Theorem aprlring 14523

Description: A ring is a local ring if and only if the relation given by df-apr 14513 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 14521	. 2 |- ( R e. LRing -> (#r ` R ) Ap ( Base ` R ) )
2		aprnzr 14522	. . . 4 |- ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) -> R e. NzRing )
3		simplll 535	. . . . . . . . . . . . 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 537	. . . . . . . . . . . . 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 538	. . . . . . . . . . . . 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 2234	. . . . . . . . . . . . . 14 |- ( Base ` R ) = ( Base ` R )
7		eqid 2234	. . . . . . . . . . . . . 14 |- ( +g ` R ) = ( +g ` R )
8	6, 7	ringcom 14259	. . . . . . . . . . . . 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 1274	. . . . . . . . . . . 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 6073	. . . . . . . . . . 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 6073	. . . . . . . . . . 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 14233	. . . . . . . . . . . 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 2234	. . . . . . . . . . . . 13 |- ( -g ` R ) = ( -g ` R )
15	6, 7, 14	grppncan 13888	. . . . . . . . . . . 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 1274	. . . . . . . . . . 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 2275	. . . . . . . . . 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 2235	. . . . . . . . . . 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 2235	. . . . . . . . . . 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 2235	. . . . . . . . . . 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 2235	. . . . . . . . . . 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 2234	. . . . . . . . . . . . 13 |- ( 1r ` R ) = ( 1r ` R )
24	6, 23	ringidcl 14248	. . . . . . . . . . . 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 14514	. . . . . . . . . 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 2312	. . . . . . . 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 742	. . . . . . 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 2234	. . . . . . . . . . . 12 |- ( 0g ` R ) = ( 0g ` R )
31	6, 30, 14	grpsubid1 13882	. . . . . . . . . . 11 |- ( ( R e. Grp /\ x e. ( Base ` R ) ) -> ( x ( -g ` R ) ( 0g ` R ) ) = x )
32	13, 4, 31	syl2anc 411	. . . . . . . . . 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 536	. . . . . . . . . . . 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 13826	. . . . . . . . . . . . 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 7576	. . . . . . . . . 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 14514	. . . . . . . . . . 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 2312	. . . . . . . 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 741	. . . . . . 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 13882	. . . . . . . . . . 11 |- ( ( R e. Grp /\ ( 1r ` R ) e. ( Base ` R ) ) -> ( ( 1r ` R ) ( -g ` R ) ( 0g ` R ) ) = ( 1r ` R ) )
48	13, 25, 47	syl2anc 411	. . . . . . . . . 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 2234	. . . . . . . . . . . 12 |- (Unit ` R ) = (Unit ` R )
50	49, 23	1unit 14337	. . . . . . . . . . 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 2311	. . . . . . . . 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 14514	. . . . . . . . 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 7577	. . . . . . 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 806	. . . . . 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 2626	. . . 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 14422	. . . 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 417	. . 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   A.wral 2522   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   Ap wap 7571   Basecbs 13296   +g cplusg 13374   0gc0g 13553   Grpcgrp 13797   -gcsg 13799   1rcur 14187   Ringcrg 14224  Unitcui 14316  NzRingcnzr 14409  LRingclring 14420  #_rcapr 14512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-tpos 6489  df-pap 7572  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-grp 13800  df-minusg 13801  df-sbg 13802  df-cmn 14087  df-abl 14088  df-mgp 14149  df-ur 14188  df-srg 14192  df-ring 14226  df-oppr 14296  df-dvdsr 14318  df-unit 14319  df-invr 14351  df-dvr 14362  df-nzr 14410  df-lring 14421  df-apr 14513

This theorem is referenced by:  drnglring  14530  opprdrng  14543