Theorem gzrngunit 19812

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 = (ℂfld ↾s ZZ[_i] )

Assertion

Ref	Expression
gzrngunit	|- ( A e. (Unit ` Z ) <-> ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) )

Proof of Theorem gzrngunit

Step	Hyp	Ref	Expression
1		gzsubrg 19800	. . . . 5 |- ZZ[_i] e. (SubRing ` ℂfld )
2		gzrng.1	. . . . . 6 |- Z = (ℂfld ↾s ZZ[_i] )
3	2	subrgbas 18789	. . . . 5 |- ( ZZ[_i] e. (SubRing ` ℂfld ) -> ZZ[_i] = ( Base ` Z ) )
4	1, 3	ax-mp 5	. . . 4 |- ZZ[_i] = ( Base ` Z )
5		eqid 2622	. . . 4 |- (Unit ` Z ) = (Unit ` Z )
6	4, 5	unitcl 18659	. . 3 |- ( A e. (Unit ` Z ) -> A e. ZZ[_i] )
7		eqid 2622	. . . . . . . . . . . 12 |- ( invr ` ℂfld ) = ( invr ` ℂfld )
8		eqid 2622	. . . . . . . . . . . 12 |- ( invr ` Z ) = ( invr ` Z )
9	2, 7, 5, 8	subrginv 18796	. . . . . . . . . . 11 |- ( ( ZZ[_i] e. (SubRing ` ℂfld ) /\ A e. (Unit ` Z ) ) -> ( ( invr ` ℂfld ) ` A ) = ( ( invr ` Z ) ` A ) )
10	1, 9	mpan 706	. . . . . . . . . 10 |- ( A e. (Unit ` Z ) -> ( ( invr ` ℂfld ) ` A ) = ( ( invr ` Z ) ` A ) )
11		gzcn 15636	. . . . . . . . . . . 12 |- ( A e. ZZ[_i] -> A e. CC )
12	6, 11	syl 17	. . . . . . . . . . 11 |- ( A e. (Unit ` Z ) -> A e. CC )
13		0red 10041	. . . . . . . . . . . . . 14 |- ( A e. (Unit ` Z ) -> 0 e. RR )
14		1re 10039	. . . . . . . . . . . . . . 15 |- 1 e. RR
15	14	a1i 11	. . . . . . . . . . . . . 14 |- ( A e. (Unit ` Z ) -> 1 e. RR )
16	12	abscld 14175	. . . . . . . . . . . . . 14 |- ( A e. (Unit ` Z ) -> ( abs ` A ) e. RR )
17		0lt1 10550	. . . . . . . . . . . . . . 15 |- 0 < 1
18	17	a1i 11	. . . . . . . . . . . . . 14 |- ( A e. (Unit ` Z ) -> 0 < 1 )
19	2	gzrngunitlem 19811	. . . . . . . . . . . . . 14 |- ( A e. (Unit ` Z ) -> 1 <_ ( abs ` A ) )
20	13, 15, 16, 18, 19	ltletrd 10197	. . . . . . . . . . . . 13 |- ( A e. (Unit ` Z ) -> 0 < ( abs ` A ) )
21	20	gt0ne0d 10592	. . . . . . . . . . . 12 |- ( A e. (Unit ` Z ) -> ( abs ` A ) =/= 0 )
22	12	abs00ad 14030	. . . . . . . . . . . . 13 |- ( A e. (Unit ` Z ) -> ( ( abs ` A ) = 0 <-> A = 0 ) )
23	22	necon3bid 2838	. . . . . . . . . . . 12 |- ( A e. (Unit ` Z ) -> ( ( abs ` A ) =/= 0 <-> A =/= 0 ) )
24	21, 23	mpbid 222	. . . . . . . . . . 11 |- ( A e. (Unit ` Z ) -> A =/= 0 )
25		cnfldinv 19777	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( invr ` ℂfld ) ` A ) = ( 1 / A ) )
26	12, 24, 25	syl2anc 693	. . . . . . . . . 10 |- ( A e. (Unit ` Z ) -> ( ( invr ` ℂfld ) ` A ) = ( 1 / A ) )
27	10, 26	eqtr3d 2658	. . . . . . . . 9 |- ( A e. (Unit ` Z ) -> ( ( invr ` Z ) ` A ) = ( 1 / A ) )
28	2	subrgring 18783	. . . . . . . . . . 11 |- ( ZZ[_i] e. (SubRing ` ℂfld ) -> Z e. Ring )
29	1, 28	ax-mp 5	. . . . . . . . . 10 |- Z e. Ring
30	5, 8	unitinvcl 18674	. . . . . . . . . 10 |- ( ( Z e. Ring /\ A e. (Unit ` Z ) ) -> ( ( invr ` Z ) ` A ) e. (Unit ` Z ) )
