Theorem aprlring 14602

Description: A ring is a local ring if and only if the relation given by df-apr 14592 is an apartness relation. (Contributed by Jim Kingdon, 28-May-2026.)

Assertion

Ref	Expression
aprlring	|- ( R e. Ring -> ( R e. LRing <-> (#r ` R ) Ap ( Base ` R ) ) )

Proof of Theorem aprlring

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

Step	Hyp	Ref	Expression
1		aprap 14600	. 2 |- ( R e. LRing -> (#r ` R ) Ap ( Base ` R ) )
2		aprnzr 14601	. . . 4 |- ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) -> R e. NzRing )
3		simplll 539	. . . . . . . . . . . . 13 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> R e. Ring )
4		simplrl 541	. . . . . . . . . . . . 13 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> x e. ( Base ` R ) )
5		simplrr 542	. . . . . . . . . . . . 13 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> y e. ( Base ` R ) )
6		eqid 2238	. . . . . . . . . . . . . 14 |- ( Base ` R ) = ( Base ` R )
7		eqid 2238	. . . . . . . . . . . . . 14 |- ( +g ` R ) = ( +g ` R )
8	6, 7	ringcom 14338	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( +g ` R ) y ) = ( y ( +g ` R ) x ) )
9	3, 4, 5, 8	syl3anc 1278	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( x ( +g ` R ) y ) = ( y ( +g ` R ) x ) )
10	9	oveq1d 6100	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( ( x ( +g ` R ) y ) ( -g ` R ) x ) = ( ( y ( +g ` R ) x ) ( -g ` R ) x ) )
11		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( x ( +g ` R ) y ) = ( 1r ` R ) )
12	11	oveq1d 6100	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( ( x ( +g ` R ) y ) ( -g ` R ) x ) = ( ( 1r ` R ) ( -g ` R ) x ) )
13	3	ringgrpd 14311	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> R e. Grp )
14		eqid 2238	. . . . . . . . . . . . 13 |- ( -g ` R ) = ( -g ` R )
15	6, 7, 14	grppncan 13898	. . . . . . . . . . . 12 |- ( ( R e. Grp /\ y e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> ( ( y ( +g ` R ) x ) ( -g ` R ) x ) = y )
16	13, 5, 4, 15	syl3anc 1278	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( ( y ( +g ` R ) x ) ( -g ` R ) x ) = y )
17	10, 12, 16	3eqtr3d 2279	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( ( 1r ` R ) ( -g ` R ) x ) = y )
18	17	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 1r ` R ) (#r ` R ) x ) -> ( ( 1r ` R ) ( -g ` R ) x ) = y )
19		eqidd 2239	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
20		eqidd 2239	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> (#r ` R ) = (#r ` R ) )
21		eqidd 2239	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( -g ` R ) = ( -g ` R ) )
22		eqidd 2239	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> (Unit ` R ) = (Unit ` R ) )
23		eqid 2238	. . . . . . . . . . . . 13 |- ( 1r ` R ) = ( 1r ` R )
24	6, 23	ringidcl 14327	. . . . . . . . . . . 12 |- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) )
25	3, 24	syl 14	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( 1r ` R ) e. ( Base ` R ) )
26	19, 20, 21, 22, 3, 25, 4	aprval 14593	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( ( 1r ` R ) (#r ` R ) x <-> ( ( 1r ` R ) ( -g ` R ) x ) e. (Unit ` R ) ) )
27	26	biimpa 296	. . . . . . . . 9 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 1r ` R ) (#r ` R ) x ) -> ( ( 1r ` R ) ( -g ` R ) x ) e. (Unit ` R ) )
28	18, 27	eqeltrrd 2316	. . . . . . . 8 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 1r ` R ) (#r ` R ) x ) -> y e. (Unit ` R ) )
29	28	olcd 746	. . . . . . 7 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 1r ` R ) (#r ` R ) x ) -> ( x e. (Unit ` R ) \/ y e. (Unit ` R ) ) )
30		eqid 2238	. . . . . . . . . . . 12 |- ( 0g ` R ) = ( 0g ` R )
31	6, 30, 14	grpsubid1 13892	. . . . . . . . . . 11 |- ( ( R e. Grp /\ x e. ( Base ` R ) ) -> ( x ( -g ` R ) ( 0g ` R ) ) = x )
32	13, 4, 31	syl2anc 415	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( x ( -g ` R ) ( 0g ` R ) ) = x )
33	32	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 0g ` R ) (#r ` R ) x ) -> ( x ( -g ` R ) ( 0g ` R ) ) = x )
34		simpllr 540	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> (#r ` R ) Ap ( Base ` R ) )
35	34	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 0g ` R ) (#r ` R ) x ) -> (#r ` R ) Ap ( Base ` R ) )
36	6, 30	grpidcl 13836	. . . . . . . . . . . . 13 |- ( R e. Grp -> ( 0g ` R ) e. ( Base ` R ) )
