Theorem isdrng2 18063

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 2471	. . 3 |- (Unit ` R ) = (Unit ` R )
3		isdrng2.z	. . 3 |- .0. = ( 0g ` R )
4	1, 2, 3	isdrng 18057	. 2 |- ( R e. DivRing <-> ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) )
5		oveq2 6316	. . . . . . 7 |- ( (Unit ` R ) = ( B \ { .0. } ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) )
6	5	adantl 473	. . . . . 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 2523	. . . . 5 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) = G )
9		eqid 2471	. . . . . . 7 |- ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s (Unit ` R ) )
10	2, 9	unitgrp 17973	. . . . . 6 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. Grp )
11	10	adantr 472	. . . . 5 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. Grp )
12	8, 11	eqeltrrd 2550	. . . 4 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> G e. Grp )
13	1, 2	unitcl 17965	. . . . . . . . 9 |- ( x e. (Unit ` R ) -> x e. B )
14	13	adantl 473	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. B )
15		difss 3549	. . . . . . . . . . . . . . 15 |- ( B \ { .0. } ) C_ B
16		eqid 2471	. . . . . . . . . . . . . . . . 17 |- (mulGrp ` R ) = (mulGrp ` R )
17	16, 1	mgpbas 17807	. . . . . . . . . . . . . . . 16 |- B = ( Base ` (mulGrp ` R ) )
18	7, 17	ressbas2 15258	. . . . . . . . . . . . . . 15 |- ( ( B \ { .0. } ) C_ B -> ( B \ { .0. } ) = ( Base ` G ) )
19	15, 18	ax-mp 5	. . . . . . . . . . . . . 14 |- ( B \ { .0. } ) = ( Base ` G )
20		eqid 2471	. . . . . . . . . . . . . 14 |- ( 0g ` G ) = ( 0g ` G )
21	19, 20	grpidcl 16772	. . . . . . . . . . . . 13 |- ( G e. Grp -> ( 0g ` G ) e. ( B \ { .0. } ) )
22	21	ad2antlr 741	. . . . . . . . . . . 12 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) e. ( B \ { .0. } ) )
23		eldifsn 4088	. . . . . . . . . . . 12 |- ( ( 0g ` G ) e. ( B \ { .0. } ) <-> ( ( 0g ` G ) e. B /\ ( 0g ` G ) =/= .0. ) )
24	22, 23	sylib 201	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( ( 0g ` G ) e. B /\ ( 0g ` G ) =/= .0. ) )
25	24	simprd 470	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) =/= .0. )
26		simpll 768	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> R e. Ring )
27	22	eldifad 3402	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) e. B )
28		simpr 468	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. (Unit ` R ) )
29		eqid 2471	. . . . . . . . . . . 12 |- (/r ` R ) = (/r ` R )
30		eqid 2471	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
31	1, 2, 29, 30	dvrcan1 17997	. . . . . . . . . . 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 1292	. . . . . . . . . 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 17992	. . . . . . . . . . . 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 1292	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( ( 0g ` G ) (/r ` R ) x ) e. B )
35	1, 30, 3	ringrz 17896	. . . . . . . . . . 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 673	. . . . . . . . . 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 2720	. . . . . . . . 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 6316	. . . . . . . . . 10 |- ( x = .0. -> ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) x ) = ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) .0. ) )
39	38	necon3i 2675	. . . . . . . . 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 4088	. . . . . . . 8 |- ( x e. ( B \ { .0. } ) <-> ( x e. B /\ x =/= .0. ) )
42	14, 40, 41	sylanbrc 677	. . . . . . 7 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. ( B \ { .0. } ) )
43	42	ex 441	. . . . . 6 |- ( ( R e. Ring /\ G e. Grp ) -> ( x e. (Unit ` R ) -> x e. ( B \ { .0. } ) ) )
44	43	ssrdv 3424	. . . . 5 |- ( ( R e. Ring /\ G e. Grp ) -> (Unit ` R ) C_ ( B \ { .0. } ) )
45		eldifi 3544	. . . . . . . . . . 11 |- ( x e. ( B \ { .0. } ) -> x e. B )
46	45	adantl 473	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x e. B )
47		eqid 2471	. . . . . . . . . . . . 13 |- ( invg ` G ) = ( invg ` G )
48	19, 47	grpinvcl 16789	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. ( B \ { .0. } ) )
49	48	adantll 728	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. ( B \ { .0. } ) )
50	49	eldifad 3402	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. B )
51		eqid 2471	. . . . . . . . . . 11 |- ( ||r ` R ) = ( ||r ` R )
52	1, 51, 30	dvdsrmul 17954	. . . . . . . . . 10 |- ( ( x e. B /\ ( ( invg ` G ) ` x ) e. B ) -> x ( ||r ` R ) ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) )
53	46, 50, 52	syl2anc 673	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` R ) ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) )
54		fvex 5889	. . . . . . . . . . . . . 14 |- ( Base ` R ) e. _V
