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Theorem zringunit 18284
Description: The units of  ZZ are the integers with norm  1, i.e.  1 and  -u 1. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
Assertion
Ref Expression
zringunit  |-  ( A  e.  (Unit ` ring )  <->  ( A  e.  ZZ  /\  ( abs `  A )  =  1 ) )

Proof of Theorem zringunit
StepHypRef Expression
1 zringbas 18259 . . . 4  |-  ZZ  =  ( Base ` ring )
2 eqid 2467 . . . 4  |-  (Unit ` ring )  =  (Unit ` ring )
31, 2unitcl 17089 . . 3  |-  ( A  e.  (Unit ` ring )  ->  A  e.  ZZ )
4 zsubrg 18236 . . . . . . 7  |-  ZZ  e.  (SubRing ` fld )
5 zgz 14303 . . . . . . . 8  |-  ( x  e.  ZZ  ->  x  e.  ZZ[_i]
)
65ssriv 3508 . . . . . . 7  |-  ZZ  C_  ZZ[_i]
7 gzsubrg 18237 . . . . . . . 8  |-  ZZ[_i]  e.  (SubRing ` fld )
8 eqid 2467 . . . . . . . . 9  |-  (flds  ZZ[_i]
)  =  (flds  ZZ[_i]
)
98subsubrg 17235 . . . . . . . 8  |-  ( ZZ[_i]  e.  (SubRing ` fld )  ->  ( ZZ  e.  (SubRing `  (flds  ZZ[_i]
) )  <->  ( ZZ  e.  (SubRing ` fld )  /\  ZZ  C_  ZZ[_i] ) ) )
107, 9ax-mp 5 . . . . . . 7  |-  ( ZZ  e.  (SubRing `  (flds  ZZ[_i] ) )  <->  ( ZZ  e.  (SubRing ` fld )  /\  ZZ  C_  ZZ[_i] ) )
114, 6, 10mpbir2an 918 . . . . . 6  |-  ZZ  e.  (SubRing `  (flds  ZZ[_i] ) )
12 df-zring 18254 . . . . . . . 8  |-ring  =  (flds  ZZ )
13 ressabs 14546 . . . . . . . . 9  |-  ( ( ZZ[_i] 
e.  (SubRing ` fld )  /\  ZZ  C_  ZZ[_i] )  ->  ( (flds  ZZ[_i]
)s 
ZZ )  =  (flds  ZZ ) )
147, 6, 13mp2an 672 . . . . . . . 8  |-  ( (flds  ZZ[_i] )s  ZZ )  =  (flds  ZZ )
1512, 14eqtr4i 2499 . . . . . . 7  |-ring  =  ( (flds  ZZ[_i] )s  ZZ )
16 eqid 2467 . . . . . . 7  |-  (Unit `  (flds  ZZ[_i] ) )  =  (Unit `  (flds  ZZ[_i] ) )
1715, 16, 2subrguss 17224 . . . . . 6  |-  ( ZZ  e.  (SubRing `  (flds  ZZ[_i] ) )  -> 
(Unit ` ring )  C_  (Unit `  (flds  ZZ[_i] ) ) )
1811, 17ax-mp 5 . . . . 5  |-  (Unit ` ring )  C_  (Unit `  (flds  ZZ[_i]
) )
1918sseli 3500 . . . 4  |-  ( A  e.  (Unit ` ring )  ->  A  e.  (Unit `  (flds  ZZ[_i]
) ) )
208gzrngunit 18248 . . . . 5  |-  ( A  e.  (Unit `  (flds  ZZ[_i] ) )  <->  ( A  e.  ZZ[_i]  /\  ( abs `  A
)  =  1 ) )
2120simprbi 464 . . . 4  |-  ( A  e.  (Unit `  (flds  ZZ[_i] ) )  -> 
( abs `  A
)  =  1 )
2219, 21syl 16 . . 3  |-  ( A  e.  (Unit ` ring )  ->  ( abs `  A )  =  1 )
233, 22jca 532 . 2  |-  ( A  e.  (Unit ` ring )  ->  ( A  e.  ZZ  /\  ( abs `  A )  =  1 ) )
24 zcn 10865 . . . . 5  |-  ( A  e.  ZZ  ->  A  e.  CC )
2524adantr 465 . . . 4  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  e.  CC )
26 simpr 461 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( abs `  A
)  =  1 )
27 ax-1ne0 9557 . . . . . . 7  |-  1  =/=  0
2827a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
1  =/=  0 )
2926, 28eqnetrd 2760 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( abs `  A
)  =/=  0 )
30 fveq2 5864 . . . . . . 7  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
31 abs0 13075 . . . . . . 7  |-  ( abs `  0 )  =  0
3230, 31syl6eq 2524 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
3332necon3i 2707 . . . . 5  |-  ( ( abs `  A )  =/=  0  ->  A  =/=  0 )
3429, 33syl 16 . . . 4  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  =/=  0 )
35 eldifsn 4152 . . . 4  |-  ( A  e.  ( CC  \  { 0 } )  <-> 
( A  e.  CC  /\  A  =/=  0 ) )
3625, 34, 35sylanbrc 664 . . 3  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  e.  ( CC  \  { 0 } ) )
37 simpl 457 . . 3  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  e.  ZZ )
