Theorem isdrng2 16840

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 2441	. . 3 |- (Unit ` R ) = (Unit ` R )
3		isdrng2.z	. . 3 |- .0. = ( 0g ` R )
4	1, 2, 3	isdrng 16834	. 2 |- ( R e. DivRing <-> ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) )
5		oveq2 6097	. . . . . . 7 |- ( (Unit ` R ) = ( B \ { .0. } ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s ( B \ { .0. } ) ) )
6	5	adantl 466	. . . . . 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 2491	. . . . 5 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) = G )
9		eqid 2441	. . . . . . 7 |- ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s (Unit ` R ) )
10	2, 9	unitgrp 16757	. . . . . 6 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. Grp )
11	10	adantr 465	. . . . 5 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. Grp )
12	8, 11	eqeltrrd 2516	. . . 4 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) -> G e. Grp )
13	1, 2	unitcl 16749	. . . . . . . . 9 |- ( x e. (Unit ` R ) -> x e. B )
14	13	adantl 466	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. B )
15		difss 3481	. . . . . . . . . . . . . . 15 |- ( B \ { .0. } ) C_ B
16		eqid 2441	. . . . . . . . . . . . . . . . 17 |- (mulGrp ` R ) = (mulGrp ` R )
17	16, 1	mgpbas 16595	. . . . . . . . . . . . . . . 16 |- B = ( Base ` (mulGrp ` R ) )
18	7, 17	ressbas2 14227	. . . . . . . . . . . . . . 15 |- ( ( B \ { .0. } ) C_ B -> ( B \ { .0. } ) = ( Base ` G ) )
19	15, 18	ax-mp 5	. . . . . . . . . . . . . 14 |- ( B \ { .0. } ) = ( Base ` G )
20		eqid 2441	. . . . . . . . . . . . . 14 |- ( 0g ` G ) = ( 0g ` G )
21	19, 20	grpidcl 15564	. . . . . . . . . . . . 13 |- ( G e. Grp -> ( 0g ` G ) e. ( B \ { .0. } ) )
22	21	ad2antlr 726	. . . . . . . . . . . 12 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) e. ( B \ { .0. } ) )
23		eldifsn 3998	. . . . . . . . . . . 12 |- ( ( 0g ` G ) e. ( B \ { .0. } ) <-> ( ( 0g ` G ) e. B /\ ( 0g ` G ) =/= .0. ) )
24	22, 23	sylib 196	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( ( 0g ` G ) e. B /\ ( 0g ` G ) =/= .0. ) )
25	24	simprd 463	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) =/= .0. )
26		simpll 753	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> R e. Ring )
27	22	eldifad 3338	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( 0g ` G ) e. B )
28		simpr 461	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. (Unit ` R ) )
29		eqid 2441	. . . . . . . . . . . 12 |- (/r ` R ) = (/r ` R )
30		eqid 2441	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
31	1, 2, 29, 30	dvrcan1 16781	. . . . . . . . . . 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 1218	. . . . . . . . . 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 16776	. . . . . . . . . . . 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 1218	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> ( ( 0g ` G ) (/r ` R ) x ) e. B )
35	1, 30, 3	rngrz 16680	. . . . . . . . . . 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 661	. . . . . . . . . 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 2633	. . . . . . . . 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 6097	. . . . . . . . . 10 |- ( x = .0. -> ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) x ) = ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) .0. ) )
39	38	necon3i 2648	. . . . . . . . 9 |- ( ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) x ) =/= ( ( ( 0g ` G ) (/r ` R ) x ) ( .r ` R ) .0. ) -> x =/= .0. )
40	37, 39	syl 16	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x =/= .0. )
41		eldifsn 3998	. . . . . . . 8 |- ( x e. ( B \ { .0. } ) <-> ( x e. B /\ x =/= .0. ) )
42	14, 40, 41	sylanbrc 664	. . . . . . 7 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. (Unit ` R ) ) -> x e. ( B \ { .0. } ) )
43	42	ex 434	. . . . . 6 |- ( ( R e. Ring /\ G e. Grp ) -> ( x e. (Unit ` R ) -> x e. ( B \ { .0. } ) ) )
44	43	ssrdv 3360	. . . . 5 |- ( ( R e. Ring /\ G e. Grp ) -> (Unit ` R ) C_ ( B \ { .0. } ) )
45		eldifi 3476	. . . . . . . . . . 11 |- ( x e. ( B \ { .0. } ) -> x e. B )
46	45	adantl 466	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x e. B )
47		eqid 2441	. . . . . . . . . . . . 13 |- ( invg ` G ) = ( invg ` G )
48	19, 47	grpinvcl 15581	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. ( B \ { .0. } ) )
49	48	adantll 713	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. ( B \ { .0. } ) )
50	49	eldifad 3338	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( invg ` G ) ` x ) e. B )
51		eqid 2441	. . . . . . . . . . 11 |- ( ||r ` R ) = ( ||r ` R )
52	1, 51, 30	dvdsrmul 16738	. . . . . . . . . 10 |- ( ( x e. B /\ ( ( invg ` G ) ` x ) e. B ) -> x ( ||r ` R ) ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) )
53	46, 50, 52	syl2anc 661	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` R ) ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) )
54		fvex 5699	. . . . . . . . . . . . . 14 |- ( Base ` R ) e. _V