31	29, 30	mpan 706	. . . . . . . . 9 |- ( A e. (Unit ` Z ) -> ( ( invr ` Z ) ` A ) e. (Unit ` Z ) )
32	27, 31	eqeltrrd 2702	. . . . . . . 8 |- ( A e. (Unit ` Z ) -> ( 1 / A ) e. (Unit ` Z ) )
33	2	gzrngunitlem 19811	. . . . . . . 8 |- ( ( 1 / A ) e. (Unit ` Z ) -> 1 <_ ( abs ` ( 1 / A ) ) )
34	32, 33	syl 17	. . . . . . 7 |- ( A e. (Unit ` Z ) -> 1 <_ ( abs ` ( 1 / A ) ) )
35		1cnd 10056	. . . . . . . 8 |- ( A e. (Unit ` Z ) -> 1 e. CC )
36	35, 12, 24	absdivd 14194	. . . . . . 7 |- ( A e. (Unit ` Z ) -> ( abs ` ( 1 / A ) ) = ( ( abs ` 1 ) / ( abs ` A ) ) )
37	34, 36	breqtrd 4679	. . . . . 6 |- ( A e. (Unit ` Z ) -> 1 <_ ( ( abs ` 1 ) / ( abs ` A ) ) )
38		1div1e1 10717	. . . . . 6 |- ( 1 / 1 ) = 1
39		abs1 14037	. . . . . . . 8 |- ( abs ` 1 ) = 1
40	39	eqcomi 2631	. . . . . . 7 |- 1 = ( abs ` 1 )
41	40	oveq1i 6660	. . . . . 6 |- ( 1 / ( abs ` A ) ) = ( ( abs ` 1 ) / ( abs ` A ) )
42	37, 38, 41	3brtr4g 4687	. . . . 5 |- ( A e. (Unit ` Z ) -> ( 1 / 1 ) <_ ( 1 / ( abs ` A ) ) )
43		lerec 10906	. . . . . 6 |- ( ( ( ( abs ` A ) e. RR /\ 0 < ( abs ` A ) ) /\ ( 1 e. RR /\ 0 < 1 ) ) -> ( ( abs ` A ) <_ 1 <-> ( 1 / 1 ) <_ ( 1 / ( abs ` A ) ) ) )
44	16, 20, 15, 18, 43	syl22anc 1327	. . . . 5 |- ( A e. (Unit ` Z ) -> ( ( abs ` A ) <_ 1 <-> ( 1 / 1 ) <_ ( 1 / ( abs ` A ) ) ) )
45	42, 44	mpbird 247	. . . 4 |- ( A e. (Unit ` Z ) -> ( abs ` A ) <_ 1 )
46		letri3 10123	. . . . 5 |- ( ( ( abs ` A ) e. RR /\ 1 e. RR ) -> ( ( abs ` A ) = 1 <-> ( ( abs ` A ) <_ 1 /\ 1 <_ ( abs ` A ) ) ) )
47	16, 14, 46	sylancl 694	. . . 4 |- ( A e. (Unit ` Z ) -> ( ( abs ` A ) = 1 <-> ( ( abs ` A ) <_ 1 /\ 1 <_ ( abs ` A ) ) ) )
48	45, 19, 47	mpbir2and 957	. . 3 |- ( A e. (Unit ` Z ) -> ( abs ` A ) = 1 )
49	6, 48	jca 554	. 2 |- ( A e. (Unit ` Z ) -> ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) )
50	11	adantr 481	. . . 4 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> A e. CC )
51		simpr 477	. . . . . 6 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> ( abs ` A ) = 1 )
52		ax-1ne0 10005	. . . . . . 7 |- 1 =/= 0
53	52	a1i 11	. . . . . 6 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> 1 =/= 0 )
54	51, 53	eqnetrd 2861	. . . . 5 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> ( abs ` A ) =/= 0 )
55		fveq2 6191	. . . . . . 7 |- ( A = 0 -> ( abs ` A ) = ( abs ` 0 ) )
56		abs0 14025	. . . . . . 7 |- ( abs ` 0 ) = 0
57	55, 56	syl6eq 2672	. . . . . 6 |- ( A = 0 -> ( abs ` A ) = 0 )
58	57	necon3i 2826	. . . . 5 |- ( ( abs ` A ) =/= 0 -> A =/= 0 )
59	54, 58	syl 17	. . . 4 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> A =/= 0 )
60		eldifsn 4317	. . . 4 |- ( A e. ( CC \ { 0 } ) <-> ( A e. CC /\ A =/= 0 ) )
