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Theorem zrngunit 17914
Description: The units of  ZZ are the integers with norm  1, i.e.  1 and  -u 1. (Contributed by Mario Carneiro, 5-Dec-2014.) Obsolete version of zringunit 17913 as of 9-Jun-2019. (New usage is discouraged.)
Hypothesis
Ref Expression
zrngunit.1  |-  Z  =  (flds  ZZ )
Assertion
Ref Expression
zrngunit  |-  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ZZ  /\  ( abs `  A )  =  1 ) )

Proof of Theorem zrngunit
StepHypRef Expression
1 zsubrg 17865 . . . . 5  |-  ZZ  e.  (SubRing ` fld )
2 zrngunit.1 . . . . . 6  |-  Z  =  (flds  ZZ )
32subrgbas 16873 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
41, 3ax-mp 5 . . . 4  |-  ZZ  =  ( Base `  Z )
5 eqid 2442 . . . 4  |-  (Unit `  Z )  =  (Unit `  Z )
64, 5unitcl 16750 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  ZZ )
7 zgz 13993 . . . . . . . 8  |-  ( x  e.  ZZ  ->  x  e.  ZZ[_i]
)
87ssriv 3359 . . . . . . 7  |-  ZZ  C_  ZZ[_i]
9 gzsubrg 17866 . . . . . . . 8  |-  ZZ[_i]  e.  (SubRing ` fld )
10 eqid 2442 . . . . . . . . 9  |-  (flds  ZZ[_i]
)  =  (flds  ZZ[_i]
)
1110subsubrg 16890 . . . . . . . 8  |-  ( ZZ[_i]  e.  (SubRing ` fld )  ->  ( ZZ  e.  (SubRing `  (flds  ZZ[_i]
) )  <->  ( ZZ  e.  (SubRing ` fld )  /\  ZZ  C_  ZZ[_i] ) ) )
129, 11ax-mp 5 . . . . . . 7  |-  ( ZZ  e.  (SubRing `  (flds  ZZ[_i] ) )  <->  ( ZZ  e.  (SubRing ` fld )  /\  ZZ  C_  ZZ[_i] ) )
131, 8, 12mpbir2an 911 . . . . . 6  |-  ZZ  e.  (SubRing `  (flds  ZZ[_i] ) )
14 ressabs 14235 . . . . . . . . 9  |-  ( ( ZZ[_i] 
e.  (SubRing ` fld )  /\  ZZ  C_  ZZ[_i] )  ->  ( (flds  ZZ[_i]
)s 
ZZ )  =  (flds  ZZ ) )
159, 8, 14mp2an 672 . . . . . . . 8  |-  ( (flds  ZZ[_i] )s  ZZ )  =  (flds  ZZ )
162, 15eqtr4i 2465 . . . . . . 7  |-  Z  =  ( (flds  ZZ[_i] )s  ZZ )
17 eqid 2442 . . . . . . 7  |-  (Unit `  (flds  ZZ[_i] ) )  =  (Unit `  (flds  ZZ[_i] ) )
1816, 17, 5subrguss 16879 . . . . . 6  |-  ( ZZ  e.  (SubRing `  (flds  ZZ[_i] ) )  -> 
(Unit `  Z )  C_  (Unit `  (flds  ZZ[_i]
) ) )
1913, 18ax-mp 5 . . . . 5  |-  (Unit `  Z )  C_  (Unit `  (flds  ZZ[_i] ) )
2019sseli 3351 . . . 4  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  (Unit `  (flds  ZZ[_i] ) ) )
2110gzrngunit 17877 . . . . 5  |-  ( A  e.  (Unit `  (flds  ZZ[_i] ) )  <->  ( A  e.  ZZ[_i]  /\  ( abs `  A
)  =  1 ) )
2221simprbi 464 . . . 4  |-  ( A  e.  (Unit `  (flds  ZZ[_i] ) )  -> 
( abs `  A
)  =  1 )
2320, 22syl 16 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  =  1 )
246, 23jca 532 . 2  |-  ( A  e.  (Unit `  Z
)  ->  ( A  e.  ZZ  /\  ( abs `  A )  =  1 ) )
25 zcn 10650 . . . . 5  |-  ( A  e.  ZZ  ->  A  e.  CC )
2625adantr 465 . . . 4  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  e.  CC )
27 simpr 461 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( abs `  A
)  =  1 )
28 ax-1ne0 9350 . . . . . . 7  |-  1  =/=  0
2928a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
1  =/=  0 )
3027, 29eqnetrd 2625 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( abs `  A
)  =/=  0 )
31 fveq2 5690 . . . . . . 7  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
32 abs0 12773 . . . . . . 7  |-  ( abs `  0 )  =  0
3331, 32syl6eq 2490 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
3433necon3i 2649 . . . . 5  |-  ( ( abs `  A )  =/=  0  ->  A  =/=  0 )
3530, 34syl 16 . . . 4  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  =/=  0 )
36 eldifsn 3999 . . . 4  |-  ( A  e.  ( CC  \  { 0 } )  <-> 
( A  e.  CC  /\  A  =/=  0 ) )
3726, 35, 36sylanbrc 664 . . 3  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  e.  ( CC  \  { 0 } ) )
