Theorem isdrng2 18757

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 2622	. . 3 |- (Unit ` R ) = (Unit ` R )
3		isdrng2.z	. . 3 |- .0. = ( 0g ` R )
4	1, 2, 3	isdrng 18751	. 2 |- ( R e. DivRing <-> ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) )
5		oveq2 6658	. . . . . . 7 |- ( (Unit ` R ) = ( B \ { .0. } ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) )
6	5	adantl 482	. . . . . 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 2674	. . . . 5 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) = G )
9		eqid 2622	. . . . . . 7 |- ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s (Unit ` R ) )
10	2, 9	unitgrp 18667	. . . . . 6 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. Grp )
11	10	adantr 481	. . . . 5 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. Grp )
12	8, 11	eqeltrrd 2702	. . . 4 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> G e. Grp )
13	1, 2	unitcl 18659	. . . . . . . . 9 |- ( x e. (Unit ` R ) -> x e. B )
14	13	adantl 482	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. B )
15		difss 3737	. . . . . . . . . . . . . . 15 |- ( B \ { .0. } ) C_ B
16		eqid 2622	. . . . . . . . . . . . . . . . 17 |- (mulGrp ` R ) = (mulGrp ` R )
17	16, 1	mgpbas 18495	. . . . . . . . . . . . . . . 16 |- B = ( Base ` (mulGrp ` R ) )
18	7, 17	ressbas2 15931	. . . . . . . . . . . . . . 15 |- ( ( B \ { .0. } ) C_ B -> ( B \ { .0. } ) = ( Base ` G ) )
19	15, 18	ax-mp 5	. . . . . . . . . . . . . 14 |- ( B \ { .0. } ) = ( Base ` G )
20		eqid 2622	. . . . . . . . . . . . . 14 |- ( 0g ` G ) = ( 0g ` G )
21	19, 20	grpidcl 17450	. . . . . . . . . . . . 13 |- ( G e. Grp -> ( 0g ` G ) e. ( B \ { .0. } ) )
22	21	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) e. ( B \ { .0. } ) )
23		eldifsn 4317	. . . . . . . . . . . 12 |- ( ( 0g ` G ) e. ( B \ { .0. } ) <-> ( ( 0g ` G ) e. B /\ ( 0g ` G ) =/= .0. ) )
24	22, 23	sylib 208	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( ( 0g ` G ) e. B /\ ( 0g ` G ) =/= .0. ) )
25	24	simprd 479	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) =/= .0. )
26		simpll 790	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> R e. Ring )
27	22	eldifad 3586	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) e. B )
28		simpr 477	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. (Unit ` R ) )
29		eqid 2622	. . . . . . . . . . . 12 |- (/r ` R ) = (/r ` R )
30		eqid 2622	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
31	1, 2, 29, 30	dvrcan1 18691	. . . . . . . . . . 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 1326	. . . . . . . . . 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 18686	. . . . . . . . . . . 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 1326	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( ( 0g ` G ) (/r ` R ) x ) e. B )
35	1, 30, 3	ringrz 18588	. . . . . . . . . . 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 693	. . . . . . . . . 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 2871	. . . . . . . . 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 6658	. . . . . . . . . 10 |- ( x = .0. -> ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) x ) = ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) .0. ) )
39	38	necon3i 2826	. . . . . . . . 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 4317	. . . . . . . 8 |- ( x e. ( B \ { .0. } ) <-> ( x e. B /\ x =/= .0. ) )
42	14, 40, 41	sylanbrc 698	. . . . . . 7 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. ( B \ { .0. } ) )
43	42	ex 450	. . . . . 6 |- ( ( R e. Ring /\ G e. Grp ) -> ( x e. (Unit ` R ) -> x e. ( B \ { .0. } ) ) )
44	43	ssrdv 3609	. . . . 5 |- ( ( R e. Ring /\ G e. Grp ) -> (Unit ` R ) C_ ( B \ { .0. } ) )
45		eldifi 3732	. . . . . . . . . . 11 |- ( x e. ( B \ { .0. } ) -> x e. B )
46	45	adantl 482	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x e. B )
47		eqid 2622	. . . . . . . . . . . . 13 |- ( invg ` G ) = ( invg ` G )
48	19, 47	grpinvcl 17467	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. ( B \ { .0. } ) )
49	48	adantll 750	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. ( B \ { .0. } ) )
50	49	eldifad 3586	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. B )
51		eqid 2622	. . . . . . . . . . 11 |- ( ||r ` R ) = ( ||r ` R )
52	1, 51, 30	dvdsrmul 18648	. . . . . . . . . 10 |- ( ( x e. B /\ ( ( invg ` G ) ` x ) e. B ) -> x ( ||r ` R ) ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) )
