Theorem isdrng2 17920

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 2429	. . 3 |- (Unit ` R ) = (Unit ` R )
3		isdrng2.z	. . 3 |- .0. = ( 0g ` R )
4	1, 2, 3	isdrng 17914	. 2 |- ( R e. DivRing <-> ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) )
5		oveq2 6313	. . . . . . 7 |- ( (Unit ` R ) = ( B \ { .0. } ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) )
6	5	adantl 467	. . . . . 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 2488	. . . . 5 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) = G )
9		eqid 2429	. . . . . . 7 |- ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s (Unit ` R ) )
10	2, 9	unitgrp 17830	. . . . . 6 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. Grp )
11	10	adantr 466	. . . . 5 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. Grp )
12	8, 11	eqeltrrd 2518	. . . 4 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> G e. Grp )
13	1, 2	unitcl 17822	. . . . . . . . 9 |- ( x e. (Unit ` R ) -> x e. B )
14	13	adantl 467	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. B )
15		difss 3598	. . . . . . . . . . . . . . 15 |- ( B \ { .0. } ) C_ B
16		eqid 2429	. . . . . . . . . . . . . . . . 17 |- (mulGrp ` R ) = (mulGrp ` R )
17	16, 1	mgpbas 17664	. . . . . . . . . . . . . . . 16 |- B = ( Base ` (mulGrp ` R ) )
18	7, 17	ressbas2 15142	. . . . . . . . . . . . . . 15 |- ( ( B \ { .0. } ) C_ B -> ( B \ { .0. } ) = ( Base ` G ) )
19	15, 18	ax-mp 5	. . . . . . . . . . . . . 14 |- ( B \ { .0. } ) = ( Base ` G )
20		eqid 2429	. . . . . . . . . . . . . 14 |- ( 0g ` G ) = ( 0g ` G )
21	19, 20	grpidcl 16645	. . . . . . . . . . . . 13 |- ( G e. Grp -> ( 0g ` G ) e. ( B \ { .0. } ) )
22	21	ad2antlr 731	. . . . . . . . . . . 12 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) e. ( B \ { .0. } ) )
23		eldifsn 4128	. . . . . . . . . . . 12 |- ( ( 0g ` G ) e. ( B \ { .0. } ) <-> ( ( 0g ` G ) e. B /\ ( 0g ` G ) =/= .0. ) )
24	22, 23	sylib 199	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( ( 0g ` G ) e. B /\ ( 0g ` G ) =/= .0. ) )
25	24	simprd 464	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) =/= .0. )
26		simpll 758	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> R e. Ring )
27	22	eldifad 3454	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) e. B )
28		simpr 462	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. (Unit ` R ) )
29		eqid 2429	. . . . . . . . . . . 12 |- (/r ` R ) = (/r ` R )
30		eqid 2429	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
31	1, 2, 29, 30	dvrcan1 17854	. . . . . . . . . . 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 1264	. . . . . . . . . 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 17849	. . . . . . . . . . . 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 1264	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( ( 0g ` G ) (/r ` R ) x ) e. B )
35	1, 30, 3	ringrz 17753	. . . . . . . . . . 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 665	. . . . . . . . . 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 2736	. . . . . . . . 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 6313	. . . . . . . . . 10 |- ( x = .0. -> ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) x ) = ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) .0. ) )
39	38	necon3i 2671	. . . . . . . . 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 4128	. . . . . . . 8 |- ( x e. ( B \ { .0. } ) <-> ( x e. B /\ x =/= .0. ) )
42	14, 40, 41	sylanbrc 668	. . . . . . 7 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. ( B \ { .0. } ) )
43	42	ex 435	. . . . . 6 |- ( ( R e. Ring /\ G e. Grp ) -> ( x e. (Unit ` R ) -> x e. ( B \ { .0. } ) ) )
44	43	ssrdv 3476	. . . . 5 |- ( ( R e. Ring /\ G e. Grp ) -> (Unit ` R ) C_ ( B \ { .0. } ) )
45		eldifi 3593	. . . . . . . . . . 11 |- ( x e. ( B \ { .0. } ) -> x e. B )
46	45	adantl 467	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x e. B )
47		eqid 2429	. . . . . . . . . . . . 13 |- ( invg ` G ) = ( invg ` G )
48	19, 47	grpinvcl 16662	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. ( B \ { .0. } ) )
49	48	adantll 718	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. ( B \ { .0. } ) )
50	49	eldifad 3454	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. B )
51		eqid 2429	. . . . . . . . . . 11 |- ( ||r ` R ) = ( ||r ` R )
52	1, 51, 30	dvdsrmul 17811	. . . . . . . . . 10 |- ( ( x e. B /\ ( ( invg ` G ) ` x ) e. B ) -> x ( ||r ` R ) ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) )
53	46, 50, 52	syl2anc 665	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` R ) ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) )