37	13, 36	syl 14	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( 0g ` R ) e. ( Base ` R ) )
38	37	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 0g ` R ) (#r ` R ) x ) -> ( 0g ` R ) e. ( Base ` R ) )
39	4	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 0g ` R ) (#r ` R ) x ) -> x e. ( Base ` R ) )
40		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 0g ` R ) (#r ` R ) x ) -> ( 0g ` R ) (#r ` R ) x )
41	35, 38, 39, 40	papsym 7612	. . . . . . . . . 10 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 0g ` R ) (#r ` R ) x ) -> x (#r ` R ) ( 0g ` R ) )
42	19, 20, 21, 22, 3, 4, 37	aprval 14593	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( x (#r ` R ) ( 0g ` R ) <-> ( x ( -g ` R ) ( 0g ` R ) ) e. (Unit ` R ) ) )
43	42	adantr 276	. . . . . . . . . 10 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 0g ` R ) (#r ` R ) x ) -> ( x (#r ` R ) ( 0g ` R ) <-> ( x ( -g ` R ) ( 0g ` R ) ) e. (Unit ` R ) ) )
44	41, 43	mpbid 147	. . . . . . . . 9 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 0g ` R ) (#r ` R ) x ) -> ( x ( -g ` R ) ( 0g ` R ) ) e. (Unit ` R ) )
45	33, 44	eqeltrrd 2316	. . . . . . . 8 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 0g ` R ) (#r ` R ) x ) -> x e. (Unit ` R ) )
46	45	orcd 745	. . . . . . 7 |- ( ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) /\ ( 0g ` R ) (#r ` R ) x ) -> ( x e. (Unit ` R ) \/ y e. (Unit ` R ) ) )
47	6, 30, 14	grpsubid1 13892	. . . . . . . . . . 11 |- ( ( R e. Grp /\ ( 1r ` R ) e. ( Base ` R ) ) -> ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) = ( 1r ` R ) )
48	13, 25, 47	syl2anc 415	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) = ( 1r ` R ) )
49		eqid 2238	. . . . . . . . . . . 12 |- (Unit ` R ) = (Unit ` R )
50	49, 23	1unit 14416	. . . . . . . . . . 11 |- ( R e. Ring -> ( 1r ` R ) e. (Unit ` R ) )
51	3, 50	syl 14	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( 1r ` R ) e. (Unit ` R ) )
52	48, 51	eqeltrd 2315	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) e. (Unit ` R ) )
53	19, 20, 21, 22, 3, 25, 37	aprval 14593	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( ( 1r ` R ) (#r ` R ) ( 0g ` R ) <-> ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) e. (Unit ` R ) ) )
54	52, 53	mpbird 167	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( 1r ` R ) (#r ` R ) ( 0g ` R ) )
55	34, 25, 37, 54, 4	papcotr 7613	. . . . . . 7 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( ( 1r ` R ) (#r ` R ) x \/ ( 0g ` R ) (#r ` R ) x ) )
56	29, 46, 55	mpjaodan 810	. . . . . 6 |- ( ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) /\ ( x ( +g ` R ) y ) = ( 1r ` R ) ) -> ( x e. (Unit ` R ) \/ y e. (Unit ` R ) ) )
57	56	ex 115	. . . . 5 |- ( ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( ( x ( +g ` R ) y ) = ( 1r ` R ) -> ( x e. (Unit ` R ) \/ y e. (Unit ` R ) ) ) )
58	57	ralrimivva 2632	. . . 4 |- ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) -> A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( +g ` R ) y ) = ( 1r ` R ) -> ( x e. (Unit ` R ) \/ y e. (Unit ` R ) ) ) )
59	6, 7, 23, 49	islring 14501	. . . 4 |- ( R e. LRing <-> ( R e. NzRing /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( +g ` R ) y ) = ( 1r ` R ) -> ( x e. (Unit ` R ) \/ y e. (Unit ` R ) ) ) ) )
60	2, 58, 59	sylanbrc 421	. . 3 |- ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) -> R e. LRing )
61	60	ex 115	. 2 |- ( R e. Ring -> ( (#r ` R ) Ap ( Base ` R ) -> R e. LRing ) )
62	1, 61	impbid2 143	1 |- ( R e. Ring -> ( R e. LRing <-> (#r ` R ) Ap ( Base ` R ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   Ap wap 7607   Basecbs 13354   +g cplusg 13433   0gc0g 13612   Grpcgrp 13807   -gcsg 13809   1rcur 14264   Ringcrg 14302  Unitcui 14395  NzRingcnzr 14488  LRingclring 14499  #_rcapr 14591

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-tpos 6516  df-pap 7608  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9306  df-2 9364  df-3 9365  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-iress 13362  df-plusg 13446  df-mulr 13447  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-minusg 13811  df-sbg 13812  df-cmn 14091  df-abl 14092  df-mgp 14220  df-ur 14265  df-srg 14270  df-ring 14304  df-oppr 14375  df-dvdsr 14397  df-unit 14398  df-invr 14430  df-dvr 14441  df-nzr 14489  df-lring 14500  df-apr 14592

This theorem is used by:  drnglring  14609  opprdrng  14622