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Theorem zrngunit 18283
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 18282 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 18234 . . . . 5  |-  ZZ  e.  (SubRing ` fld )
2 zrngunit.1 . . . . . 6  |-  Z  =  (flds  ZZ )
32subrgbas 17216 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
41, 3ax-mp 5 . . . 4  |-  ZZ  =  ( Base `  Z )
5 eqid 2462 . . . 4  |-  (Unit `  Z )  =  (Unit `  Z )
64, 5unitcl 17087 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  ZZ )
7 zgz 14301 . . . . . . . 8  |-  ( x  e.  ZZ  ->  x  e.  ZZ[_i]
)
87ssriv 3503 . . . . . . 7  |-  ZZ  C_  ZZ[_i]
9 gzsubrg 18235 . . . . . . . 8  |-  ZZ[_i]  e.  (SubRing ` fld )
10 eqid 2462 . . . . . . . . 9  |-  (flds  ZZ[_i]
)  =  (flds  ZZ[_i]
)
1110subsubrg 17233 . . . . . . . 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 913 . . . . . 6  |-  ZZ  e.  (SubRing `  (flds  ZZ[_i] ) )
14 ressabs 14544 . . . . . . . . 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 2494 . . . . . . 7  |-  Z  =  ( (flds  ZZ[_i] )s  ZZ )
17 eqid 2462 . . . . . . 7  |-  (Unit `  (flds  ZZ[_i] ) )  =  (Unit `  (flds  ZZ[_i] ) )
1816, 17, 5subrguss 17222 . . . . . 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 3495 . . . 4  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  (Unit `  (flds  ZZ[_i] ) ) )
2110gzrngunit 18246 . . . . 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 10860 . . . . 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 9552 . . . . . . 7  |-  1  =/=  0
2928a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
1  =/=  0 )
3027, 29eqnetrd 2755 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( abs `  A
)  =/=  0 )
31 fveq2 5859 . . . . . . 7  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
32 abs0 13070 . . . . . . 7  |-  ( abs `  0 )  =  0
3331, 32syl6eq 2519 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
3433necon3i 2702 . . . . 5  |-  ( ( abs `  A )  =/=  0  ->  A  =/=  0 )
3530, 34syl 16 . . . 4  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  =/=  0 )
36 eldifsn 4147 . . . 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 18215 . . . . . 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 10859 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  RR )
4241adantr 465 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  ->  A  e.  RR )
43 absresq 13087 . . . . . . . 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 6292 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( abs `  A
) ^ 2 )  =  ( 1 ^ 2 ) )
46 sq1 12219 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
4745, 46syl6eq 2519 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( abs `  A
) ^ 2 )  =  1 )
4826sqvald 12264 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( A ^ 2 )  =  ( A  x.  A ) )
4944, 47, 483eqtr3rd 2512 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( A  x.  A
)  =  1 )
50 ax-1cn 9541 . . . . . . . 8  |-  1  e.  CC
5150a1i 11 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
1  e.  CC )
5251, 26, 26, 35divmuld 10333 . . . . . 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 2503 . . . 4  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( invr ` fld ) `  A )  =  A )
5554, 38eqeltrd 2550 . . 3  |-  ( ( A  e.  ZZ  /\  ( abs `  A )  =  1 )  -> 
( ( invr ` fld ) `  A )  e.  ZZ )
56 cnfldbas 18190 . . . . . 6  |-  CC  =  ( Base ` fld )
57 cnfld0 18208 . . . . . 6  |-  0  =  ( 0g ` fld )
58 cndrng 18213 . . . . . 6  |-fld  e.  DivRing
5956, 57, 58drngui 17180 . . . . 5  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
60 eqid 2462 . . . . 5  |-  ( invr ` fld )  =  ( invr ` fld )
612, 59, 5, 60subrgunit 17225 . . . 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 1175 . 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 968    = wceq 1374    e. wcel 1762    =/= wne 2657    \ cdif 3468    C_ wss 3471   {csn 4022   ` cfv 5581  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    x. cmul 9488    / cdiv 10197   2c2 10576   ZZcz 10855   ^cexp 12124   abscabs 13019   ZZ[_i]cgz 14297   Basecbs 14481   ↾s cress 14482  Unitcui 17067   invrcinvr 17099  SubRingcsubrg 17203  ℂfldccnfld 18186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-rp 11212  df-fz 11664  df-seq 12066  df-exp 12125  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-gz 14298  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-0g 14688  df-mnd 15723  df-grp 15853  df-minusg 15854  df-subg 15988  df-cmn 16591  df-mgp 16927  df-ur 16939  df-rng 16983  df-cring 16984  df-oppr 17051  df-dvdsr 17069  df-unit 17070  df-invr 17100  df-dvr 17111  df-drng 17176  df-subrg 17205  df-cnfld 18187
This theorem is referenced by:  prmirredlemOLD  18288
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