Theorem unitgrp 18667

Description: The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
unitmulcl.1	|- U = (Unit ` R )
unitgrp.2	|- G = ( (mulGrp ` R ) ↾s U )

Assertion

Ref	Expression
unitgrp	|- ( R e. Ring -> G e. Grp )

Proof of Theorem unitgrp

Dummy variables x y z m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unitmulcl.1	. . . 4 |- U = (Unit ` R )
2		unitgrp.2	. . . 4 |- G = ( (mulGrp ` R ) ↾s U )
3	1, 2	unitgrpbas 18666	. . 3 |- U = ( Base ` G )
4	3	a1i 11	. 2 |- ( R e. Ring -> U = ( Base ` G ) )
5		fvex 6201	. . . 4 |- ( Base ` G ) e. _V
6	3, 5	eqeltri 2697	. . 3 |- U e. _V
7		eqid 2622	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
8		eqid 2622	. . . . 5 |- ( .r ` R ) = ( .r ` R )
9	7, 8	mgpplusg 18493	. . . 4 |- ( .r ` R ) = ( +g ` (mulGrp ` R ) )
10	2, 9	ressplusg 15993	. . 3 |- ( U e. _V -> ( .r ` R ) = ( +g ` G ) )
11	6, 10	mp1i 13	. 2 |- ( R e. Ring -> ( .r ` R ) = ( +g ` G ) )
12	1, 8	unitmulcl 18664	. 2 |- ( ( R e. Ring /\ x e. U /\ y e. U ) -> ( x ( .r ` R ) y ) e. U )
13		eqid 2622	. . . . 5 |- ( Base ` R ) = ( Base ` R )
14	13, 1	unitcl 18659	. . . 4 |- ( x e. U -> x e. ( Base ` R ) )
15	13, 1	unitcl 18659	. . . 4 |- ( y e. U -> y e. ( Base ` R ) )
16	13, 1	unitcl 18659	. . . 4 |- ( z e. U -> z e. ( Base ` R ) )
17	14, 15, 16	3anim123i 1247	. . 3 |- ( ( x e. U /\ y e. U /\ z e. U ) -> ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) )
18	13, 8	ringass 18564	. . 3 |- ( ( R e. Ring /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( .r ` R ) y ) ( .r ` R ) z ) = ( x ( .r ` R ) ( y ( .r ` R ) z ) ) )
19	17, 18	sylan2 491	. 2 |- ( ( R e. Ring /\ ( x e. U /\ y e. U /\ z e. U ) ) -> ( ( x ( .r ` R ) y ) ( .r ` R ) z ) = ( x ( .r ` R ) ( y ( .r ` R ) z ) ) )
20		eqid 2622	. . 3 |- ( 1r ` R ) = ( 1r ` R )
21	1, 20	1unit 18658	. 2 |- ( R e. Ring -> ( 1r ` R ) e. U )
22	13, 8, 20	ringlidm 18571	. . 3 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) x ) = x )
23	14, 22	sylan2 491	. 2 |- ( ( R e. Ring /\ x e. U ) -> ( ( 1r ` R ) ( .r ` R ) x ) = x )
24		simpr 477	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> x e. U )
25		eqid 2622	. . . . 5 |- ( ||r ` R ) = ( ||r ` R )
26		eqid 2622	. . . . 5 |- (oppr ` R ) = (oppr ` R )
27		eqid 2622	. . . . 5 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
28	1, 20, 25, 26, 27	isunit 18657	. . . 4 |- ( x e. U <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
29	24, 28	sylib 208	. . 3 |- ( ( R e. Ring /\ x e. U ) -> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
30	14	adantl 482	. . . . . 6 |- ( ( R e. Ring /\ x e. U ) -> x e. ( Base ` R ) )
31	13, 25, 8	dvdsr2 18647	. . . . . 6 |- ( x e. ( Base ` R ) -> ( x ( ||r ` R ) ( 1r ` R ) <-> E. y e. ( Base ` R ) ( y ( .r ` R ) x ) = ( 1r ` R ) ) )
32	30, 31	syl 17	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( x ( ||r ` R ) ( 1r ` R ) <-> E. y e. ( Base ` R ) ( y ( .r ` R ) x ) = ( 1r ` R ) ) )
33	26, 13	opprbas 18629	. . . . . . 7 |- ( Base ` R ) = ( Base ` (oppr ` R ) )
34		eqid 2622	. . . . . . 7 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
35	33, 27, 34	dvdsr2 18647	. . . . . 6 |- ( x e. ( Base ` R ) -> ( x ( ||r ` (oppr ` R ) ) ( 1r ` R ) <-> E. m e. ( Base ` R ) ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) )
