Theorem zringunit 19836

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 ` ℤring ) <-> ( A e. ZZ /\ ( abs ` A ) = 1 ) )

Proof of Theorem zringunit

Step	Hyp	Ref	Expression
1		zringbas 19824	. . . 4 |- ZZ = ( Base ` ℤring )
2		eqid 2622	. . . 4 |- (Unit ` ℤring ) = (Unit ` ℤring )
3	1, 2	unitcl 18659	. . 3 |- ( A e. (Unit ` ℤring ) -> A e. ZZ )
4		zsubrg 19799	. . . . . . 7 |- ZZ e. (SubRing ` ℂfld )
5		zgz 15637	. . . . . . . 8 |- ( x e. ZZ -> x e. ZZ[_i] )
6	5	ssriv 3607	. . . . . . 7 |- ZZ C_ ZZ[_i]
7		gzsubrg 19800	. . . . . . . 8 |- ZZ[_i] e. (SubRing ` ℂfld )
8		eqid 2622	. . . . . . . . 9 |- (ℂfld ↾s ZZ[_i] ) = (ℂfld ↾s ZZ[_i] )
9	8	subsubrg 18806	. . . . . . . 8 |- ( ZZ[_i] e. (SubRing ` ℂfld ) -> ( ZZ e. (SubRing ` (ℂfld ↾s ZZ[_i] ) ) <-> ( ZZ e. (SubRing ` ℂfld ) /\ ZZ C_ ZZ[_i] ) ) )
10	7, 9	ax-mp 5	. . . . . . 7 |- ( ZZ e. (SubRing ` (ℂfld ↾s ZZ[_i] ) ) <-> ( ZZ e. (SubRing ` ℂfld ) /\ ZZ C_ ZZ[_i] ) )
11	4, 6, 10	mpbir2an 955	. . . . . 6 |- ZZ e. (SubRing ` (ℂfld ↾s ZZ[_i] ) )
12		df-zring 19819	. . . . . . . 8 |- ℤring = (ℂfld ↾s ZZ )
13		ressabs 15939	. . . . . . . . 9 |- ( ( ZZ[_i] e. (SubRing ` ℂfld ) /\ ZZ C_ ZZ[_i] ) -> ( (ℂfld ↾s ZZ[_i] ) ↾s ZZ ) = (ℂfld ↾s ZZ ) )
14	7, 6, 13	mp2an 708	. . . . . . . 8 |- ( (ℂfld ↾s ZZ[_i] ) ↾s ZZ ) = (ℂfld ↾s ZZ )
15	12, 14	eqtr4i 2647	. . . . . . 7 |- ℤring = ( (ℂfld ↾s ZZ[_i] ) ↾s ZZ )
16		eqid 2622	. . . . . . 7 |- (Unit ` (ℂfld ↾s ZZ[_i] ) ) = (Unit ` (ℂfld ↾s ZZ[_i] ) )
17	15, 16, 2	subrguss 18795	. . . . . 6 |- ( ZZ e. (SubRing ` (ℂfld ↾s ZZ[_i] ) ) -> (Unit ` ℤring ) C_ (Unit ` (ℂfld ↾s ZZ[_i] ) ) )
18	11, 17	ax-mp 5	. . . . 5 |- (Unit ` ℤring ) C_ (Unit ` (ℂfld ↾s ZZ[_i] ) )
19	18	sseli 3599	. . . 4 |- ( A e. (Unit ` ℤring ) -> A e. (Unit ` (ℂfld ↾s ZZ[_i] ) ) )
20	8	gzrngunit 19812	. . . . 5 |- ( A e. (Unit ` (ℂfld ↾s ZZ[_i] ) ) <-> ( A e. ZZ[_i] /\ ( abs ` A ) = 1 ) )
21	20	simprbi 480	. . . 4 |- ( A e. (Unit ` (ℂfld ↾s ZZ[_i] ) ) -> ( abs ` A ) = 1 )
22	19, 21	syl 17	. . 3 |- ( A e. (Unit ` ℤring ) -> ( abs ` A ) = 1 )
23	3, 22	jca 554	. 2 |- ( A e. (Unit ` ℤring ) -> ( A e. ZZ /\ ( abs ` A ) = 1 ) )
24		zcn 11382	. . . . 5 |- ( A e. ZZ -> A e. CC )
25	24	adantr 481	. . . 4 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> A e. CC )
26		simpr 477	. . . . . 6 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> ( abs ` A ) = 1 )
27		ax-1ne0 10005	. . . . . . 7 |- 1 =/= 0
28	27	a1i 11	. . . . . 6 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> 1 =/= 0 )
29	26, 28	eqnetrd 2861	. . . . 5 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> ( abs ` A ) =/= 0 )
30		fveq2 6191	. . . . . . 7 |- ( A = 0 -> ( abs ` A ) = ( abs ` 0 ) )
31		abs0 14025	. . . . . . 7 |- ( abs ` 0 ) = 0
32	30, 31	syl6eq 2672	. . . . . 6 |- ( A = 0 -> ( abs ` A ) = 0 )
33	32	necon3i 2826	. . . . 5 |- ( ( abs ` A ) =/= 0 -> A =/= 0 )
34	29, 33	syl 17	. . . 4 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> A =/= 0 )
35		eldifsn 4317	. . . 4 |- ( A e. ( CC \ { 0 } ) <-> ( A e. CC /\ A =/= 0 ) )
36	25, 34, 35	sylanbrc 698	. . 3 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> A e. ( CC \ { 0 } ) )