55	1, 54	eqeltri 2545	. . . . . . . . . . . . 13 |- B e. _V
56		difexg 4545	. . . . . . . . . . . . 13 |- ( B e. _V -> ( B \ { .0. } ) e. _V )
57	16, 30	mgpplusg 17805	. . . . . . . . . . . . . 14 |- ( .r ` R ) = ( +g ` (mulGrp ` R ) )
58	7, 57	ressplusg 15317	. . . . . . . . . . . . 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 16790	. . . . . . . . . . 11 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 0g ` G ) )
61	60	adantll 728	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 0g ` G ) )
62		eqid 2471	. . . . . . . . . . . . . . 15 |- ( 1r ` R ) = ( 1r ` R )
63	1, 62	ringidcl 17879	. . . . . . . . . . . . . 14 |- ( R e. Ring -> ( 1r ` R ) e. B )
64	1, 30, 62	ringlidm 17882	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ ( 1r ` R ) e. B ) -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
65	63, 64	mpdan 681	. . . . . . . . . . . . 13 |- ( R e. Ring -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
66	65	adantr 472	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ G e. Grp ) -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
67		simpr 468	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ G e. Grp ) -> G e. Grp )
68	2, 62	1unit 17964	. . . . . . . . . . . . . . 15 |- ( R e. Ring -> ( 1r ` R ) e. (Unit ` R ) )
69	68	adantr 472	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ G e. Grp ) -> ( 1r ` R ) e. (Unit ` R ) )
70	44, 69	sseldd 3419	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ G e. Grp ) -> ( 1r ` R ) e. ( B \ { .0. } ) )
71	19, 59, 20	grpid 16779	. . . . . . . . . . . . 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 673	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ G e. Grp ) -> ( ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) <-> ( 0g ` G ) = ( 1r ` R ) ) )
73	66, 72	mpbid 215	. . . . . . . . . . 11 |- ( ( R e. Ring /\ G e. Grp ) -> ( 0g ` G ) = ( 1r ` R ) )
74	73	adantr 472	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( 0g ` G ) = ( 1r ` R ) )
75	61, 74	eqtrd 2505	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 1r ` R ) )
76	53, 75	breqtrd 4420	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` R ) ( 1r ` R ) )
77		eqid 2471	. . . . . . . . . . . 12 |- (oppr ` R ) = (oppr ` R )
78	77, 1	opprbas 17935	. . . . . . . . . . 11 |- B = ( Base ` (oppr ` R ) )
79		eqid 2471	. . . . . . . . . . 11 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
80		eqid 2471	. . . . . . . . . . 11 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
81	78, 79, 80	dvdsrmul 17954	. . . . . . . . . 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 673	. . . . . . . . 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 17932	. . . . . . . . . 10 |- ( ( ( invg ` G ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) ( ( invg ` G ) ` x ) )
84	19, 59, 20, 47	grprinv 16791	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 0g ` G ) )
85	84	adantll 728	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 0g ` G ) )
86	85, 74	eqtrd 2505	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 1r ` R ) )
87	83, 86	syl5eq 2517	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) )
88	82, 87	breqtrd 4420	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
89	2, 62, 51, 77, 79	isunit 17963	. . . . . . . 8 |- ( x e. (Unit ` R ) <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
90	76, 88, 89	sylanbrc 677	. . . . . . 7 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x e. (Unit ` R ) )
91	90	ex 441	. . . . . 6 |- ( ( R e. Ring /\ G e. Grp ) -> ( x e. ( B \ { .0. } ) -> x e. (Unit ` R ) ) )
92	91	ssrdv 3424	. . . . 5 |- ( ( R e. Ring /\ G e. Grp ) -> ( B \ { .0. } ) C_ (Unit ` R ) )
93	44, 92	eqssd 3435	. . . 4 |- ( ( R e. Ring /\ G e. Grp ) -> (Unit ` R ) = ( B \ { .0. } ) )
94	12, 93	impbida 850	. . 3 |- ( R e. Ring -> ( (Unit ` R ) = ( B \ { .0. } ) <-> G e. Grp ) )
95	94	pm5.32i 649	. 2 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) <-> ( R e. Ring /\ G e. Grp ) )
96	4, 95	bitri 257	1 |- ( R e. DivRing <-> ( R e. Ring /\ G e. Grp ) )

Syntax hints:   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   _Vcvv 3031   \ cdif 3387   C_ wss 3390   {csn 3959   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   ↾_s cress 15200   +g cplusg 15268   .rcmulr 15269   0gc0g 15416   Grpcgrp 16747   invgcminusg 16748  mulGrpcmgp 17801   1rcur 17813   Ringcrg 17858  opp_rcoppr 17928   ||_rcdsr 17944  Unitcui 17945  /_rcdvr 17988   DivRingcdr 18053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752  df-mgp 17802  df-ur 17814  df-ring 17860  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-drng 18055

This theorem is referenced by:  drngmgp  18065  isdrngd  18078