38 simpl 457 . . 3  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  e.  ZZ )
39 cnfldinv 17846 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( invr ` fld ) `  A )  =  ( 1  /  A ) )
4026, 35, 39syl2anc 661 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( invr ` fld ) `  A )  =  ( 1  /  A ) )
41 zre 10649 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  RR )
4241adantr 465 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  e.  RR )
43 absresq 12790 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( abs `  A
) ^ 2 )  =  ( A ^
2 ) )
4442, 43syl 16 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( abs `  A
) ^ 2 )  =  ( A ^
2 ) )
4527oveq1d 6105 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( abs `  A
) ^ 2 )  =  ( 1 ^ 2 ) )
46 sq1 11959 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
4745, 46syl6eq 2490 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( abs `  A
) ^ 2 )  =  1 )
4826sqvald 12004 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( A ^ 2 )  =  ( A  x.  A ) )
4944, 47, 483eqtr3rd 2483 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( A  x.  A
)  =  1 )
50 ax-1cn 9339 . . . . . . . 8  |-  1  e.  CC
5150a1i 11 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
1  e.  CC )
5251, 26, 26, 35divmuld 10128 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( 1  /  A )  =  A  <-> 
( A  x.  A
)  =  1 ) )
5349, 52mpbird 232 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( 1  /  A
)  =  A )
5440, 53eqtrd 2474 . . . 4  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( invr ` fld ) `  A )  =  A )
5554, 38eqeltrd 2516 . . 3  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( invr ` fld ) `  A )  e.  ZZ )
56 cnfldbas 17821 . . . . . 6  |-  CC  =  ( Base ` fld )
57 cnfld0 17839 . . . . . 6  |-  0  =  ( 0g ` fld )
58 cndrng 17844 . . . . . 6  |-fld  e.  DivRing
5956, 57, 58drngui 16837 . . . . 5  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
60 eqid 2442 . . . . 5  |-  ( invr ` fld )  =  ( invr ` fld )
612, 59, 5, 60subrgunit 16882 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ( CC  \  {
0 } )  /\  A  e.  ZZ  /\  (
( invr ` fld ) `  A )  e.  ZZ ) ) )
621, 61ax-mp 5 . . 3  |-  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ( CC  \  {
0 } )  /\  A  e.  ZZ  /\  (
( invr ` fld ) `  A )  e.  ZZ ) )
6337, 38, 55, 62syl3anbrc 1172 . 2  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  e.  (Unit `  Z
) )
6424, 63impbii 188 1  |-  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ZZ  /\  ( abs `  A )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2605    \ cdif 3324    C_ wss 3327   {csn 3876   ` cfv 5417  (class class class)co 6090   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282    x. cmul 9286    / cdiv 9992   2c2 10370   ZZcz 10645   ^cexp 11864   abscabs 12722   ZZ[_i]cgz 13989   Basecbs 14173   ↾s cress 14174  Unitcui 16730   invrcinvr 16762  SubRingcsubrg 16860  ℂfldccnfld 17817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-rp 10991  df-fz 11437  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-gz 13990  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-0g 14379  df-mnd 15414  df-grp 15544  df-minusg 15545  df-subg 15677  df-cmn 16278  df-mgp 16591  df-ur 16603  df-rng 16646  df-cring 16647  df-oppr 16714  df-dvdsr 16732  df-unit 16733  df-invr 16763  df-dvr 16774  df-drng 16833  df-subrg 16862  df-cnfld 17818
This theorem is referenced by:  prmirredlemOLD  17919
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