37		simpl 473	. . 3 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> A e. ZZ )
38		cnfldinv 19777	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( invr ` ℂfld ) ` A ) = ( 1 / A ) )
39	25, 34, 38	syl2anc 693	. . . . 5 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> ( ( invr ` ℂfld ) ` A ) = ( 1 / A ) )
40		zre 11381	. . . . . . . . 9 |- ( A e. ZZ -> A e. RR )
41	40	adantr 481	. . . . . . . 8 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> A e. RR )
42		absresq 14042	. . . . . . . 8 |- ( A e. RR -> ( ( abs ` A ) ^ 2 ) = ( A ^ 2 ) )
43	41, 42	syl 17	. . . . . . 7 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> ( ( abs ` A ) ^ 2 ) = ( A ^ 2 ) )
44	26	oveq1d 6665	. . . . . . . 8 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> ( ( abs ` A ) ^ 2 ) = ( 1 ^ 2 ) )
45		sq1 12958	. . . . . . . 8 |- ( 1 ^ 2 ) = 1
46	44, 45	syl6eq 2672	. . . . . . 7 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> ( ( abs ` A ) ^ 2 ) = 1 )
47	25	sqvald 13005	. . . . . . 7 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> ( A ^ 2 ) = ( A x. A ) )
48	43, 46, 47	3eqtr3rd 2665	. . . . . 6 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> ( A x. A ) = 1 )
49		1cnd 10056	. . . . . . 7 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> 1 e. CC )
50	49, 25, 25, 34	divmuld 10823	. . . . . 6 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> ( ( 1 / A ) = A <-> ( A x. A ) = 1 ) )
51	48, 50	mpbird 247	. . . . 5 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> ( 1 / A ) = A )
52	39, 51	eqtrd 2656	. . . 4 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> ( ( invr ` ℂfld ) ` A ) = A )
53	52, 37	eqeltrd 2701	. . 3 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> ( ( invr ` ℂfld ) ` A ) e. ZZ )
54		cnfldbas 19750	. . . . . 6 |- CC = ( Base ` ℂfld )
55		cnfld0 19770	. . . . . 6 |- 0 = ( 0g ` ℂfld )
56		cndrng 19775	. . . . . 6 |- ℂfld e. DivRing
57	54, 55, 56	drngui 18753	. . . . 5 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
58		eqid 2622	. . . . 5 |- ( invr ` ℂfld ) = ( invr ` ℂfld )
59	12, 57, 2, 58	subrgunit 18798	. . . 4 |- ( ZZ e. (SubRing ` ℂfld ) -> ( A e. (Unit ` ℤring ) <-> ( A e. ( CC \ { 0 } ) /\ A e. ZZ /\ ( ( invr ` ℂfld ) ` A ) e. ZZ ) ) )
60	4, 59	ax-mp 5	. . 3 |- ( A e. (Unit ` ℤring ) <-> ( A e. ( CC \ { 0 } ) /\ A e. ZZ /\ ( ( invr ` ℂfld ) ` A ) e. ZZ ) )
61	36, 37, 53, 60	syl3anbrc 1246	. 2 |- ( ( A e. ZZ /\ ( abs ` A ) = 1 ) -> A e. (Unit ` ℤring ) )
62	23, 61	impbii 199	1 |- ( A e. (Unit ` ℤring ) <-> ( A e. ZZ /\ ( abs ` A ) = 1 ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   {csn 4177   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   / cdiv 10684   2c2 11070   ZZcz 11377   ^cexp 12860   abscabs 13974   ZZ[_i]cgz 15633   ↾_s cress 15858  Unitcui 18639   invrcinvr 18671  SubRingcsubrg 18776  ℂ_fldccnfld 19746  ℤ_ringzring 19818

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  df-zring 19819

This theorem is referenced by:  zringndrg  19838  prmirredlem  19841  qqhval2lem  30025