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Theorem zringunit 17919
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 17894 . . . 4  |-  ZZ  =  ( Base ` ring )
2 eqid 2443 . . . 4  |-  (Unit ` ring )  =  (Unit ` ring )
31, 2unitcl 16756 . . 3  |-  ( A  e.  (Unit ` ring )  ->  A  e.  ZZ )
4 zsubrg 17871 . . . . . . 7  |-  ZZ  e.  (SubRing ` fld )
5 zgz 13999 . . . . . . . 8  |-  ( x  e.  ZZ  ->  x  e.  ZZ[_i]
)
65ssriv 3365 . . . . . . 7  |-  ZZ  C_  ZZ[_i]
7 gzsubrg 17872 . . . . . . . 8  |-  ZZ[_i]  e.  (SubRing ` fld )
8 eqid 2443 . . . . . . . . 9  |-  (flds  ZZ[_i]
)  =  (flds  ZZ[_i]
)
98subsubrg 16896 . . . . . . . 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 911 . . . . . 6  |-  ZZ  e.  (SubRing `  (flds  ZZ[_i] ) )
12 df-zring 17889 . . . . . . . 8  |-ring  =  (flds  ZZ )
13 ressabs 14241 . . . . . . . . 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 2466 . . . . . . 7  |-ring  =  ( (flds  ZZ[_i] )s  ZZ )
16 eqid 2443 . . . . . . 7  |-  (Unit `  (flds  ZZ[_i] ) )  =  (Unit `  (flds  ZZ[_i] ) )
1715, 16, 2subrguss 16885 . . . . . 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 3357 . . . 4  |-  ( A  e.  (Unit ` ring )  ->  A  e.  (Unit `  (flds  ZZ[_i]
) ) )
208gzrngunit 17883 . . . . 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 10656 . . . . 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 9356 . . . . . . 7  |-  1  =/=  0
2827a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
1  =/=  0 )
2926, 28eqnetrd 2631 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( abs `  A
)  =/=  0 )
30 fveq2 5696 . . . . . . 7  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
31 abs0 12779 . . . . . . 7  |-  ( abs `  0 )  =  0
3230, 31syl6eq 2491 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
3332necon3i 2655 . . . . 5  |-  ( ( abs `  A )  =/=  0  ->  A  =/=  0 )
3429, 33syl 16 . . . 4  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  =/=  0 )
35 eldifsn 4005 . . . 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 17852 . . . . . 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 10655 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  RR )
4140adantr 465 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  e.  RR )
42 absresq 12796 . . . . . . . 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 6111 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( abs `  A
) ^ 2 )  =  ( 1 ^ 2 ) )
45 sq1 11965 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
4644, 45syl6eq 2491 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( abs `  A
) ^ 2 )  =  1 )
4725sqvald 12010 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( A ^ 2 )  =  ( A  x.  A ) )
4843, 46, 473eqtr3rd 2484 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( A  x.  A
)  =  1 )
49 1cnd 9407 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
1  e.  CC )
5049, 25, 25, 34divmuld 10134 . . . . . 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 2475 . . . 4  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( invr ` fld ) `  A )  =  A )
5352, 37eqeltrd 2517 . . 3  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( invr ` fld ) `  A )  e.  ZZ )
54 cnfldbas 17827 . . . . . 6  |-  CC  =  ( Base ` fld )
55 cnfld0 17845 . . . . . 6  |-  0  =  ( 0g ` fld )
56 cndrng 17850 . . . . . 6  |-fld  e.  DivRing
5754, 55, 56drngui 16843 . . . . 5  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
58 eqid 2443 . . . . 5  |-  ( invr ` fld )  =  ( invr ` fld )
5912, 57, 2, 58subrgunit 16888 . . . 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 1172 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611    \ cdif 3330    C_ wss 3333   {csn 3882   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    x. cmul 9292    / cdiv 9998   2c2 10376   ZZcz 10651   ^cexp 11870   abscabs 12728   ZZ[_i]cgz 13995   ↾s cress 14180  Unitcui 16736   invrcinvr 16768  SubRingcsubrg 16866  ℂfldccnfld 17823  ℤringzring 17888
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-tpos 6750  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-rp 10997  df-fz 11443  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-gz 13996  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-0g 14385  df-mnd 15420  df-grp 15550  df-minusg 15551  df-subg 15683  df-cmn 16284  df-mgp 16597  df-ur 16609  df-rng 16652  df-cring 16653  df-oppr 16720  df-dvdsr 16738  df-unit 16739  df-invr 16769  df-dvr 16780  df-drng 16839  df-subrg 16868  df-cnfld 17824  df-zring 17889
This theorem is referenced by:  prmirredlem  17922  qqhval2lem  26415
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