53	46, 50, 52	syl2anc 693	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` R ) ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) )
54		fvex 6201	. . . . . . . . . . . . . 14 |- ( Base ` R ) e. _V
55	1, 54	eqeltri 2697	. . . . . . . . . . . . 13 |- B e. _V
56		difexg 4808	. . . . . . . . . . . . 13 |- ( B e. _V -> ( B \ { .0. } ) e. _V )
57	16, 30	mgpplusg 18493	. . . . . . . . . . . . . 14 |- ( .r ` R ) = ( +g ` (mulGrp ` R ) )
58	7, 57	ressplusg 15993	. . . . . . . . . . . . 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 17468	. . . . . . . . . . 11 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 0g ` G ) )
61	60	adantll 750	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 0g ` G ) )
62		eqid 2622	. . . . . . . . . . . . . . 15 |- ( 1r ` R ) = ( 1r ` R )
63	1, 62	ringidcl 18568	. . . . . . . . . . . . . 14 |- ( R e. Ring -> ( 1r ` R ) e. B )
64	1, 30, 62	ringlidm 18571	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ ( 1r ` R ) e. B ) -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
65	63, 64	mpdan 702	. . . . . . . . . . . . 13 |- ( R e. Ring -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
66	65	adantr 481	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ G e. Grp ) -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
67		simpr 477	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ G e. Grp ) -> G e. Grp )
68	2, 62	1unit 18658	. . . . . . . . . . . . . . 15 |- ( R e. Ring -> ( 1r ` R ) e. (Unit ` R ) )
69	68	adantr 481	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ G e. Grp ) -> ( 1r ` R ) e. (Unit ` R ) )
70	44, 69	sseldd 3604	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ G e. Grp ) -> ( 1r ` R ) e. ( B \ { .0. } ) )
71	19, 59, 20	grpid 17457	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ G e. Grp ) -> ( ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) <-> ( 0g ` G ) = ( 1r ` R ) ) )
73	66, 72	mpbid 222	. . . . . . . . . . 11 |- ( ( R e. Ring /\ G e. Grp ) -> ( 0g ` G ) = ( 1r ` R ) )
74	73	adantr 481	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( 0g ` G ) = ( 1r ` R ) )
75	61, 74	eqtrd 2656	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 1r ` R ) )
76	53, 75	breqtrd 4679	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` R ) ( 1r ` R ) )
77		eqid 2622	. . . . . . . . . . . 12 |- (oppr ` R ) = (oppr ` R )
78	77, 1	opprbas 18629	. . . . . . . . . . 11 |- B = ( Base ` (oppr ` R ) )
79		eqid 2622	. . . . . . . . . . 11 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
80		eqid 2622	. . . . . . . . . . 11 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
81	78, 79, 80	dvdsrmul 18648	. . . . . . . . . 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 693	. . . . . . . . 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 18626	. . . . . . . . . 10 |- ( ( ( invg ` G ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) ( ( invg ` G ) ` x ) )
84	19, 59, 20, 47	grprinv 17469	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 0g ` G ) )
85	84	adantll 750	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 0g ` G ) )
86	85, 74	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 1r ` R ) )
87	83, 86	syl5eq 2668	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) )
88	82, 87	breqtrd 4679	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
89	2, 62, 51, 77, 79	isunit 18657	. . . . . . . 8 |- ( x e. (Unit ` R ) <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
90	76, 88, 89	sylanbrc 698	. . . . . . 7 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x e. (Unit ` R ) )
91	90	ex 450	. . . . . 6 |- ( ( R e. Ring /\ G e. Grp ) -> ( x e. ( B \ { .0. } ) -> x e. (Unit ` R ) ) )
92	91	ssrdv 3609	. . . . 5 |- ( ( R e. Ring /\ G e. Grp ) -> ( B \ { .0. } ) C_ (Unit ` R ) )
93	44, 92	eqssd 3620	. . . 4 |- ( ( R e. Ring /\ G e. Grp ) -> (Unit ` R ) = ( B \ { .0. } ) )
94	12, 93	impbida 877	. . 3 |- ( R e. Ring -> ( (Unit ` R ) = ( B \ { .0. } ) <-> G e. Grp ) )
95	94	pm5.32i 669	. 2 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) <-> ( R e. Ring /\ G e. Grp ) )
96	4, 95	bitri 264	1 |- ( R e. DivRing <-> ( R e. Ring /\ G e. Grp ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547  opp_rcoppr 18622   ||_rcdsr 18638  Unitcui 18639  /_rcdvr 18682   DivRingcdr 18747

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

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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749

This theorem is referenced by:  drngmgp  18759  isdrngd  18772