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Theorem gzrngunit 18610
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 18599 . . . . 5  |-  ZZ[_i]  e.  (SubRing ` fld )
2 gzrng.1 . . . . . 6  |-  Z  =  (flds  ZZ[_i] )
32subrgbas 17565 . . . . 5  |-  ( ZZ[_i]  e.  (SubRing ` fld )  ->  ZZ[_i]  =  ( Base `  Z ) )
41, 3ax-mp 5 . . . 4  |-  ZZ[_i]  =  ( Base `  Z )
5 eqid 2457 . . . 4  |-  (Unit `  Z )  =  (Unit `  Z )
64, 5unitcl 17435 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  ZZ[_i] )
7 eqid 2457 . . . . . . . . . . . 12  |-  ( invr ` fld )  =  ( invr ` fld )
8 eqid 2457 . . . . . . . . . . . 12  |-  ( invr `  Z )  =  (
invr `  Z )
92, 7, 5, 8subrginv 17572 . . . . . . . . . . 11  |-  ( ( ZZ[_i] 
e.  (SubRing ` fld )  /\  A  e.  (Unit `  Z )
)  ->  ( ( invr ` fld ) `  A )  =  ( ( invr `  Z ) `  A
) )
101, 9mpan 670 . . . . . . . . . 10  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr ` fld ) `  A )  =  ( ( invr `  Z ) `  A
) )
11 gzcn 14462 . . . . . . . . . . . 12  |-  ( A  e.  ZZ[_i]  ->  A  e.  CC )
126, 11syl 16 . . . . . . . . . . 11  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  CC )
13 0red 9614 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  0  e.  RR )
14 1re 9612 . . . . . . . . . . . . . . 15  |-  1  e.  RR
1514a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  1  e.  RR )
1612abscld 13279 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  e.  RR )
17 0lt1 10096 . . . . . . . . . . . . . . 15  |-  0  <  1
1817a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  0  <  1 )
192gzrngunitlem 18609 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( abs `  A ) )
2013, 15, 16, 18, 19ltletrd 9759 . . . . . . . . . . . . 13  |-  ( A  e.  (Unit `  Z
)  ->  0  <  ( abs `  A ) )
2120gt0ne0d 10138 . . . . . . . . . . . 12  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  =/=  0
)
2212abs00ad 13135 . . . . . . . . . . . . 13  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =  0  <->  A  =  0
) )
2322necon3bid 2715 . . . . . . . . . . . 12  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =/=  0  <->  A  =/=  0
) )
2421, 23mpbid 210 . . . . . . . . . . 11  |-  ( A  e.  (Unit `  Z
)  ->  A  =/=  0 )
25 cnfldinv 18576 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( invr ` fld ) `  A )  =  ( 1  /  A ) )
2612, 24, 25syl2anc 661 . . . . . . . . . 10  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr ` fld ) `  A )  =  ( 1  /  A ) )
2710, 26eqtr3d 2500 . . . . . . . . 9  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr `  Z ) `  A )  =  ( 1  /  A ) )
282subrgring 17559 . . . . . . . . . . 11  |-  ( ZZ[_i]  e.  (SubRing ` fld )  ->  Z  e.  Ring )
291, 28ax-mp 5 . . . . . . . . . 10  |-  Z  e. 
Ring
305, 8unitinvcl 17450 . . . . . . . . . 10  |-  ( ( Z  e.  Ring  /\  A  e.  (Unit `  Z )
)  ->  ( ( invr `  Z ) `  A )  e.  (Unit `  Z ) )
3129, 30mpan 670 . . . . . . . . 9  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr `  Z ) `  A )  e.  (Unit `  Z ) )
3227, 31eqeltrrd 2546 . . . . . . . 8  |-  ( A  e.  (Unit `  Z
)  ->  ( 1  /  A )  e.  (Unit `  Z )
)
332gzrngunitlem 18609 . . . . . . . 8  |-  ( ( 1  /  A )  e.  (Unit `  Z
)  ->  1  <_  ( abs `  ( 1  /  A ) ) )
3432, 33syl 16 . . . . . . 7  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( abs `  ( 1  /  A ) ) )
35 1cnd 9629 . . . . . . . 8  |-  ( A  e.  (Unit `  Z
)  ->  1  e.  CC )
3635, 12, 24absdivd 13298 . . . . . . 7  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  ( 1  /  A
) )  =  ( ( abs `  1
)  /  ( abs `  A ) ) )
3734, 36breqtrd 4480 . . . . . 6  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( ( abs `  1
)  /  ( abs `  A ) ) )
38 1div1e1 10258 . . . . . 6  |-  ( 1  /  1 )  =  1
39 abs1 13142 . . . . . . . 8  |-  ( abs `  1 )  =  1
4039eqcomi 2470 . . . . . . 7  |-  1  =  ( abs `  1
)
4140oveq1i 6306 . . . . . 6  |-  ( 1  /  ( abs `  A
) )  =  ( ( abs `  1
)  /  ( abs `  A ) )
4237, 38, 413brtr4g 4488 . . . . 5  |-  ( A  e.  (Unit `  Z
)  ->  ( 1  /  1 )  <_ 
( 1  /  ( abs `  A ) ) )
43 lerec 10447 . . . . . 6  |-  ( ( ( ( abs `  A
)  e.  RR  /\  0  <  ( abs `  A
) )  /\  (
1  e.  RR  /\  0  <  1 ) )  ->  ( ( abs `  A )  <_  1  <->  ( 1  /  1 )  <_  ( 1  / 
( abs `  A
) ) ) )
4416, 20, 15, 18, 43syl22anc 1229 . . . . 5  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  <_ 
1  <->  ( 1  / 
1 )  <_  (
1  /  ( abs `  A ) ) ) )
4542, 44mpbird 232 . . . 4  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  <_  1
)
46 letri3 9687 . . . . 5  |-  ( ( ( abs `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( abs `  A
