld)β€˜π΄) ∈ β„€[i])
76 cnfldbas 21148 . . . . . 6 β„‚ = (Baseβ€˜β„‚fld)
77 cnfld0 21169 . . . . . 6 0 = (0gβ€˜β„‚fld)
78 cndrng 21174 . . . . . 6 β„‚fld ∈ DivRing
7976, 77, 78drngui 20506 . . . . 5 (β„‚ βˆ– {0}) = (Unitβ€˜β„‚fld)
802, 79, 5, 7subrgunit 20480 . . . 4 (β„€[i] ∈ (SubRingβ€˜β„‚fld) β†’ (𝐴 ∈ (Unitβ€˜π‘) ↔ (𝐴 ∈ (β„‚ βˆ– {0}) ∧ 𝐴 ∈ β„€[i] ∧ ((invrβ€˜β„‚fld)β€˜π΄) ∈ β„€[i])))
811, 80ax-mp 5 . . 3 (𝐴 ∈ (Unitβ€˜π‘) ↔ (𝐴 ∈ (β„‚ βˆ– {0}) ∧ 𝐴 ∈ β„€[i] ∧ ((invrβ€˜β„‚fld)β€˜π΄) ∈ β„€[i]))
8261, 62, 75, 81syl3anbrc 1341 . 2 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ 𝐴 ∈ (Unitβ€˜π‘))
8349, 82impbii 208 1 (𝐴 ∈ (Unitβ€˜π‘) ↔ (𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938   βˆ– cdif 3944  {csn 4627   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113   Β· cmul 11117   < clt 11252   ≀ cle 11253   / cdiv 11875  2c2 12271  β†‘cexp 14031  βˆ—ccj 15047  abscabs 15185  β„€[i]cgz 16866  Basecbs 17148   β†Ύs cress 17177  Ringcrg 20127  Unitcui 20246  invrcinvr 20278  SubRingcsubrg 20457  β„‚fldccnfld 21144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-rp 12979  df-fz 13489  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-gz 16867  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-starv 17216  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-subg 19039  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-cring 20130  df-oppr 20225  df-dvdsr 20248  df-unit 20249  df-invr 20279  df-dvr 20292  df-subrng 20434  df-subrg 20459  df-drng 20502  df-cnfld 21145
This theorem is referenced by:  zringunit  21237
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