61	50, 59, 60	sylanbrc 698	. . 3 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> A e. ( CC \ { 0 } ) )
62		simpl 473	. . 3 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> A e. ZZ[_i] )
63	50, 59, 25	syl2anc 693	. . . . 5 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> ( ( invr ` ℂfld ) ` A ) = ( 1 / A ) )
64	50	absvalsqd 14181	. . . . . . 7 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) )
65	51	oveq1d 6665	. . . . . . . 8 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> ( ( abs ` A ) ^ 2 ) = ( 1 ^ 2 ) )
66		sq1 12958	. . . . . . . 8 |- ( 1 ^ 2 ) = 1
67	65, 66	syl6eq 2672	. . . . . . 7 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> ( ( abs ` A ) ^ 2 ) = 1 )
68	64, 67	eqtr3d 2658	. . . . . 6 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> ( A x. ( * ` A ) ) = 1 )
69	68	oveq1d 6665	. . . . 5 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> ( ( A x. ( * ` A ) ) / A ) = ( 1 / A ) )
70	50	cjcld 13936	. . . . . 6 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> ( * ` A ) e. CC )
71	70, 50, 59	divcan3d 10806	. . . . 5 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> ( ( A x. ( * ` A ) ) / A ) = ( * ` A ) )
72	63, 69, 71	3eqtr2d 2662	. . . 4 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> ( ( invr ` ℂfld ) ` A ) = ( * ` A ) )
73		gzcjcl 15640	. . . . 5 |- ( A e. ZZ[_i] -> ( * ` A ) e. ZZ[_i] )
74	73	adantr 481	. . . 4 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> ( * ` A ) e. ZZ[_i] )
75	72, 74	eqeltrd 2701	. . 3 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> ( ( invr ` ℂfld ) ` A ) e. ZZ[_i] )
76		cnfldbas 19750	. . . . . 6 |- CC = ( Base ` ℂfld )
77		cnfld0 19770	. . . . . 6 |- 0 = ( 0g ` ℂfld )
78		cndrng 19775	. . . . . 6 |- ℂfld e. DivRing
79	76, 77, 78	drngui 18753	. . . . 5 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
80	2, 79, 5, 7	subrgunit 18798	. . . 4 |- ( ZZ[_i] e. (SubRing ` ℂfld ) -> ( A e. (Unit ` Z ) <-> ( A e. ( CC \ { 0 } ) /\ A e. ZZ[_i] /\ ( ( invr ` ℂfld ) ` A ) e. ZZ[_i] ) ) )
81	1, 80	ax-mp 5	. . 3 |- ( A e. (Unit ` Z ) <-> ( A e. ( CC \ { 0 } ) /\ A e. ZZ[_i] /\ ( ( invr ` ℂfld ) ` A ) e. ZZ[_i] ) )
82	61, 62, 75, 81	syl3anbrc 1246	. 2 |- ( ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) -> A e. (Unit ` Z ) )
83	49, 82	impbii 199	1 |- ( A e. (Unit ` Z ) <-> ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684   2c2 11070   ^cexp 12860   *ccj 13836   abscabs 13974   ZZ[_i]cgz 15633   Basecbs 15857   ↾_s cress 15858   Ringcrg 18547  Unitcui 18639   invrcinvr 18671  SubRingcsubrg 18776  ℂ_fldccnfld 19746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-gz 15634  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-subrg 18778  df-cnfld 19747

This theorem is referenced by:  zringunit  19836