36	30, 35	syl 17	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( x ( ||r ` (oppr ` R ) ) ( 1r ` R ) <-> E. m e. ( Base ` R ) ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) )
37	32, 36	anbi12d 747	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> ( ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) <-> ( E. y e. ( Base ` R ) ( y ( .r ` R ) x ) = ( 1r ` R ) /\ E. m e. ( Base ` R ) ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) )
38		reeanv 3107	. . . . 5 |- ( E. y e. ( Base ` R ) E. m e. ( Base ` R ) ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) <-> ( E. y e. ( Base ` R ) ( y ( .r ` R ) x ) = ( 1r ` R ) /\ E. m e. ( Base ` R ) ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) )
39		simprl 794	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> m e. ( Base ` R ) )
40	30	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> x e. ( Base ` R ) )
41	13, 25, 8	dvdsrmul 18648	. . . . . . . . . . . 12 |- ( ( m e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> m ( ||r ` R ) ( x ( .r ` R ) m ) )
42	39, 40, 41	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> m ( ||r ` R ) ( x ( .r ` R ) m ) )
43		simplll 798	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> R e. Ring )
44		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> y e. ( Base ` R ) )
45	13, 8	ringass 18564	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ ( y e. ( Base ` R ) /\ x e. ( Base ` R ) /\ m e. ( Base ` R ) ) ) -> ( ( y ( .r ` R ) x ) ( .r ` R ) m ) = ( y ( .r ` R ) ( x ( .r ` R ) m ) ) )
46	43, 44, 40, 39, 45	syl13anc 1328	. . . . . . . . . . . . 13 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( ( y ( .r ` R ) x ) ( .r ` R ) m ) = ( y ( .r ` R ) ( x ( .r ` R ) m ) ) )
47		simprrl 804	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( y ( .r ` R ) x ) = ( 1r ` R ) )
48	47	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( ( y ( .r ` R ) x ) ( .r ` R ) m ) = ( ( 1r ` R ) ( .r ` R ) m ) )
49	13, 8, 26, 34	opprmul 18626	. . . . . . . . . . . . . . 15 |- ( m ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) m )
50		simprrr 805	. . . . . . . . . . . . . . 15 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) )
51	49, 50	syl5eqr 2670	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( x ( .r ` R ) m ) = ( 1r ` R ) )
52	51	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( y ( .r ` R ) ( x ( .r ` R ) m ) ) = ( y ( .r ` R ) ( 1r ` R ) ) )
53	46, 48, 52	3eqtr3d 2664	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( ( 1r ` R ) ( .r ` R ) m ) = ( y ( .r ` R ) ( 1r ` R ) ) )
54	13, 8, 20	ringlidm 18571	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ m e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) m ) = m )
55	43, 39, 54	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( ( 1r ` R ) ( .r ` R ) m ) = m )
56	13, 8, 20	ringridm 18572	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ y e. ( Base ` R ) ) -> ( y ( .r ` R ) ( 1r ` R ) ) = y )
57	43, 44, 56	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( y ( .r ` R ) ( 1r ` R ) ) = y )
58	53, 55, 57	3eqtr3d 2664	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> m = y )
59	42, 58, 51	3brtr3d 4684	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> y ( ||r ` R ) ( 1r ` R ) )