)  =  1  <->  (
( abs `  A
)  <_  1  /\  1  <_  ( abs `  A
) ) ) )
4716, 14, 46sylancl 662 . . . 4  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =  1  <->  ( ( abs `  A )  <_  1  /\  1  <_  ( abs `  A ) ) ) )
4845, 19, 47mpbir2and 922 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  =  1 )
496, 48jca 532 . 2  |-  ( A  e.  (Unit `  Z
)  ->  ( A  e.  ZZ[_i]  /\  ( abs `  A
)  =  1 ) )
5011adantr 465 . . . 4  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  e.  CC )
51 simpr 461 . . . . . 6  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  ( abs `  A )  =  1 )
52 ax-1ne0 9578 . . . . . . 7  |-  1  =/=  0
5352a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  1  =/=  0 )
5451, 53eqnetrd 2750 . . . . 5  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  ( abs `  A )  =/=  0 )
55 fveq2 5872 . . . . . . 7  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
56 abs0 13130 . . . . . . 7  |-  ( abs `  0 )  =  0
5755, 56syl6eq 2514 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
5857necon3i 2697 . . . . 5  |-  ( ( abs `  A )  =/=  0  ->  A  =/=  0 )
5954, 58syl 16 . . . 4  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  =/=  0 )
60 eldifsn 4157 . . . 4  |-  ( A  e.  ( CC  \  { 0 } )  <-> 
( A  e.  CC  /\  A  =/=  0 ) )
6150, 59, 60sylanbrc 664 . . 3  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  e.  ( CC  \  {
0 } ) )
62 simpl 457 . . 3  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  e.  ZZ[_i]
)
6350, 59, 25syl2anc 661 . . . . 5  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  =  ( 1  /  A ) )
6450absvalsqd 13285 . . . . . . 7  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
6551oveq1d 6311 . . . . . . . 8  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  ( 1 ^ 2 ) )
66 sq1 12265 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
6765, 66syl6eq 2514 . . . . . . 7  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  1 )
6864, 67eqtr3d 2500 . . . . . 6  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  ( A  x.  ( * `  A ) )  =  1 )
6968oveq1d 6311 . . . . 5  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( A  x.  (
* `  A )
)  /  A )  =  ( 1  /  A ) )
7050cjcld 13041 . . . . . 6  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
* `  A )  e.  CC )
7170, 50, 59divcan3d 10346 . . . . 5  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( A  x.  (
* `  A )
)  /  A )  =  ( * `  A ) )
7263, 69, 713eqtr2d 2504 . . . 4  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  =  ( * `  A ) )
73 gzcjcl 14466 . . . . 5  |-  ( A  e.  ZZ[_i]  ->  ( * `  A )  e.  ZZ[_i] )
7473adantr 465 . . . 4  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
* `  A )  e.  ZZ[_i]
)
7572, 74eqeltrd 2545 . . 3  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  e.  ZZ[_i] )
76 cnfldbas 18551 . . . . . 6  |-  CC  =  ( Base ` fld )
77 cnfld0 18569 . . . . . 6  |-  0  =  ( 0g ` fld )
78 cndrng 18574 . . . . . 6  |-fld  e.  DivRing
7976, 77, 78drngui 17529 . . . . 5  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
802, 79, 5, 7subrgunit 17574 . . . 4  |-  ( ZZ[_i]  e.  (SubRing ` fld )  ->  ( A  e.  (Unit `  Z )  <->  ( A  e.  ( CC 
\  { 0 } )  /\  A  e.  ZZ[_i]  /\  ( ( invr ` fld ) `  A )  e.  ZZ[_i] ) ) )
811, 80ax-mp 5 . . 3  |-  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ( CC  \  {
0 } )  /\  A  e.  ZZ[_i]  /\  ( (
invr ` fld ) `  A )  e.  ZZ[_i] ) )
8261, 62, 75, 81syl3anbrc 1180 . 2  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  e.  (Unit `  Z )
)
8349, 82impbii 188 1  |-  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ZZ[_i]  /\  ( abs `  A
)  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652    \ cdif 3468   {csn 4032   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514    < clt 9645    <_ cle 9646    / cdiv 10227   2c2 10606   ^cexp 12169   *ccj 12941   abscabs 13079   ZZ[_i]cgz 14459   Basecbs 14644   ↾s cress 14645   Ringcrg 17325  Unitcui 17415   invrcinvr 17447  SubRingcsubrg 17552  ℂfldccnfld 18547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-rp 11246  df-fz 11698  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-gz 14460  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185  df-subg 16325  df-cmn 16927  df-mgp 17269  df-ur 17281  df-ring 17327  df-cring 17328  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-dvr 17459  df-drng 17525  df-subrg 17554  df-cnfld 18548
This theorem is referenced by:  zringunit  18647  zrngunit  18648
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