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Theorem gzrngunit 16437
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 16426 . . . . 5  |-  ZZ [
_i ]  e.  (SubRing ` fld )
2 gzrng.1 . . . . . 6  |-  Z  =  (flds  ZZ [ _i ] )
32subrgbas 15554 . . . . 5  |-  ( ZZ [ _i ]  e.  (SubRing ` fld )  ->  ZZ [ _i ]  =  ( Base `  Z ) )
41, 3ax-mp 8 . . . 4  |-  ZZ [
_i ]  =  (
Base `  Z )
5 eqid 2283 . . . 4  |-  (Unit `  Z )  =  (Unit `  Z )
64, 5unitcl 15441 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  ZZ [ _i ] )
7 eqid 2283 . . . . . . . . . . . 12  |-  ( invr ` fld )  =  ( invr ` fld )
8 eqid 2283 . . . . . . . . . . . 12  |-  ( invr `  Z )  =  (
invr `  Z )
92, 7, 5, 8subrginv 15561 . . . . . . . . . . 11  |-  ( ( ZZ [ _i ]  e.  (SubRing ` fld )  /\  A  e.  (Unit `  Z )
)  ->  ( ( invr ` fld ) `  A )  =  ( ( invr `  Z ) `  A
) )
101, 9mpan 651 . . . . . . . . . 10  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr ` fld ) `  A )  =  ( ( invr `  Z ) `  A
) )
11 gzcn 12979 . . . . . . . . . . . 12  |-  ( A  e.  ZZ [ _i ]  ->  A  e.  CC )
126, 11syl 15 . . . . . . . . . . 11  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  CC )
13 0re 8838 . . . . . . . . . . . . . . 15  |-  0  e.  RR
1413a1i 10 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  0  e.  RR )
15 1re 8837 . . . . . . . . . . . . . . 15  |-  1  e.  RR
1615a1i 10 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  1  e.  RR )
1712abscld 11918 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  e.  RR )
18 0lt1 9296 . . . . . . . . . . . . . . 15  |-  0  <  1
1918a1i 10 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  0  <  1 )
202gzrngunitlem 16436 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( abs `  A ) )
2114, 16, 17, 19, 20ltletrd 8976 . . . . . . . . . . . . 13  |-  ( A  e.  (Unit `  Z
)  ->  0  <  ( abs `  A ) )
2221gt0ne0d 9337 . . . . . . . . . . . 12  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  =/=  0
)
23 abs00 11774 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( abs `  A
)  =  0  <->  A  =  0 ) )
2412, 23syl 15 . . . . . . . . . . . . 13  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =  0  <->  A  =  0
) )
2524necon3bid 2481 . . . . . . . . . . . 12  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =/=  0  <->  A  =/=  0
) )
2622, 25mpbid 201 . . . . . . . . . . 11  |-  ( A  e.  (Unit `  Z
)  ->  A  =/=  0 )
27 cnfldinv 16405 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( invr ` fld ) `  A )  =  ( 1  /  A ) )
2812, 26, 27syl2anc 642 . . . . . . . . . 10  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr ` fld ) `  A )  =  ( 1  /  A ) )
2910, 28eqtr3d 2317 . . . . . . . . 9  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr `  Z ) `  A )  =  ( 1  /  A ) )
302subrgrng 15548 . . . . . . . . . . 11  |-  ( ZZ [ _i ]  e.  (SubRing ` fld )  ->  Z  e.  Ring )
311, 30ax-mp 8 . . . . . . . . . 10  |-  Z  e. 
Ring
325, 8unitinvcl 15456 . . . . . . . . . 10  |-  ( ( Z  e.  Ring  /\  A  e.  (Unit `  Z )
)  ->  ( ( invr `  Z ) `  A )  e.  (Unit `  Z ) )
3331, 32mpan 651 . . . . . . . . 9  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr `  Z ) `  A )  e.  (Unit `  Z ) )
3429, 33eqeltrrd 2358 . . . . . . . 8  |-  ( A  e.  (Unit `  Z
)  ->  ( 1  /  A )  e.  (Unit `  Z )
)
352gzrngunitlem 16436 . . . . . . . 8  |-  ( ( 1  /  A )  e.  (Unit `  Z
)  ->  1  <_  ( abs `  ( 1  /  A ) ) )
3634, 35syl 15 . . . . . . 7  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( abs `  ( 1  /  A ) ) )
37 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
3837a1i 10 . . . . . . . 8  |-  ( A  e.  (Unit `  Z
)  ->  1  e.  CC )
3938, 12, 26absdivd 11937 . . . . . . 7  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  ( 1  /  A
) )  =  ( ( abs `  1
)  /  ( abs `  A ) ) )
4036, 39breqtrd 4047 . . . . . 6  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( ( abs `  1
)  /  ( abs `  A ) ) )
4137div1i 9488 . . . . . 6  |-  ( 1  /  1 )  =  1
42 abs1 11782 . . . . . . . 8  |-  ( abs `  1 )  =  1
4342eqcomi 2287 . . . . . . 7  |-  1  =  ( abs `  1
)
4443oveq1i 5868 . . . . . 6  |-  ( 1  /  ( abs `  A
) )  =  ( ( abs `  1
)  /  ( abs `  A ) )
4540, 41, 443brtr4g 4055 . . . . 5  |-  ( A  e.  (Unit `  Z
)  ->  ( 1  /  1 )  <_ 
( 1  /  ( abs `  A ) ) )
46 lerec 9638 . . . . . 6  |-  ( ( ( ( abs `  A
)  e.  RR  /\  0  <  ( abs `  A
) )  /\  (
1  e.  RR  /\  0  <  1 ) )  ->  ( ( abs `  A )  <_  1  <->  ( 1  /  1 )  <_  ( 1  / 
( abs `  A
) ) ) )
