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Theorem gzrngunit 16719
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 16708 . . . . 5  |-  ZZ [
_i ]  e.  (SubRing ` fld )
2 gzrng.1 . . . . . 6  |-  Z  =  (flds  ZZ [ _i ] )
32subrgbas 15832 . . . . 5  |-  ( ZZ [ _i ]  e.  (SubRing ` fld )  ->  ZZ [ _i ]  =  ( Base `  Z ) )
41, 3ax-mp 8 . . . 4  |-  ZZ [
_i ]  =  (
Base `  Z )
5 eqid 2404 . . . 4  |-  (Unit `  Z )  =  (Unit `  Z )
64, 5unitcl 15719 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  ZZ [ _i ] )
7 eqid 2404 . . . . . . . . . . . 12  |-  ( invr ` fld )  =  ( invr ` fld )
8 eqid 2404 . . . . . . . . . . . 12  |-  ( invr `  Z )  =  (
invr `  Z )
92, 7, 5, 8subrginv 15839 . . . . . . . . . . 11  |-  ( ( ZZ [ _i ]  e.  (SubRing ` fld )  /\  A  e.  (Unit `  Z )
)  ->  ( ( invr ` fld ) `  A )  =  ( ( invr `  Z ) `  A
) )
101, 9mpan 652 . . . . . . . . . 10  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr ` fld ) `  A )  =  ( ( invr `  Z ) `  A
) )
11 gzcn 13255 . . . . . . . . . . . 12  |-  ( A  e.  ZZ [ _i ]  ->  A  e.  CC )
126, 11syl 16 . . . . . . . . . . 11  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  CC )
13 0re 9047 . . . . . . . . . . . . . . 15  |-  0  e.  RR
1413a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  0  e.  RR )
15 1re 9046 . . . . . . . . . . . . . . 15  |-  1  e.  RR
1615a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  1  e.  RR )
1712abscld 12193 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  e.  RR )
18 0lt1 9506 . . . . . . . . . . . . . . 15  |-  0  <  1
1918a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  0  <  1 )
202gzrngunitlem 16718 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( abs `  A ) )
2114, 16, 17, 19, 20ltletrd 9186 . . . . . . . . . . . . 13  |-  ( A  e.  (Unit `  Z
)  ->  0  <  ( abs `  A ) )
2221gt0ne0d 9547 . . . . . . . . . . . 12  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  =/=  0
)
2312abs00ad 12050 . . . . . . . . . . . . 13  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =  0  <->  A  =  0
) )
2423necon3bid 2602 . . . . . . . . . . . 12  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =/=  0  <->  A  =/=  0
) )
2522, 24mpbid 202 . . . . . . . . . . 11  |-  ( A  e.  (Unit `  Z
)  ->  A  =/=  0 )
26 cnfldinv 16687 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( invr ` fld ) `  A )  =  ( 1  /  A ) )
2712, 25, 26syl2anc 643 . . . . . . . . . 10  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr ` fld ) `  A )  =  ( 1  /  A ) )
2810, 27eqtr3d 2438 . . . . . . . . 9  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr `  Z ) `  A )  =  ( 1  /  A ) )
292subrgrng 15826 . . . . . . . . . . 11  |-  ( ZZ [ _i ]  e.  (SubRing ` fld )  ->  Z  e.  Ring )
301, 29ax-mp 8 . . . . . . . . . 10  |-  Z  e. 
Ring
315, 8unitinvcl 15734 . . . . . . . . . 10  |-  ( ( Z  e.  Ring  /\  A  e.  (Unit `  Z )
)  ->  ( ( invr `  Z ) `  A )  e.  (Unit `  Z ) )
3230, 31mpan 652 . . . . . . . . 9  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr `  Z ) `  A )  e.  (Unit `  Z ) )
3328, 32eqeltrrd 2479 . . . . . . . 8  |-  ( A  e.  (Unit `  Z
)  ->  ( 1  /  A )  e.  (Unit `  Z )
)
342gzrngunitlem 16718 . . . . . . . 8  |-  ( ( 1  /  A )  e.  (Unit `  Z
)  ->  1  <_  ( abs `  ( 1  /  A ) ) )
3533, 34syl 16 . . . . . . 7  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( abs `  ( 1  /  A ) ) )
36 ax-1cn 9004 . . . . . . . . 9  |-  1  e.  CC
3736a1i 11 . . . . . . . 8  |-  ( A  e.  (Unit `  Z
)  ->  1  e.  CC )
3837, 12, 25absdivd 12212 . . . . . . 7  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  ( 1  /  A
) )  =  ( ( abs `  1
)  /  ( abs `  A ) ) )
3935, 38breqtrd 4196 . . . . . 6  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( ( abs `  1
)  /  ( abs `  A ) ) )
4036div1i 9698 . . . . . 6  |-  ( 1  /  1 )  =  1
41 abs1 12057 . . . . . . . 8  |-  ( abs `  1 )  =  1
4241eqcomi 2408 . . . . . . 7  |-  1  =  ( abs `  1
)
4342oveq1i 6050 . . . . . 6  |-  ( 1  /  ( abs `  A
) )  =  ( ( abs `  1
)  /  ( abs `  A ) )
4439, 40, 433brtr4g 4204 . . . . 5  |-  ( A  e.  (Unit `  Z
)  ->  ( 1  /  1 )  <_ 
( 1  /  ( abs `  A ) ) )
45 lerec 9848 . . . . . 6  |-  ( ( ( ( abs `  A
)  e.  RR  /\  0  <  ( abs `  A
) )  /\  (
1  e.  RR  /\  0  <  1 ) )  ->  ( ( abs `  A )  <_  1  <->  ( 1  /  1 )  <_  ( 1  / 
( abs `  A
) ) ) )
4617, 21, 16, 19, 45syl22anc 1185 . . . . 5  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  <_ 
1  <->  ( 1  / 
1 )  <_  (
1  /  ( abs `  A ) ) ) )
4744, 46mpbird 224 . . . 4  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  <_  1
