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Theorem gzrngunit 17900
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 17889 . . . . 5  |-  ZZ[_i]  e.  (SubRing ` fld )
2 gzrng.1 . . . . . 6  |-  Z  =  (flds  ZZ[_i] )
32subrgbas 16896 . . . . 5  |-  ( ZZ[_i]  e.  (SubRing ` fld )  ->  ZZ[_i]  =  ( Base `  Z ) )
41, 3ax-mp 5 . . . 4  |-  ZZ[_i]  =  ( Base `  Z )
5 eqid 2443 . . . 4  |-  (Unit `  Z )  =  (Unit `  Z )
64, 5unitcl 16773 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  ZZ[_i] )
7 eqid 2443 . . . . . . . . . . . 12  |-  ( invr ` fld )  =  ( invr ` fld )
8 eqid 2443 . . . . . . . . . . . 12  |-  ( invr `  Z )  =  (
invr `  Z )
92, 7, 5, 8subrginv 16903 . . . . . . . . . . 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 14014 . . . . . . . . . . . 12  |-  ( A  e.  ZZ[_i]  ->  A  e.  CC )
126, 11syl 16 . . . . . . . . . . 11  |-  ( A  e.  (Unit `  Z
)  ->  A  e.  CC )
13 0red 9408 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  0  e.  RR )
14 1re 9406 . . . . . . . . . . . . . . 15  |-  1  e.  RR
1514a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  1  e.  RR )
1612abscld 12943 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  e.  RR )
17 0lt1 9883 . . . . . . . . . . . . . . 15  |-  0  <  1
1817a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  0  <  1 )
192gzrngunitlem 17899 . . . . . . . . . . . . . 14  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( abs `  A ) )
2013, 15, 16, 18, 19ltletrd 9552 . . . . . . . . . . . . 13  |-  ( A  e.  (Unit `  Z
)  ->  0  <  ( abs `  A ) )
2120gt0ne0d 9925 . . . . . . . . . . . 12  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  =/=  0
)
2212abs00ad 12800 . . . . . . . . . . . . 13  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =  0  <->  A  =  0
) )
2322necon3bid 2637 . . . . . . . . . . . 12  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =/=  0  <->  A  =/=  0
) )
2421, 23mpbid 210 . . . . . . . . . . 11  |-  ( A  e.  (Unit `  Z
)  ->  A  =/=  0 )
25 cnfldinv 17869 . . . . . . . . . . 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 2477 . . . . . . . . 9  |-  ( A  e.  (Unit `  Z
)  ->  ( ( invr `  Z ) `  A )  =  ( 1  /  A ) )
282subrgrng 16890 . . . . . . . . . . 11  |-  ( ZZ[_i]  e.  (SubRing ` fld )  ->  Z  e.  Ring )
291, 28ax-mp 5 . . . . . . . . . 10  |-  Z  e. 
Ring
305, 8unitinvcl 16788 . . . . . . . . . 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 2518 . . . . . . . 8  |-  ( A  e.  (Unit `  Z
)  ->  ( 1  /  A )  e.  (Unit `  Z )
)
332gzrngunitlem 17899 . . . . . . . 8  |-  ( ( 1  /  A )  e.  (Unit `  Z
)  ->  1  <_  ( abs `  ( 1  /  A ) ) )
3432, 33syl 16 . . . . . . 7  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( abs `  ( 1  /  A ) ) )
35 ax-1cn 9361 . . . . . . . . 9  |-  1  e.  CC
3635a1i 11 . . . . . . . 8  |-  ( A  e.  (Unit `  Z
)  ->  1  e.  CC )
3736, 12, 24absdivd 12962 . . . . . . 7  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  ( 1  /  A
) )  =  ( ( abs `  1
)  /  ( abs `  A ) ) )
3834, 37breqtrd 4337 . . . . . 6  |-  ( A  e.  (Unit `  Z
)  ->  1  <_  ( ( abs `  1
)  /  ( abs `  A ) ) )
39 1div1e1 10045 . . . . . 6  |-  ( 1  /  1 )  =  1
40 abs1 12807 . . . . . . . 8  |-  ( abs `  1 )  =  1
4140eqcomi 2447 . . . . . . 7  |-  1  =  ( abs `  1
)
4241oveq1i 6122 . . . . . 6  |-  ( 1  /  ( abs `  A
) )  =  ( ( abs `  1
)  /  ( abs `  A ) )
4338, 39, 423brtr4g 4345 . . . . 5  |-  ( A  e.  (Unit `  Z
)  ->  ( 1  /  1 )  <_ 
( 1  /  ( abs `  A ) ) )
44 lerec 10235 . . . . . 6  |-  ( ( ( ( abs `  A
)  e.  RR  /\  0  <  ( abs `  A
) )  /\  (
1  e.  RR  /\  0  <  1 ) )  ->  ( ( abs `  A )  <_  1  <->  ( 1  /  1 )  <_  ( 1  / 
( abs `  A
) ) ) )
4516, 20, 15, 18, 44syl22anc 1219 . . . . 5  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  <_ 
1  <->  ( 1  / 
1 )  <_  (
1  /  ( abs `  A ) ) ) )
4643, 45mpbird 232 . . . 4  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  <_  1
)
47 letri3 9481 . . . . 5  |-  ( ( ( abs `  A
)  e.  RR  /\  1  e.  RR )  ->  ( ( abs `  A
)  =  1  <->  (
( abs `  A
)  <_  1  /\  1  <_  ( abs `  A
) ) ) )
4816, 14, 47sylancl 662 . . . 4  |-  ( A  e.  (Unit `  Z
)  ->  ( ( abs `  A )  =  1  <->  ( ( abs `  A )  <_  1  /\  1  <_  ( abs `  A ) ) ) )
4946, 19, 48mpbir2and 913 . . 3  |-  ( A  e.  (Unit `  Z
)  ->  ( abs `  A )  =  1 )
506, 49jca 532 . 2  |-  ( A  e.  (Unit `  Z
