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Theorem gzrngunit 21212
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 21200 . . . . 5 β„€[i] ∈ (SubRingβ€˜β„‚fld)
2 gzrng.1 . . . . . 6 𝑍 = (β„‚fld β†Ύs β„€[i])
32subrgbas 20472 . . . . 5 (β„€[i] ∈ (SubRingβ€˜β„‚fld) β†’ β„€[i] = (Baseβ€˜π‘))
41, 3ax-mp 5 . . . 4 β„€[i] = (Baseβ€˜π‘)
5 eqid 2731 . . . 4 (Unitβ€˜π‘) = (Unitβ€˜π‘)
64, 5unitcl 20267 . . 3 (𝐴 ∈ (Unitβ€˜π‘) β†’ 𝐴 ∈ β„€[i])
7 eqid 2731 . . . . . . . . . . . 12 (invrβ€˜β„‚fld) = (invrβ€˜β„‚fld)
8 eqid 2731 . . . . . . . . . . . 12 (invrβ€˜π‘) = (invrβ€˜π‘)
92, 7, 5, 8subrginv 20479 . . . . . . . . . . 11 ((β„€[i] ∈ (SubRingβ€˜β„‚fld) ∧ 𝐴 ∈ (Unitβ€˜π‘)) β†’ ((invrβ€˜β„‚fld)β€˜π΄) = ((invrβ€˜π‘)β€˜π΄))
101, 9mpan 687 . . . . . . . . . 10 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((invrβ€˜β„‚fld)β€˜π΄) = ((invrβ€˜π‘)β€˜π΄))
11 gzcn 16870 . . . . . . . . . . . 12 (𝐴 ∈ β„€[i] β†’ 𝐴 ∈ β„‚)
126, 11syl 17 . . . . . . . . . . 11 (𝐴 ∈ (Unitβ€˜π‘) β†’ 𝐴 ∈ β„‚)
13 0red 11222 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unitβ€˜π‘) β†’ 0 ∈ ℝ)
14 1re 11219 . . . . . . . . . . . . . . 15 1 ∈ ℝ
1514a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unitβ€˜π‘) β†’ 1 ∈ ℝ)
1612abscld 15388 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unitβ€˜π‘) β†’ (absβ€˜π΄) ∈ ℝ)
17 0lt1 11741 . . . . . . . . . . . . . . 15 0 < 1
1817a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unitβ€˜π‘) β†’ 0 < 1)
192gzrngunitlem 21211 . . . . . . . . . . . . . 14 (𝐴 ∈ (Unitβ€˜π‘) β†’ 1 ≀ (absβ€˜π΄))
2013, 15, 16, 18, 19ltletrd 11379 . . . . . . . . . . . . 13 (𝐴 ∈ (Unitβ€˜π‘) β†’ 0 < (absβ€˜π΄))
2120gt0ne0d 11783 . . . . . . . . . . . 12 (𝐴 ∈ (Unitβ€˜π‘) β†’ (absβ€˜π΄) β‰  0)
2212abs00ad 15242 . . . . . . . . . . . . 13 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((absβ€˜π΄) = 0 ↔ 𝐴 = 0))
2322necon3bid 2984 . . . . . . . . . . . 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 21177 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) β†’ ((invrβ€˜β„‚fld)β€˜π΄) = (1 / 𝐴))
2612, 24, 25syl2anc 583 . . . . . . . . . 10 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((invrβ€˜β„‚fld)β€˜π΄) = (1 / 𝐴))
2710, 26eqtr3d 2773 . . . . . . . . 9 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((invrβ€˜π‘)β€˜π΄) = (1 / 𝐴))
282subrgring 20465 . . . . . . . . . . 11 (β„€[i] ∈ (SubRingβ€˜β„‚fld) β†’ 𝑍 ∈ Ring)
291, 28ax-mp 5 . . . . . . . . . 10 𝑍 ∈ Ring
305, 8unitinvcl 20282 . . . . . . . . . 10 ((𝑍 ∈ Ring ∧ 𝐴 ∈ (Unitβ€˜π‘)) β†’ ((invrβ€˜π‘)β€˜π΄) ∈ (Unitβ€˜π‘))
3129, 30mpan 687 . . . . . . . . 9 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((invrβ€˜π‘)β€˜π΄) ∈ (Unitβ€˜π‘))