4717, 21, 16, 19, 46syl22anc 1183 . . . . 5  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  <_ 
1  <->  ( 1  / 
1 )  <_  (
1  /  ( abs `  A ) ) ) )
4845, 47mpbird 223 . . . 4  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  <_  1
)
49 letri3 8907 . . . . 5  |-  ( ( ( abs `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( abs `  A
)  =  1  <->  (
( abs `  A
)  <_  1  /\  1  <_  ( abs `  A
) ) ) )
5017, 15, 49sylancl 643 . . . 4  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =  1  <->  ( ( abs `  A )  <_  1  /\  1  <_  ( abs `  A ) ) ) )
5148, 20, 50mpbir2and 888 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  =  1 )
526, 51jca 518 . 2  |-  ( A  e.  (Unit `  Z
)  ->  ( A  e.  ZZ [ _i ]  /\  ( abs `  A
)  =  1 ) )
5311adantr 451 . . . 4  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  A  e.  CC )
54 simpr 447 . . . . . 6  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  ( abs `  A )  =  1 )
55 ax-1ne0 8806 . . . . . . 7  |-  1  =/=  0
5655a1i 10 . . . . . 6  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  1  =/=  0 )
5754, 56eqnetrd 2464 . . . . 5  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  ( abs `  A )  =/=  0 )
58 fveq2 5525 . . . . . . 7  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
59 abs0 11770 . . . . . . 7  |-  ( abs `  0 )  =  0
6058, 59syl6eq 2331 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
6160necon3i 2485 . . . . 5  |-  ( ( abs `  A )  =/=  0  ->  A  =/=  0 )
6257, 61syl 15 . . . 4  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  A  =/=  0 )
63 eldifsn 3749 . . . 4  |-  ( A  e.  ( CC  \  { 0 } )  <-> 
( A  e.  CC  /\  A  =/=  0 ) )
6453, 62, 63sylanbrc 645 . . 3  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  A  e.  ( CC  \  {
0 } ) )
65 simpl 443 . . 3  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  A  e.  ZZ [ _i ]
)
6653, 62, 27syl2anc 642 . . . . 5  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  =  ( 1  /  A ) )
6753absvalsqd 11924 . . . . . . 7  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
6854oveq1d 5873 . . . . . . . 8  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  ( 1 ^ 2 ) )
69 sq1 11198 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
7068, 69syl6eq 2331 . . . . . . 7  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  1 )
7167, 70eqtr3d 2317 . . . . . 6  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  ( A  x.  ( * `  A ) )  =  1 )
7271oveq1d 5873 . . . . 5  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( A  x.  (
* `  A )
)  /  A )  =  ( 1  /  A ) )
7353cjcld 11681 . . . . . 6  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
* `  A )  e.  CC )
7473, 53, 62divcan3d 9541 . . . . 5  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( A  x.  (
* `  A )
)  /  A )  =  ( * `  A ) )
7566, 72, 743eqtr2d 2321 . . . 4  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  =  ( * `  A ) )
76 gzcjcl 12983 . . . . 5  |-  ( A  e.  ZZ [ _i ]  ->  ( * `  A )  e.  ZZ [ _i ] )
7776adantr 451 . . . 4  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
* `  A )  e.  ZZ [ _i ]
)
7875, 77eqeltrd 2357 . . 3  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  e.  ZZ [ _i ] )
79 cnfldbas 16383 . . . . . 6  |-  CC  =  ( Base ` fld )
80 cnfld0 16398 . . . . . 6  |-  0  =  ( 0g ` fld )
81 cndrng 16403 . . . . . 6  |-fld  e.  DivRing
8279, 80, 81drngui 15518 . . . . 5  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
832, 82, 5, 7subrgunit 15563 . . . 4  |-  ( ZZ [ _i ]  e.  (SubRing ` fld )  ->  ( A  e.  (Unit `  Z )  <->  ( A  e.  ( CC 
\  { 0 } )  /\  A  e.  ZZ [ _i ]  /\  ( ( invr ` fld ) `  A )  e.  ZZ [ _i ] ) ) )
841, 83ax-mp 8 . . 3  |-  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ( CC  \  {
0 } )  /\  A  e.  ZZ [ _i ]  /\  ( ( invr ` fld ) `  A )  e.  ZZ [ _i ]
) )
8564, 65, 78, 84syl3anbrc 1136 . 2  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  A  e.  (Unit `  Z )
)
8652, 85impbii 180 1  |-  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ZZ [ _i ]  /\  ( abs `  A
)  =  1 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423   2c2 9795   ^cexp 11104   *ccj 11581   abscabs 11719   ZZ [ _i ]cgz 12976   Basecbs 13148   ↾s cress 13149   Ringcrg 15337  Unitcui 15421   invrcinvr 15453  SubRingcsubrg 15541  ℂfldccnfld 16377
This theorem is referenced by:  zrngunit  16438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-gz 12977  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-subrg 15543  df-cnfld 16378
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