)  ->  ( A  e.  ZZ[_i]  /\  ( abs `  A
)  =  1 ) )
5111adantr 465 . . . 4  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  e.  CC )
52 simpr 461 . . . . . 6  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  ( abs `  A )  =  1 )
53 ax-1ne0 9372 . . . . . . 7  |-  1  =/=  0
5453a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  1  =/=  0 )
5552, 54eqnetrd 2654 . . . . 5  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  ( abs `  A )  =/=  0 )
56 fveq2 5712 . . . . . . 7  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
57 abs0 12795 . . . . . . 7  |-  ( abs `  0 )  =  0
5856, 57syl6eq 2491 . . . . . 6  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
5958necon3i 2674 . . . . 5  |-  ( ( abs `  A )  =/=  0  ->  A  =/=  0 )
6055, 59syl 16 . . . 4  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  =/=  0 )
61 eldifsn 4021 . . . 4  |-  ( A  e.  ( CC  \  { 0 } )  <-> 
( A  e.  CC  /\  A  =/=  0 ) )
6251, 60, 61sylanbrc 664 . . 3  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  e.  ( CC  \  {
0 } ) )
63 simpl 457 . . 3  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  e.  ZZ[_i]
)
6451, 60, 25syl2anc 661 . . . . 5  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  =  ( 1  /  A ) )
6551absvalsqd 12949 . . . . . . 7  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
6652oveq1d 6127 . . . . . . . 8  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  ( 1 ^ 2 ) )
67 sq1 11981 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
6866, 67syl6eq 2491 . . . . . . 7  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( abs `  A
) ^ 2 )  =  1 )
6965, 68eqtr3d 2477 . . . . . 6  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  ( A  x.  ( * `  A ) )  =  1 )
7069oveq1d 6127 . . . . 5  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( A  x.  (
* `  A )
)  /  A )  =  ( 1  /  A ) )
7151cjcld 12706 . . . . . 6  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
* `  A )  e.  CC )
7271, 51, 60divcan3d 10133 . . . . 5  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( A  x.  (
* `  A )
)  /  A )  =  ( * `  A ) )
7364, 70, 723eqtr2d 2481 . . . 4  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  =  ( * `  A ) )
74 gzcjcl 14018 . . . . 5  |-  ( A  e.  ZZ[_i]  ->  ( * `  A )  e.  ZZ[_i] )
7574adantr 465 . . . 4  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
* `  A )  e.  ZZ[_i]
)
7673, 75eqeltrd 2517 . . 3  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  (
( invr ` fld ) `  A )  e.  ZZ[_i] )
77 cnfldbas 17844 . . . . . 6  |-  CC  =  ( Base ` fld )
78 cnfld0 17862 . . . . . 6  |-  0  =  ( 0g ` fld )
79 cndrng 17867 . . . . . 6  |-fld  e.  DivRing
8077, 78, 79drngui 16860 . . . . 5  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
812, 80, 5, 7subrgunit 16905 . . . 4  |-  ( ZZ[_i]  e.  (SubRing ` fld )  ->  ( A  e.  (Unit `  Z )  <->  ( A  e.  ( CC 
\  { 0 } )  /\  A  e.  ZZ[_i]  /\  ( ( invr ` fld ) `  A )  e.  ZZ[_i] ) ) )
821, 81ax-mp 5 . . 3  |-  ( A  e.  (Unit `  Z
)  <->  ( A  e.  ( CC  \  {
0 } )  /\  A  e.  ZZ[_i]  /\  ( (
invr ` fld ) `  A )  e.  ZZ[_i] ) )
8362, 63, 76, 82syl3anbrc 1172 . 2  |-  ( ( A  e.  ZZ[_i]  /\  ( abs `  A )  =  1 )  ->  A  e.  (Unit `  Z )
)
8450, 83impbii 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620    \ cdif 3346   {csn 3898   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   CCcc 9301   RRcr 9302   0cc0 9303   1c1 9304    x. cmul 9308    < clt 9439    <_ cle 9440    / cdiv 10014   2c2 10392   ^cexp 11886   *ccj 12606   abscabs 12744   ZZ[_i]cgz 14011   Basecbs 14195   ↾s cress 14196   Ringcrg 16667  Unitcui 16753   invrcinvr 16785  SubRingcsubrg 16883  ℂfldccnfld 17840
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
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 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-rp 11013  df-fz 11459  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-gz 14012  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-starv 14274  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-subg 15699  df-cmn 16300  df-mgp 16614  df-ur 16626  df-rng 16669  df-cring 16670  df-oppr 16737  df-dvdsr 16755  df-unit 16756  df-invr 16786  df-dvr 16797  df-drng 16856  df-subrg 16885  df-cnfld 17841
This theorem is referenced by:  zringunit  17936  zrngunit  17937
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