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Theorem gzrngunit 18968
Description: The units on  ZZ [
_i ] are the gaussian integers with norm  1. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypothesis
Ref Expression
gzrng.1  |-  Z  =  (flds  ZZ[_i] )
Assertion
Ref Expression
gzrngunit  |-  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ZZ[_i]  /\  ( abs `  A
)  =  1 ) )

Proof of Theorem gzrngunit
StepHypRef Expression
1 gzsubrg 18957 . . . . 5  |-  ZZ[_i]  e.  (SubRing ` fld )
2 gzrng.1 . . . . . 6  |-  Z  =  (flds  ZZ[_i] )
32subrgbas 17952 . . . . 5  |-  ( ZZ[_i]  e.  (SubRing ` fld )  ->  ZZ[_i]  =  ( Base `  Z ) )
41, 3ax-mp 5 . . . 4  |-  ZZ[_i]  =  ( Base `  Z )
5 eqid 2429 . . . 4  |-  (Unit `  Z )  =  (Unit `  Z )
64, 5unitcl 17822 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  ZZ[_i] )
7 eqid 2429 . . . . . . . . . . . 12  |-  ( invr ` fld )  =  ( invr ` fld )
8 eqid 2429 . . . . . . . . . . . 12  |-  ( invr `  Z )  =  (
invr `  Z )
92, 7, 5, 8subrginv 17959 . . . . . . . . . . 11  |-  ( ( ZZ[_i] 
e.  (SubRing ` fld )  /\  A  e.  (Unit `  Z )
)  ->  ( ( invr ` fld ) `  A )  =  ( ( invr `  Z ) `  A
) )
101, 9mpan 674 . . . . . . . . . 10  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr ` fld ) `  A )  =  ( ( invr `  Z ) `  A
) )
11 gzcn 14839 . . . . . . . . . . . 12  |-  ( A  e.  ZZ[_i]  ->  A  e.  CC )
126, 11syl 17 . . . . . . . . . . 11  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  CC )
13 0red 9643 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  0  e.  RR )
14 1re 9641 . . . . . . . . . . . . . . 15  |-  1  e.  RR
1514a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  1  e.  RR )
1612abscld 13476 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  e.  RR )
17 0lt1 10135 . . . . . . . . . . . . . . 15  |-  0  <  1
1817a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  0  <  1 )
192gzrngunitlem 18967 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( abs `  A ) )
2013, 15, 16, 18, 19ltletrd 9794 . . . . . . . . . . . . 13  |-  ( A  e.  (Unit `  Z
)  ->  0  <  ( abs `  A ) )
2120gt0ne0d 10177 . . . . . . . . . . . 12  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  =/=  0
)
2212abs00ad 13332 . . . . . . . . . . . . 13  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =  0  <->  A  =  0
) )
2322necon3bid 2689 . . . . . . . . . . . 12  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =/=  0  <->  A  =/=  0
) )
2421, 23mpbid 213 . . . . . . . . . . 11  |-  ( A  e.  (Unit `  Z
)  ->  A  =/=  0 )
25 cnfldinv 18934 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( invr ` fld ) `  A )  =  ( 1  /  A ) )
2612, 24, 25syl2anc 665 . . . . . . . . . 10  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr ` fld ) `  A )  =  ( 1  /  A ) )
2710, 26eqtr3d 2472 . . . . . . . . 9  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr `  Z ) `  A )  =  ( 1  /  A ) )
282subrgring 17946 . . . . . . . . . . 11  |-  ( ZZ[_i]  e.  (SubRing ` fld )  ->  Z  e.  Ring )
291, 28ax-mp 5 . . . . . . . . . 10  |-  Z  e. 
Ring
305, 8unitinvcl 17837 . . . . . . . . . 10  |-  ( ( Z  e.  Ring  /\  A  e.  (Unit `  Z )
)  ->  ( ( invr `  Z ) `  A )  e.  (Unit `  Z ) )
3129, 30mpan 674 . . . . . . . . 9  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr `  Z ) `  A )  e.  (Unit `  Z ) )
3227, 31eqeltrrd 2518 . . . . . . . 8  |-  ( A  e.  (Unit `  Z
)  ->  ( 1  /  A )  e.  (Unit `  Z )
)
332gzrngunitlem 18967 . . . . . . . 8  |-  ( ( 1  /  A )  e.  (Unit `  Z
)  ->  1  <_  ( abs `  ( 1  /  A ) ) )
3432, 33syl 17 . . . . . . 7  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( abs `  ( 1  /  A ) ) )
35 1cnd 9658 . . . . . . . 8  |-  ( A  e.  (Unit `  Z
)  ->  1  e.  CC )
3635, 12, 24absdivd 13495 . . . . . . 7  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  ( 1  /  A
) )  =  ( ( abs `  1
)  /  ( abs `  A ) ) )
3734, 36breqtrd 4450 . . . . . 6  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( ( abs `  1
)  /  ( abs `  A ) ) )
38 1div1e1 10299 . . . . . 6  |-  ( 1  /  1 )  =  1
39 abs1 13339 . . . . . . . 8  |-  ( abs `  1 )  =  1
4039eqcomi 2442 . . . . . . 7  |-  1  =  ( abs `  1
)
4140oveq1i 6315 . . . . . 6  |-  ( 1  /  ( abs `  A
) )  =  ( ( abs `  1
)  /  ( abs `  A ) )
4237, 38, 413brtr4g 4458 . . . . 5  |-  ( A  e.  (Unit `  Z
)  ->  ( 1  /  1 )  <_ 
( 1  /  ( abs `  A ) ) )
43 lerec 10488 . . . . . 6  |-  ( ( ( ( abs `  A
)  e.  RR  /\  0  <  ( abs `  A
) )  /\  (
1  e.  RR  /\  0  <  1 ) )  ->  ( ( abs `  A )  <_  1  <->  ( 1  /  1 )  <_  ( 1  / 
( abs `  A
) ) ) )
4416, 20, 15, 18, 43syl22anc 1265 . . . . 5  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  <_ 
1  <->  ( 1  / 
1 )  <_  (