3227, 31eqeltrrd 2833 . . . . . . . 8 (𝐴 ∈ (Unitβ€˜π‘) β†’ (1 / 𝐴) ∈ (Unitβ€˜π‘))
332gzrngunitlem 21211 . . . . . . . 8 ((1 / 𝐴) ∈ (Unitβ€˜π‘) β†’ 1 ≀ (absβ€˜(1 / 𝐴)))
3432, 33syl 17 . . . . . . 7 (𝐴 ∈ (Unitβ€˜π‘) β†’ 1 ≀ (absβ€˜(1 / 𝐴)))
35 1cnd 11214 . . . . . . . 8 (𝐴 ∈ (Unitβ€˜π‘) β†’ 1 ∈ β„‚)
3635, 12, 24absdivd 15407 . . . . . . 7 (𝐴 ∈ (Unitβ€˜π‘) β†’ (absβ€˜(1 / 𝐴)) = ((absβ€˜1) / (absβ€˜π΄)))
3734, 36breqtrd 5174 . . . . . 6 (𝐴 ∈ (Unitβ€˜π‘) β†’ 1 ≀ ((absβ€˜1) / (absβ€˜π΄)))
38 1div1e1 11909 . . . . . 6 (1 / 1) = 1
39 abs1 15249 . . . . . . . 8 (absβ€˜1) = 1
4039eqcomi 2740 . . . . . . 7 1 = (absβ€˜1)
4140oveq1i 7422 . . . . . 6 (1 / (absβ€˜π΄)) = ((absβ€˜1) / (absβ€˜π΄))
4237, 38, 413brtr4g 5182 . . . . 5 (𝐴 ∈ (Unitβ€˜π‘) β†’ (1 / 1) ≀ (1 / (absβ€˜π΄)))
43 lerec 12102 . . . . . 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 836 . . . . 5 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((absβ€˜π΄) ≀ 1 ↔ (1 / 1) ≀ (1 / (absβ€˜π΄))))
4542, 44mpbird 257 . . . 4 (𝐴 ∈ (Unitβ€˜π‘) β†’ (absβ€˜π΄) ≀ 1)
46 letri3 11304 . . . . 5 (((absβ€˜π΄) ∈ ℝ ∧ 1 ∈ ℝ) β†’ ((absβ€˜π΄) = 1 ↔ ((absβ€˜π΄) ≀ 1 ∧ 1 ≀ (absβ€˜π΄))))
4716, 14, 46sylancl 585 . . . 4 (𝐴 ∈ (Unitβ€˜π‘) β†’ ((absβ€˜π΄) = 1 ↔ ((absβ€˜π΄) ≀ 1 ∧ 1 ≀ (absβ€˜π΄))))
4845, 19, 47mpbir2and 710 . . 3 (𝐴 ∈ (Unitβ€˜π‘) β†’ (absβ€˜π΄) = 1)
496, 48jca 511 . 2 (𝐴 ∈ (Unitβ€˜π‘) β†’ (𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1))
5011adantr 480 . . . 4 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ 𝐴 ∈ β„‚)
51 simpr 484 . . . . . 6 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ (absβ€˜π΄) = 1)
52 ax-1ne0 11183 . . . . . . 7 1 β‰  0
5352a1i 11 . . . . . 6 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ 1 β‰  0)
5451, 53eqnetrd 3007 . . . . 5 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ (absβ€˜π΄) β‰  0)
55 fveq2 6891 . . . . . . 7 (𝐴 = 0 β†’ (absβ€˜π΄) = (absβ€˜0))
56 abs0 15237 . . . . . . 7 (absβ€˜0) = 0
5755, 56eqtrdi 2787 . . . . . 6 (𝐴 = 0 β†’ (absβ€˜π΄) = 0)
5857necon3i 2972 . . . . 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 4790 . . . 4 (𝐴 ∈ (β„‚ βˆ– {0}) ↔ (𝐴 ∈ β„‚ ∧ 𝐴 β‰  0))
6150, 59, 60sylanbrc 582 . . 3 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ 𝐴 ∈ (β„‚ βˆ– {0}))
62 simpl 482 . . 3 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ 𝐴 ∈ β„€[i])
6350, 59, 25syl2anc 583 . . . . 5 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((invrβ€˜β„‚fld)β€˜π΄) = (1 / 𝐴))
6450absvalsqd 15394 . . . . . . 7 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((absβ€˜π΄)↑2) = (𝐴 Β· (βˆ—β€˜π΄)))
6551oveq1d 7427 . . . . . . . 8 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((absβ€˜π΄)↑2) = (1↑2))
66 sq1 14164 . . . . . . . 8 (1↑2) = 1
