MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gzrngunit Unicode version

Theorem gzrngunit 16543
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 16532 . . . . 5  |-  ZZ [
_i ]  e.  (SubRing ` fld )
2 gzrng.1 . . . . . 6  |-  Z  =  (flds  ZZ [ _i ] )
32subrgbas 15653 . . . . 5  |-  ( ZZ [ _i ]  e.  (SubRing ` fld )  ->  ZZ [ _i ]  =  ( Base `  Z ) )
41, 3ax-mp 8 . . . 4  |-  ZZ [
_i ]  =  (
Base `  Z )
5 eqid 2358 . . . 4  |-  (Unit `  Z )  =  (Unit `  Z )
64, 5unitcl 15540 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  ZZ [ _i ] )
7 eqid 2358 . . . . . . . . . . . 12  |-  ( invr ` fld )  =  ( invr ` fld )
8 eqid 2358 . . . . . . . . . . . 12  |-  ( invr `  Z )  =  (
invr `  Z )
92, 7, 5, 8subrginv 15660 . . . . . . . . . . 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 13076 . . . . . . . . . . . 12  |-  ( A  e.  ZZ [ _i ]  ->  A  e.  CC )
126, 11syl 15 . . . . . . . . . . 11  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  CC )
13 0re 8928 . . . . . . . . . . . . . . 15  |-  0  e.  RR
1413a1i 10 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  0  e.  RR )
15 1re 8927 . . . . . . . . . . . . . . 15  |-  1  e.  RR
1615a1i 10 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  1  e.  RR )
1712abscld 12014 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  e.  RR )
18 0lt1 9386 . . . . . . . . . . . . . . 15  |-  0  <  1
1918a1i 10 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  0  <  1 )
202gzrngunitlem 16542 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( abs `  A ) )
2114, 16, 17, 19, 20ltletrd 9066 . . . . . . . . . . . . 13  |-  ( A  e.  (Unit `  Z
)  ->  0  <  ( abs `  A ) )
2221gt0ne0d 9427 . . . . . . . . . . . 12  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  =/=  0
)
23 abs00 11870 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( abs `  A
)  =  0  <->  A  =  0 ) )
2412, 23syl 15 . . . . . . . . . . . . 13  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =  0  <->  A  =  0
) )
2524necon3bid 2556 . . . . . . . . . . . 12  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =/=  0  <->  A  =/=  0
) )
2622, 25mpbid 201 . . . . . . . . . . 11  |-  ( A  e.  (Unit `  Z
)  ->  A  =/=  0 )
27 cnfldinv 16511 . . . . . . . . . . 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 2392 . . . . . . . . 9  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr `  Z ) `  A )  =  ( 1  /  A ) )
302subrgrng 15647 . . . . . . . . . . 11  |-  ( ZZ [ _i ]  e.  (SubRing ` fld )  ->  Z  e.  Ring )
311, 30ax-mp 8 . . . . . . . . . 10  |-  Z  e. 
Ring
325, 8unitinvcl 15555 . . . . . . . . . 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 2433 . . . . . . . 8  |-  ( A  e.  (Unit `  Z
)  ->  ( 1  /  A )  e.  (Unit `  Z )
)
352gzrngunitlem 16542 . . . . . . . 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 8885 . . . . . . . . 9  |-  1  e.  CC
3837a1i 10 . . . . . . . 8  |-  ( A  e.  (Unit `  Z
)  ->  1  e.  CC )
3938, 12, 26absdivd 12033 . . . . . . 7  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  ( 1  /  A
) )  =  ( ( abs `  1
)  /  ( abs `  A ) ) )
4036, 39breqtrd 4128 . . . . . 6  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( ( abs `  1
)  /  ( abs `  A ) ) )
4137div1i 9578 . . . . . 6  |-  ( 1  /  1 )  =  1
42 abs1 11878 . . . . . . . 8  |-  ( abs `  1 )  =  1
4342eqcomi 2362 . . . . . . 7  |-  1  =  ( abs `  1
)
4443oveq1i 5955 . . . . . 6  |-  ( 1  /  ( abs `  A
) )  =  ( ( abs `  1
)  /  ( abs `  A ) )
4540, 41, 443brtr4g 4136 . . . . 5  |-  ( A  e.  (Unit `  Z
)  ->  ( 1  /  1 )  <_ 
( 1  /  ( abs `  A ) ) )
46 lerec 9728 . . . . . 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 8997 . . . . 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 8896 . . . . . . 7  |-  1  =/=  0
5655a1i 10 . . . . . 6  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  1  =/=  0 )
5754, 56eqnetrd 2539 . . . . 5  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  ( abs `  A )  =/=  0 )
58 fveq2 5608 . . . . . . 7  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
59 abs0 11866 . . . . . . 7  |-  ( abs `  0 )  =  0
6058, 59syl6eq 2406 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
6160necon3i 2560 . . . . 5  |-  ( ( abs `  A )  =/=  0  ->  A  =/=  0 )
6257, 61syl 15 . . . 4  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  A  =/=  0 )
63 eldifsn 3825 . . . 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 12020 . . . . . . 7  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
6854oveq1d 5960 . . . . . . . 8  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  ( 1 ^ 2 ) )
69 sq1 11291 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
7068, 69syl6eq 2406 . . . . . . 7  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  1 )
7167, 70eqtr3d 2392 . . . . . 6  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  ( A  x.  ( * `  A ) )  =  1 )
7271oveq1d 5960 . . . . 5  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( A  x.  (
* `  A )
)  /  A )  =  ( 1  /  A ) )
7353cjcld 11777 . . . . . 6  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
* `  A )  e.  CC )
7473, 53, 62divcan3d 9631 . . . . 5  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( A  x.  (
* `  A )
)  /  A )  =  ( * `  A ) )
7566, 72, 743eqtr2d 2396 . . . 4  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  =  ( * `  A ) )
76 gzcjcl 13080 . . . . 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 2432 . . 3  |-  ( ( A  e.  ZZ [
_i ]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  e.  ZZ [ _i ] )
79 cnfldbas 16486 . . . . . 6  |-  CC  =  ( Base ` fld )
80 cnfld0 16504 . . . . . 6  |-  0  =  ( 0g ` fld )
81 cndrng 16509 . . . . . 6  |-fld  e.  DivRing
8279, 80, 81drngui 15617 . . . . 5  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
832, 82, 5, 7subrgunit 15662 . . . 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 1642    e. wcel 1710    =/= wne 2521    \ cdif 3225   {csn 3716   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   CCcc 8825   RRcr 8826   0cc0 8827   1c1 8828    x. cmul 8832    < clt 8957    <_ cle 8958    / cdiv 9513   2c2 9885   ^cexp 11197   *ccj 11677   abscabs 11815   ZZ [ _i ]cgz 13073   Basecbs 13245   ↾s cress 13246   Ringcrg 15436  Unitcui 15520   invrcinvr 15552  SubRingcsubrg 15640  ℂfldccnfld 16482
This theorem is referenced by:  zrngunit  16544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-tpos 6321  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-rp 10447  df-fz 10875  df-seq 11139  df-exp 11198  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-gz 13074  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-0g 13503  df-mnd 14466  df-grp 14588  df-minusg 14589  df-subg 14717  df-cmn 15190  df-mgp 15425  df-rng 15439  df-cring 15440  df-ur 15441  df-oppr 15504  df-dvdsr 15522  df-unit 15523  df-invr 15553  df-dvr 15564  df-drng 15613  df-subrg 15642  df-cnfld 16483
  Copyright terms: Public domain W3C validator