Theorem isdrng2 17985

Description: A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
isdrng2.b	|- B = ( Base ` R )
isdrng2.z	|- .0. = ( 0g ` R )
isdrng2.g	|- G = ( (mulGrp ` R ) ↾s ( B \ { .0. } ) )

Assertion

Ref	Expression
isdrng2	|- ( R e. DivRing <-> ( R e. Ring /\ G e. Grp ) )

Proof of Theorem isdrng2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isdrng2.b	. . 3 |- B = ( Base ` R )
2		eqid 2451	. . 3 |- (Unit ` R ) = (Unit ` R )
3		isdrng2.z	. . 3 |- .0. = ( 0g ` R )
4	1, 2, 3	isdrng 17979	. 2 |- ( R e. DivRing <-> ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) )
5		oveq2 6298	. . . . . . 7 |- ( (Unit ` R ) = ( B \ { .0. } ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) )
6	5	adantl 468	. . . . . 6 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) )
7		isdrng2.g	. . . . . 6 |- G = ( (mulGrp ` R ) ↾s ( B \ { .0. } ) )
8	6, 7	syl6eqr 2503	. . . . 5 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) = G )
9		eqid 2451	. . . . . . 7 |- ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s (Unit ` R ) )
10	2, 9	unitgrp 17895	. . . . . 6 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. Grp )
11	10	adantr 467	. . . . 5 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. Grp )
12	8, 11	eqeltrrd 2530	. . . 4 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> G e. Grp )
13	1, 2	unitcl 17887	. . . . . . . . 9 |- ( x e. (Unit ` R ) -> x e. B )
14	13	adantl 468	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. B )
15		difss 3560	. . . . . . . . . . . . . . 15 |- ( B \ { .0. } ) C_ B
16		eqid 2451	. . . . . . . . . . . . . . . . 17 |- (mulGrp ` R ) = (mulGrp ` R )
17	16, 1	mgpbas 17729	. . . . . . . . . . . . . . . 16 |- B = ( Base ` (mulGrp ` R ) )
18	7, 17	ressbas2 15180	. . . . . . . . . . . . . . 15 |- ( ( B \ { .0. } ) C_ B -> ( B \ { .0. } ) = ( Base ` G ) )
19	15, 18	ax-mp 5	. . . . . . . . . . . . . 14 |- ( B \ { .0. } ) = ( Base ` G )
20		eqid 2451	. . . . . . . . . . . . . 14 |- ( 0g ` G ) = ( 0g ` G )
21	19, 20	grpidcl 16694	. . . . . . . . . . . . 13 |- ( G e. Grp -> ( 0g ` G ) e. ( B \ { .0. } ) )
22	21	ad2antlr 733	. . . . . . . . . . . 12 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) e. ( B \ { .0. } ) )
23		eldifsn 4097	. . . . . . . . . . . 12 |- ( ( 0g ` G ) e. ( B \ { .0. } ) <-> ( ( 0g ` G ) e. B /\ ( 0g ` G ) =/= .0. ) )
24	22, 23	sylib 200	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( ( 0g ` G ) e. B /\ ( 0g ` G ) =/= .0. ) )
25	24	simprd 465	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) =/= .0. )
26		simpll 760	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> R e. Ring )
27	22	eldifad 3416	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) e. B )
28		simpr 463	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. (Unit ` R ) )
29		eqid 2451	. . . . . . . . . . . 12 |- (/r ` R ) = (/r ` R )
30		eqid 2451	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
31	1, 2, 29, 30	dvrcan1 17919	. . . . . . . . . . 11 |- ( ( R e. Ring /\ ( 0g ` G ) e. B /\ x e. (Unit ` R ) ) -> ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) x ) = ( 0g ` G ) )
32	26, 27, 28, 31	syl3anc 1268	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) x ) = ( 0g ` G ) )
33	1, 2, 29	dvrcl 17914	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ ( 0g ` G ) e. B /\ x e. (Unit ` R ) ) -> ( ( 0g ` G ) (/r ` R ) x ) e. B )
34	26, 27, 28, 33	syl3anc 1268	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( ( 0g ` G ) (/r ` R ) x ) e. B )
35	1, 30, 3	ringrz 17818	. . . . . . . . . . 11 |- ( ( R e. Ring /\ ( ( 0g ` G ) (/r ` R ) x ) e. B ) -> ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) .0. ) = .0. )