6765, 66eqtrdi 2787 . . . . . . 7 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((absβ€˜π΄)↑2) = 1)
6864, 67eqtr3d 2773 . . . . . 6 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ (𝐴 Β· (βˆ—β€˜π΄)) = 1)
6968oveq1d 7427 . . . . 5 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((𝐴 Β· (βˆ—β€˜π΄)) / 𝐴) = (1 / 𝐴))
7050cjcld 15148 . . . . . 6 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ (βˆ—β€˜π΄) ∈ β„‚)
7170, 50, 59divcan3d 12000 . . . . 5 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((𝐴 Β· (βˆ—β€˜π΄)) / 𝐴) = (βˆ—β€˜π΄))
7263, 69, 713eqtr2d 2777 . . . 4 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((invrβ€˜β„‚fld)β€˜π΄) = (βˆ—β€˜π΄))
73 gzcjcl 16874 . . . . 5 (𝐴 ∈ β„€[i] β†’ (βˆ—β€˜π΄) ∈ β„€[i])
7473adantr 480 . . . 4 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ (βˆ—β€˜π΄) ∈ β„€[i])
7572, 74eqeltrd 2832 . . 3 ((𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1) β†’ ((invrβ€˜β„‚fld)β€˜π΄) ∈ β„€[i])
76 cnfldbas 21149 . . . . . 6 β„‚ = (Baseβ€˜β„‚fld)
77 cnfld0 21170 . . . . . 6 0 = (0gβ€˜β„‚fld)
78 cndrng 21175 . . . . . 6 β„‚fld ∈ DivRing
7976, 77, 78drngui 20507 . . . . 5 (β„‚ βˆ– {0}) = (Unitβ€˜β„‚fld)
802, 79, 5, 7subrgunit 20481 . . . 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 1342 . 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939   βˆ– cdif 3945  {csn 4628   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  β„‚cc 11112  β„cr 11113  0cc0 11114  1c1 11115   Β· cmul 11119   < clt 11253   ≀ cle 11254   / cdiv 11876  2c2 12272  β†‘cexp 14032  βˆ—ccj 15048  abscabs 15186  β„€[i]cgz 16867  Basecbs 17149   β†Ύs cress 17178  Ringcrg 20128  Unitcui 20247  invrcinvr 20279  SubRingcsubrg 20458  β„‚fldccnfld 21145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192  ax-addf 11193  ax-mulf 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-tpos 8215  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9441  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-uz 12828  df-rp 12980  df-fz 13490  df-seq 13972  df-exp 14033  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-gz 16868  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-starv 17217  df-tset 17221  df-ple 17222  df-ds 17224  df-unif 17225  df-0g 17392  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-grp 18859  df-minusg 18860  df-subg 19040  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-ring 20130  df-cring 20131  df-oppr 20226  df-dvdsr 20249  df-unit 20250  df-invr 20280  df-dvr 20293  df-subrng 20435  df-subrg 20460  df-drng 20503  df-cnfld 21146
This theorem is referenced by:  zringunit  21238
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