)
48 letri3 9116 . . . . 5  |-  ( ( ( abs `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( abs `  A
)  =  1  <->  (
( abs `  A
)  <_  1  /\  1  <_  ( abs `  A
) ) ) )
4917, 15, 48sylancl 644 . . . 4  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =  1  <->  ( ( abs `  A )  <_  1  /\  1  <_  ( abs `  A ) ) ) )
5047, 20, 49mpbir2and 889 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  =  1 )
516, 50jca 519 . 2  |-  ( A  e.  (Unit `  Z
)  ->  ( A  e.  ZZ [ _i ]  /\  ( abs `  A
)  =  1 ) )
5211adantr 452 . . . 4  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  A  e.  CC )
53 simpr 448 . . . . . 6  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  ( abs `  A )  =  1 )
54 ax-1ne0 9015 . . . . . . 7  |-  1  =/=  0
5554a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  1  =/=  0 )
5653, 55eqnetrd 2585 . . . . 5  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  ( abs `  A )  =/=  0 )
57 fveq2 5687 . . . . . . 7  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
58 abs0 12045 . . . . . . 7  |-  ( abs `  0 )  =  0
5957, 58syl6eq 2452 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
6059necon3i 2606 . . . . 5  |-  ( ( abs `  A )  =/=  0  ->  A  =/=  0 )
6156, 60syl 16 . . . 4  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  A  =/=  0 )
62 eldifsn 3887 . . . 4  |-  ( A  e.  ( CC  \  { 0 } )  <-> 
( A  e.  CC  /\  A  =/=  0 ) )
6352, 61, 62sylanbrc 646 . . 3  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  A  e.  ( CC  \  {
0 } ) )
64 simpl 444 . . 3  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  A  e.  ZZ [ _i ]
)
6552, 61, 26syl2anc 643 . . . . 5  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  =  ( 1  /  A ) )
6652absvalsqd 12199 . . . . . . 7  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
6753oveq1d 6055 . . . . . . . 8  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  ( 1 ^ 2 ) )
68 sq1 11431 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
6967, 68syl6eq 2452 . . . . . . 7  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  1 )
7066, 69eqtr3d 2438 . . . . . 6  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  ( A  x.  ( * `  A ) )  =  1 )
7170oveq1d 6055 . . . . 5  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( A  x.  (
* `  A )
)  /  A )  =  ( 1  /  A ) )
7252cjcld 11956 . . . . . 6  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
* `  A )  e.  CC )
7372, 52, 61divcan3d 9751 . . . . 5  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( A  x.  (
* `  A )
)  /  A )  =  ( * `  A ) )
7465, 71, 733eqtr2d 2442 . . . 4  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  =  ( * `  A ) )
75 gzcjcl 13259 . . . . 5  |-  ( A  e.  ZZ [ _i ]  ->  ( * `  A )  e.  ZZ [ _i ] )
7675adantr 452 . . . 4  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
* `  A )  e.  ZZ [ _i ]
)
7774, 76eqeltrd 2478 . . 3  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  e.  ZZ [ _i ] )
78 cnfldbas 16662 . . . . . 6  |-  CC  =  ( Base ` fld )
79 cnfld0 16680 . . . . . 6  |-  0  =  ( 0g ` fld )
80 cndrng 16685 . . . . . 6  |-fld  e.  DivRing
8178, 79, 80drngui 15796 . . . . 5  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
822, 81, 5, 7subrgunit 15841 . . . 4  |-  ( ZZ [ _i ]  e.  (SubRing ` fld )  ->  ( A  e.  (Unit `  Z )  <->  ( A  e.  ( CC 
\  { 0 } )  /\  A  e.  ZZ [ _i ]  /\  ( ( invr ` fld ) `  A )  e.  ZZ [ _i ] ) ) )
831, 82ax-mp 8 . . 3  |-  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ( CC  \  {
0 } )  /\  A  e.  ZZ [ _i ]  /\  ( ( invr ` fld ) `  A )  e.  ZZ [ _i ]
) )
8463, 64, 77, 83syl3anbrc 1138 . 2  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  A  e.  (Unit `  Z )
)
8551, 84impbii 181 1  |-  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ZZ [ _i ]  /\  ( abs `  A
)  =  1 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277   {csn 3774   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    < clt 9076    <_ cle 9077    / cdiv 9633   2c2 10005   ^cexp 11337   *ccj 11856   abscabs 11994   ZZ [ _i ]cgz 13252   Basecbs 13424   ↾s cress 13425   Ringcrg 15615  Unitcui 15699   invrcinvr 15731  SubRingcsubrg 15819  ℂfldccnfld 16658
This theorem is referenced by:  zrngunit  16720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-rp 10569  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-gz 13253  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-subg 14896  df-cmn 15369  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-subrg 15821  df-cnfld 16659
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