1  /  ( abs `  A ) ) ) )
4542, 44mpbird 235 . . . 4  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  <_  1
)
46 letri3 9718 . . . . 5  |-  ( ( ( abs `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( abs `  A
)  =  1  <->  (
( abs `  A
)  <_  1  /\  1  <_  ( abs `  A
) ) ) )
4716, 14, 46sylancl 666 . . . 4  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =  1  <->  ( ( abs `  A )  <_  1  /\  1  <_  ( abs `  A ) ) ) )
4845, 19, 47mpbir2and 930 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  =  1 )
496, 48jca 534 . 2  |-  ( A  e.  (Unit `  Z
)  ->  ( A  e.  ZZ[_i]  /\  ( abs `  A
)  =  1 ) )
5011adantr 466 . . . 4  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  e.  CC )
51 simpr 462 . . . . . 6  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  ( abs `  A )  =  1 )
52 ax-1ne0 9607 . . . . . . 7  |-  1  =/=  0
5352a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  1  =/=  0 )
5451, 53eqnetrd 2724 . . . . 5  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  ( abs `  A )  =/=  0 )
55 fveq2 5881 . . . . . . 7  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
56 abs0 13327 . . . . . . 7  |-  ( abs `  0 )  =  0
5755, 56syl6eq 2486 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
5857necon3i 2671 . . . . 5  |-  ( ( abs `  A )  =/=  0  ->  A  =/=  0 )
5954, 58syl 17 . . . 4  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  =/=  0 )
60 eldifsn 4128 . . . 4  |-  ( A  e.  ( CC  \  { 0 } )  <-> 
( A  e.  CC  /\  A  =/=  0 ) )
6150, 59, 60sylanbrc 668 . . 3  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  e.  ( CC  \  {
0 } ) )
62 simpl 458 . . 3  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  e.  ZZ[_i]
)
6350, 59, 25syl2anc 665 . . . . 5  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  =  ( 1  /  A ) )
6450absvalsqd 13482 . . . . . . 7  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
6551oveq1d 6320 . . . . . . . 8  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  ( 1 ^ 2 ) )
66 sq1 12366 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
6765, 66syl6eq 2486 . . . . . . 7  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  1 )
6864, 67eqtr3d 2472 . . . . . 6  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  ( A  x.  ( * `  A ) )  =  1 )
6968oveq1d 6320 . . . . 5  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( A  x.  (
* `  A )
)  /  A )  =  ( 1  /  A ) )
7050cjcld 13238 . . . . . 6  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
* `  A )  e.  CC )
7170, 50, 59divcan3d 10387 . . . . 5  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( A  x.  (
* `  A )
)  /  A )  =  ( * `  A ) )
7263, 69, 713eqtr2d 2476 . . . 4  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  =  ( * `  A ) )
73 gzcjcl 14843 . . . . 5  |-  ( A  e.  ZZ[_i]  ->  ( * `  A )  e.  ZZ[_i] )
7473adantr 466 . . . 4  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
* `  A )  e.  ZZ[_i]
)
7572, 74eqeltrd 2517 . . 3  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  e.  ZZ[_i] )
76 cnfldbas 18909 . . . . . 6  |-  CC  =  ( Base ` fld )
77 cnfld0 18927 . . . . . 6  |-  0  =  ( 0g ` fld )
78 cndrng 18932 . . . . . 6  |-fld  e.  DivRing
7976, 77, 78drngui 17916 . . . . 5  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
802, 79, 5, 7subrgunit 17961 . . . 4  |-  ( ZZ[_i]  e.  (SubRing ` fld )  ->  ( A  e.  (Unit `  Z )  <->  ( A  e.  ( CC 
\  { 0 } )  /\  A  e.  ZZ[_i]  /\  ( ( invr ` fld ) `  A )  e.  ZZ[_i] ) ) )
811, 80ax-mp 5 . . 3  |-  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ( CC  \  {
0 } )  /\  A  e.  ZZ[_i]  /\  ( (
invr ` fld ) `  A )  e.  ZZ[_i] ) )
8261, 62, 75, 81syl3anbrc 1189 . 2  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  e.  (Unit `  Z )
)
8349, 82impbii 190 1  |-  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ZZ[_i]  /\  ( abs `  A
)  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625    \ cdif 3439   {csn 4002   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543    < clt 9674    <_ cle 9675    / cdiv 10268   2c2 10659   ^cexp 12269   *ccj 13138   abscabs 13276   ZZ[_i]cgz 14836   Basecbs 15084   ↾s cress 15085   Ringcrg 17715  Unitcui 17802   invrcinvr 17834  SubRingcsubrg 17939  ℂfldccnfld 18905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-fz 11783  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-gz 14837  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-subg 16765  df-cmn 17367  df-mgp 17659  df-ur 17671  df-ring 17717  df-cring 17718  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-drng 17912  df-subrg 17941  df-cnfld 18906
This theorem is referenced by:  zringunit  18993
  Copyright terms: Public domain W3C validator