54		fvex 5891	. . . . . . . . . . . . . 14 |- ( Base ` R ) e. _V
55	1, 54	eqeltri 2513	. . . . . . . . . . . . 13 |- B e. _V
56		difexg 4573	. . . . . . . . . . . . 13 |- ( B e. _V -> ( B \ { .0. } ) e. _V )
57	16, 30	mgpplusg 17662	. . . . . . . . . . . . . 14 |- ( .r ` R ) = ( +g ` (mulGrp ` R ) )
58	7, 57	ressplusg 15198	. . . . . . . . . . . . 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 16663	. . . . . . . . . . 11 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 0g ` G ) )
61	60	adantll 718	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 0g ` G ) )
62		eqid 2429	. . . . . . . . . . . . . . 15 |- ( 1r ` R ) = ( 1r ` R )
63	1, 62	ringidcl 17736	. . . . . . . . . . . . . 14 |- ( R e. Ring -> ( 1r ` R ) e. B )
64	1, 30, 62	ringlidm 17739	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ ( 1r ` R ) e. B ) -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
65	63, 64	mpdan 672	. . . . . . . . . . . . 13 |- ( R e. Ring -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
66	65	adantr 466	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ G e. Grp ) -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
67		simpr 462	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ G e. Grp ) -> G e. Grp )
68	2, 62	1unit 17821	. . . . . . . . . . . . . . 15 |- ( R e. Ring -> ( 1r ` R ) e. (Unit ` R ) )
69	68	adantr 466	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ G e. Grp ) -> ( 1r ` R ) e. (Unit ` R ) )
70	44, 69	sseldd 3471	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ G e. Grp ) -> ( 1r ` R ) e. ( B \ { .0. } ) )
71	19, 59, 20	grpid 16652	. . . . . . . . . . . . 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 665	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ G e. Grp ) -> ( ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) <-> ( 0g ` G ) = ( 1r ` R ) ) )
73	66, 72	mpbid 213	. . . . . . . . . . 11 |- ( ( R e. Ring /\ G e. Grp ) -> ( 0g ` G ) = ( 1r ` R ) )
74	73	adantr 466	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( 0g ` G ) = ( 1r ` R ) )
75	61, 74	eqtrd 2470	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 1r ` R ) )
76	53, 75	breqtrd 4450	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` R ) ( 1r ` R ) )
77		eqid 2429	. . . . . . . . . . . 12 |- (oppr ` R ) = (oppr ` R )
78	77, 1	opprbas 17792	. . . . . . . . . . 11 |- B = ( Base ` (oppr ` R ) )
79		eqid 2429	. . . . . . . . . . 11 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
80		eqid 2429	. . . . . . . . . . 11 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
81	78, 79, 80	dvdsrmul 17811	. . . . . . . . . 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 665	. . . . . . . . 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 17789	. . . . . . . . . 10 |- ( ( ( invg ` G ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) ( ( invg ` G ) ` x ) )
84	19, 59, 20, 47	grprinv 16664	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 0g ` G ) )
85	84	adantll 718	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 0g ` G ) )
86	85, 74	eqtrd 2470	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 1r ` R ) )
87	83, 86	syl5eq 2482	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) )
88	82, 87	breqtrd 4450	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
89	2, 62, 51, 77, 79	isunit 17820	. . . . . . . 8 |- ( x e. (Unit ` R ) <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
90	76, 88, 89	sylanbrc 668	. . . . . . 7 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x e. (Unit ` R ) )
91	90	ex 435	. . . . . 6 |- ( ( R e. Ring /\ G e. Grp ) -> ( x e. ( B \ { .0. } ) -> x e. (Unit ` R ) ) )
92	91	ssrdv 3476	. . . . 5 |- ( ( R e. Ring /\ G e. Grp ) -> ( B \ { .0. } ) C_ (Unit ` R ) )
93	44, 92	eqssd 3487	. . . 4 |- ( ( R e. Ring /\ G e. Grp ) -> (Unit ` R ) = ( B \ { .0. } ) )
94	12, 93	impbida 840	. . 3 |- ( R e. Ring -> ( (Unit ` R ) = ( B \ { .0. } ) <-> G e. Grp ) )
95	94	pm5.32i 641	. 2 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) <-> ( R e. Ring /\ G e. Grp ) )
96	4, 95	bitri 252	1 |- ( R e. DivRing <-> ( R e. Ring /\ G e. Grp ) )

Syntax hints:   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1870   =/= wne 2625   _Vcvv 3087   \ cdif 3439   C_ wss 3442   {csn 4002   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   ↾_s cress 15085   +g cplusg 15152   .rcmulr 15153   0gc0g 15297   Grpcgrp 16620   invgcminusg 16621  mulGrpcmgp 17658   1rcur 17670   Ringcrg 17715  opp_rcoppr 17785   ||_rcdsr 17801  Unitcui 17802  /_rcdvr 17845   DivRingcdr 17910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-drng 17912

This theorem is referenced by:  drngmgp  17922  isdrngd  17935