60	33, 27, 34	dvdsrmul 18648	. . . . . . . . . . . 12 |- ( ( y e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> y ( ||r ` (oppr ` R ) ) ( x ( .r ` (oppr ` R ) ) y ) )
61	44, 40, 60	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> y ( ||r ` (oppr ` R ) ) ( x ( .r ` (oppr ` R ) ) y ) )
62	13, 8, 26, 34	opprmul 18626	. . . . . . . . . . . 12 |- ( x ( .r ` (oppr ` R ) ) y ) = ( y ( .r ` R ) x )
63	62, 47	syl5eq 2668	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( 1r ` R ) )
64	61, 63	breqtrd 4679	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> y ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
65	1, 20, 25, 26, 27	isunit 18657	. . . . . . . . . 10 |- ( y e. U <-> ( y ( ||r ` R ) ( 1r ` R ) /\ y ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
66	59, 64, 65	sylanbrc 698	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> y e. U )
67	66, 47	jca 554	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) /\ ( m e. ( Base ` R ) /\ ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) ) -> ( y e. U /\ ( y ( .r ` R ) x ) = ( 1r ` R ) ) )
68	67	rexlimdvaa 3032	. . . . . . 7 |- ( ( ( R e. Ring /\ x e. U ) /\ y e. ( Base ` R ) ) -> ( E. m e. ( Base ` R ) ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) -> ( y e. U /\ ( y ( .r ` R ) x ) = ( 1r ` R ) ) ) )
69	68	expimpd 629	. . . . . 6 |- ( ( R e. Ring /\ x e. U ) -> ( ( y e. ( Base ` R ) /\ E. m e. ( Base ` R ) ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) ) -> ( y e. U /\ ( y ( .r ` R ) x ) = ( 1r ` R ) ) ) )
70	69	reximdv2 3014	. . . . 5 |- ( ( R e. Ring /\ x e. U ) -> ( E. y e. ( Base ` R ) E. m e. ( Base ` R ) ( ( y ( .r ` R ) x ) = ( 1r ` R ) /\ ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) -> E. y e. U ( y ( .r ` R ) x ) = ( 1r ` R ) ) )
71	38, 70	syl5bir 233	. . . 4 |- ( ( R e. Ring /\ x e. U ) -> ( ( E. y e. ( Base ` R ) ( y ( .r ` R ) x ) = ( 1r ` R ) /\ E. m e. ( Base ` R ) ( m ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) ) -> E. y e. U ( y ( .r ` R ) x ) = ( 1r ` R ) ) )
72	37, 71	sylbid 230	. . 3 |- ( ( R e. Ring /\ x e. U ) -> ( ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) -> E. y e. U ( y ( .r ` R ) x ) = ( 1r ` R ) ) )
73	29, 72	mpd 15	. 2 |- ( ( R e. Ring /\ x e. U ) -> E. y e. U ( y ( .r ` R ) x ) = ( 1r ` R ) )
74	4, 11, 12, 19, 21, 23, 73	isgrpde 17443	1 |- ( R e. Ring -> G e. Grp )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942   Grpcgrp 17422  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547  opp_rcoppr 18622   ||_rcdsr 18638  Unitcui 18639

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-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-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642

This theorem is referenced by:  unitabl  18668  unitsubm  18670  unitinvcl  18674  unitinvinv  18675  unitlinv  18677  unitrinv  18678  isdrng2  18757  subrgugrp  18799  expghm  19844  invrvald  20482  nrginvrcn  22496  nrgtdrg  22497  dchrfi  24980  dchrghm  24981  dchrabs  24985  dchrptlem1  24989  dchrptlem2  24990  dchrptlem3  24991  dchrsum2  24993  rdivmuldivd  29791  dvrcan5  29793  rhmunitinv  29822  idomodle  37774  proot1mul  37777  proot1hash  37778  proot1ex  37779