36	26, 34, 35	syl2anc 667	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) .0. ) = .0. )
37	25, 32, 36	3netr4d 2701	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) x ) =/= ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) .0. ) )
38		oveq2 6298	. . . . . . . . . 10 |- ( x = .0. -> ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) x ) = ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) .0. ) )
39	38	necon3i 2656	. . . . . . . . 9 |- ( ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) x ) =/= ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) .0. ) -> x =/= .0. )
40	37, 39	syl 17	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x =/= .0. )
41		eldifsn 4097	. . . . . . . 8 |- ( x e. ( B \ { .0. } ) <-> ( x e. B /\ x =/= .0. ) )
42	14, 40, 41	sylanbrc 670	. . . . . . 7 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. ( B \ { .0. } ) )
43	42	ex 436	. . . . . 6 |- ( ( R e. Ring /\ G e. Grp ) -> ( x e. (Unit ` R ) -> x e. ( B \ { .0. } ) ) )
44	43	ssrdv 3438	. . . . 5 |- ( ( R e. Ring /\ G e. Grp ) -> (Unit ` R ) C_ ( B \ { .0. } ) )
45		eldifi 3555	. . . . . . . . . . 11 |- ( x e. ( B \ { .0. } ) -> x e. B )
46	45	adantl 468	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x e. B )
47		eqid 2451	. . . . . . . . . . . . 13 |- ( invg ` G ) = ( invg ` G )
48	19, 47	grpinvcl 16711	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. ( B \ { .0. } ) )
49	48	adantll 720	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. ( B \ { .0. } ) )
50	49	eldifad 3416	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. B )
51		eqid 2451	. . . . . . . . . . 11 |- ( ||r ` R ) = ( ||r ` R )
52	1, 51, 30	dvdsrmul 17876	. . . . . . . . . 10 |- ( ( x e. B /\ ( ( invg ` G ) ` x ) e. B ) -> x ( ||r ` R ) ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) )
53	46, 50, 52	syl2anc 667	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` R ) ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) )
54		fvex 5875	. . . . . . . . . . . . . 14 |- ( Base ` R ) e. _V
55	1, 54	eqeltri 2525	. . . . . . . . . . . . 13 |- B e. _V
56		difexg 4551	. . . . . . . . . . . . 13 |- ( B e. _V -> ( B \ { .0. } ) e. _V )
57	16, 30	mgpplusg 17727	. . . . . . . . . . . . . 14 |- ( .r ` R ) = ( +g ` (mulGrp ` R ) )
58	7, 57	ressplusg 15239	. . . . . . . . . . . . 13 |- ( ( B \ { .0. } ) e. _V -> ( .r ` R ) = ( +g ` G ) )
59	55, 56, 58	mp2b 10	. . . . . . . . . . . 12 |- ( .r ` R ) = ( +g ` G )
60	19, 59, 20, 47	grplinv 16712	. . . . . . . . . . 11 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 0g ` G ) )
61	60	adantll 720	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 0g ` G ) )
62		eqid 2451	. . . . . . . . . . . . . . 15 |- ( 1r ` R ) = ( 1r ` R )
63	1, 62	ringidcl 17801	. . . . . . . . . . . . . 14 |- ( R e. Ring -> ( 1r ` R ) e. B )
64	1, 30, 62	ringlidm 17804	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ ( 1r ` R ) e. B ) -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
65	63, 64	mpdan 674	. . . . . . . . . . . . 13 |- ( R e. Ring -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
66	65	adantr 467	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ G e. Grp ) -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
67		simpr 463	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ G e. Grp ) -> G e. Grp )
68	2, 62	1unit 17886	. . . . . . . . . . . . . . 15 |- ( R e. Ring -> ( 1r ` R ) e. (Unit ` R ) )
69	68	adantr 467	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ G e. Grp ) -> ( 1r ` R ) e. (Unit ` R ) )