55	1, 54	eqeltri 2511	. . . . . . . . . . . . 13 |- B e. _V
56		difexg 4438	. . . . . . . . . . . . 13 |- ( B e. _V -> ( B \ { .0. } ) e. _V )
57	16, 30	mgpplusg 16593	. . . . . . . . . . . . . 14 |- ( .r ` R ) = ( +g ` (mulGrp ` R ) )
58	7, 57	ressplusg 14278	. . . . . . . . . . . . 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 15582	. . . . . . . . . . 11 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 0g ` G ) )
61	60	adantll 713	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 0g ` G ) )
62		eqid 2441	. . . . . . . . . . . . . . 15 |- ( 1r ` R ) = ( 1r ` R )
63	1, 62	rngidcl 16663	. . . . . . . . . . . . . 14 |- ( R e. Ring -> ( 1r ` R ) e. B )
64	1, 30, 62	rnglidm 16666	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ ( 1r ` R ) e. B ) -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
65	63, 64	mpdan 668	. . . . . . . . . . . . 13 |- ( R e. Ring -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
66	65	adantr 465	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ G e. Grp ) -> ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) )
67		simpr 461	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ G e. Grp ) -> G e. Grp )
68	2, 62	1unit 16748	. . . . . . . . . . . . . . 15 |- ( R e. Ring -> ( 1r ` R ) e. (Unit ` R ) )
69	68	adantr 465	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ G e. Grp ) -> ( 1r ` R ) e. (Unit ` R ) )
70	44, 69	sseldd 3355	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ G e. Grp ) -> ( 1r ` R ) e. ( B \ { .0. } ) )
71	19, 59, 20	grpid 15571	. . . . . . . . . . . . 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 661	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ G e. Grp ) -> ( ( ( 1r ` R ) ( .r ` R ) ( 1r ` R ) ) = ( 1r ` R ) <-> ( 0g ` G ) = ( 1r ` R ) ) )
73	66, 72	mpbid 210	. . . . . . . . . . 11 |- ( ( R e. Ring /\ G e. Grp ) -> ( 0g ` G ) = ( 1r ` R ) )
74	73	adantr 465	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( 0g ` G ) = ( 1r ` R ) )
75	61, 74	eqtrd 2473	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` R ) x ) = ( 1r ` R ) )
76	53, 75	breqtrd 4314	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` R ) ( 1r ` R ) )
77		eqid 2441	. . . . . . . . . . . 12 |- (oppr ` R ) = (oppr ` R )
78	77, 1	opprbas 16719	. . . . . . . . . . 11 |- B = ( Base ` (oppr ` R ) )
79		eqid 2441	. . . . . . . . . . 11 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
80		eqid 2441	. . . . . . . . . . 11 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
81	78, 79, 80	dvdsrmul 16738	. . . . . . . . . 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 661	. . . . . . . . 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 16716	. . . . . . . . . 10 |- ( ( ( invg ` G ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) ( ( invg ` G ) ` x ) )
84	19, 59, 20, 47	grprinv 15583	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 0g ` G ) )
85	84	adantll 713	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 0g ` G ) )
86	85, 74	eqtrd 2473	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( x ( .r ` R ) ( ( invg ` G ) ` x ) ) = ( 1r ` R ) )
87	83, 86	syl5eq 2485	. . . . . . . . 9 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> ( ( ( invg ` G ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) )
88	82, 87	breqtrd 4314	. . . . . . . 8 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
89	2, 62, 51, 77, 79	isunit 16747	. . . . . . . 8 |- ( x e. (Unit ` R ) <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
90	76, 88, 89	sylanbrc 664	. . . . . . 7 |- ( ( ( R e. Ring /\ G e. Grp ) /\ x e. ( B \ { .0. } ) ) -> x e. (Unit ` R ) )
91	90	ex 434	. . . . . 6 |- ( ( R e. Ring /\ G e. Grp ) -> ( x e. ( B \ { .0. } ) -> x e. (Unit ` R ) ) )
92	91	ssrdv 3360	. . . . 5 |- ( ( R e. Ring /\ G e. Grp ) -> ( B \ { .0. } ) C_ (Unit ` R ) )
93	44, 92	eqssd 3371	. . . 4 |- ( ( R e. Ring /\ G e. Grp ) -> (Unit ` R ) = ( B \ { .0. } ) )
94	12, 93	impbida 828	. . 3 |- ( R e. Ring -> ( (Unit ` R ) = ( B \ { .0. } ) <-> G e. Grp ) )
95	94	pm5.32i 637	. 2 |- ( ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) <-> ( R e. Ring /\ G e. Grp ) )
96	4, 95	bitri 249	1 |- ( R e. DivRing <-> ( R e. Ring /\ G e. Grp ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2604   _Vcvv 2970   \ cdif 3323   C_ wss 3326   {csn 3875   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   Basecbs 14172   ↾_s cress 14173   +g cplusg 14236   .rcmulr 14237   0gc0g 14376   Grpcgrp 15408   invgcminusg 15409  mulGrpcmgp 16589   1rcur 16601   Ringcrg 16643  opp_rcoppr 16712   ||_rcdsr 16728  Unitcui 16729  /_rcdvr 16772   DivRingcdr 16830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-tpos 6743  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-0g 14378  df-mnd 15413  df-grp 15543  df-minusg 15544  df-mgp 16590  df-ur 16602  df-rng 16645  df-oppr 16713  df-dvdsr 16731  df-unit 16732  df-invr 16762  df-dvr 16773  df-drng 16832

This theorem is referenced by:  drngmgp  16842  isdrngd  16855