38 cnfldinv 18217 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( invr ` fld ) `  A )  =  ( 1  /  A ) )
3925, 34, 38syl2anc 661 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( invr ` fld ) `  A )  =  ( 1  /  A ) )
40 zre 10864 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  RR )
4140adantr 465 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  e.  RR )
42 absresq 13092 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( abs `  A
) ^ 2 )  =  ( A ^
2 ) )
4341, 42syl 16 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( abs `  A
) ^ 2 )  =  ( A ^
2 ) )
4426oveq1d 6297 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( abs `  A
) ^ 2 )  =  ( 1 ^ 2 ) )
45 sq1 12224 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
4644, 45syl6eq 2524 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( abs `  A
) ^ 2 )  =  1 )
4725sqvald 12269 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( A ^ 2 )  =  ( A  x.  A ) )
4843, 46, 473eqtr3rd 2517 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( A  x.  A
)  =  1 )
49 1cnd 9608 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
1  e.  CC )
5049, 25, 25, 34divmuld 10338 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( 1  /  A )  =  A  <-> 
( A  x.  A
)  =  1 ) )
5148, 50mpbird 232 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( 1  /  A
)  =  A )
5239, 51eqtrd 2508 . . . 4  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( invr ` fld ) `  A )  =  A )
5352, 37eqeltrd 2555 . . 3  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( invr ` fld ) `  A )  e.  ZZ )
54 cnfldbas 18192 . . . . . 6  |-  CC  =  ( Base ` fld )
55 cnfld0 18210 . . . . . 6  |-  0  =  ( 0g ` fld )
56 cndrng 18215 . . . . . 6  |-fld  e.  DivRing
5754, 55, 56drngui 17182 . . . . 5  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
58 eqid 2467 . . . . 5  |-  ( invr ` fld )  =  ( invr ` fld )
5912, 57, 2, 58subrgunit 17227 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  ( A  e.  (Unit ` ring )  <->  ( A  e.  ( CC  \  {
0 } )  /\  A  e.  ZZ  /\  (
( invr ` fld ) `  A )  e.  ZZ ) ) )
604, 59ax-mp 5 . . 3  |-  ( A  e.  (Unit ` ring )  <->  ( A  e.  ( CC  \  {
0 } )  /\  A  e.  ZZ  /\  (
( invr ` fld ) `  A )  e.  ZZ ) )
6136, 37, 53, 60syl3anbrc 1180 . 2  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  e.  (Unit ` ring ) )
6223, 61impbii 188 1  |-  ( A  e.  (Unit ` ring )  <->  ( A  e.  ZZ  /\  ( abs `  A )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473    C_ wss 3476   {csn 4027   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    x. cmul 9493    / cdiv 10202   2c2 10581   ZZcz 10860   ^cexp 12129   abscabs 13024   ZZ[_i]cgz 14299   ↾s cress 14484  Unitcui 17069   invrcinvr 17101  SubRingcsubrg 17205  ℂfldccnfld 18188  ℤringzring 18253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-rp 11217  df-fz 11669  df-seq 12071  df-exp 12130  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-gz 14300  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-0g 14690  df-mnd 15725  df-grp 15855  df-minusg 15856  df-subg 15990  df-cmn 16593  df-mgp 16929  df-ur 16941  df-rng 16985  df-cring 16986  df-oppr 17053  df-dvdsr 17071  df-unit 17072  df-invr 17102  df-dvr 17113  df-drng 17178  df-subrg 17207  df-cnfld 18189  df-zring 18254
This theorem is referenced by:  prmirredlem  18287  qqhval2lem  27595
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