70	44, 69	sseldd 3433	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ G e. Grp ) -> ( 1r ` R ) e. ( B \ { .0. } ) )
71	19, 59, 20	grpid 16701	. . . . . . . . . . . . 13 |- ( ( G e. Grp /\ ( 1r ` R ) e. ( B \ { .0. } ) ) -> ( ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) <-> ( 0g ` G ) = ( 1r ` R ) ) )
72	67, 70, 71	syl2anc 667	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ G e. Grp ) -> ( ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) <-> ( 0g ` G ) = ( 1r ` R ) ) )
73	66, 72	mpbid 214	. . . . . . . . . . 11 |- ( ( R e. Ring /\ G e. Grp ) -> ( 0g ` G ) = ( 1r ` R ) )
74	73	adantr 467	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( 0g ` G ) = ( 1r ` R ) )
75	61, 74	eqtrd 2485	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 1r ` R ) )
76	53, 75	breqtrd 4427	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` R ) ( 1r ` R ) )
77		eqid 2451	. . . . . . . . . . . 12 |- (oppr ` R ) = (oppr ` R )
78	77, 1	opprbas 17857	. . . . . . . . . . 11 |- B = ( Base ` (oppr ` R ) )
79		eqid 2451	. . . . . . . . . . 11 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
80		eqid 2451	. . . . . . . . . . 11 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
81	78, 79, 80	dvdsrmul 17876	. . . . . . . . . 10 |- ( ( x e. B /\ ( ( invg ` G ) ` x ) e. B ) -> x ( ||r ` (oppr ` R ) ) ( ( ( invg ` G ) ` x ) ( .r ` (oppr ` R ) ) x ) )
82	46, 50, 81	syl2anc 667	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` (oppr ` R ) ) ( ( ( invg ` G ) ` x ) ( .r ` (oppr ` R ) ) x ) )
83	1, 30, 77, 80	opprmul 17854	. . . . . . . . . 10 |- ( ( ( invg ` G ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) ( ( invg ` G ) ` x ) )
84	19, 59, 20, 47	grprinv 16713	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 0g ` G ) )
85	84	adantll 720	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 0g ` G ) )
86	85, 74	eqtrd 2485	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 1r ` R ) )
87	83, 86	syl5eq 2497	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) )
88	82, 87	breqtrd 4427	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
89	2, 62, 51, 77, 79	isunit 17885	. . . . . . . 8 |- ( x e. (Unit ` R ) <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
90	76, 88, 89	sylanbrc 670	. . . . . . 7 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x e. (Unit ` R ) )
91	90	ex 436	. . . . . 6 |- ( ( R e. Ring /\ G e. Grp ) -> ( x e. ( B \ { .0. } ) -> x e. (Unit ` R ) ) )
92	91	ssrdv 3438	. . . . 5 |- ( ( R e. Ring /\ G e. Grp ) -> ( B \ { .0. } ) C_ (Unit ` R ) )
93	44, 92	eqssd 3449	. . . 4 |- ( ( R e. Ring /\ G e. Grp ) -> (Unit ` R ) = ( B \ { .0. } ) )
94	12, 93	impbida 843	. . 3 |- ( R e. Ring -> ( (Unit ` R ) = ( B \ { .0. } ) <-> G e. Grp ) )
95	94	pm5.32i 643	. 2 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) <-> ( R e. Ring /\ G e. Grp ) )
96	4, 95	bitri 253	1 |- ( R e. DivRing <-> ( R e. Ring /\ G e. Grp ) )

Syntax hints:   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   =/= wne 2622   _Vcvv 3045   \ cdif 3401   C_ wss 3404   {csn 3968   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   Basecbs 15121   ↾_s cress 15122   +g cplusg 15190   .rcmulr 15191   0gc0g 15338   Grpcgrp 16669   invgcminusg 16670  mulGrpcmgp 17723   1rcur 17735   Ringcrg 17780  opp_rcoppr 17850   ||_rcdsr 17866  Unitcui 17867  /_rcdvr 17910   DivRingcdr 17975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-minusg 16674  df-mgp 17724  df-ur 17736  df-ring 17782  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-dvr 17911  df-drng 17977

This theorem is referenced by:  drngmgp  17987  isdrngd  18000