Theorem rnglidlmsgrp 14573

Description: The multiplicative group of a (left) ideal of a non-unital ring is a semigroup. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption .0. e. U is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)

Hypotheses

Ref	Expression
rnglidlabl.l	|- L = (LIdeal ` R )
rnglidlabl.i	|- I = ( R ↾s U )
rnglidlabl.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
rnglidlmsgrp	|- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Smgrp )

Proof of Theorem rnglidlmsgrp

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnglidlabl.l	. . 3 |- L = (LIdeal ` R )
2		rnglidlabl.i	. . 3 |- I = ( R ↾s U )
3		rnglidlabl.z	. . 3 |- .0. = ( 0g ` R )
4	1, 2, 3	rnglidlmmgm 14572	. 2 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Mgm )
5		eqid 2231	. . . . . . . . . 10 |- (mulGrp ` R ) = (mulGrp ` R )
6	5	rngmgp 14011	. . . . . . . . 9 |- ( R e. Rng -> (mulGrp ` R ) e. Smgrp )
7	6	3ad2ant1 1045	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` R ) e. Smgrp )
8	7	adantr 276	. . . . . . 7 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> (mulGrp ` R ) e. Smgrp )
9	1, 2	lidlssbas 14553	. . . . . . . . . . . . 13 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
10	9	sseld 3227	. . . . . . . . . . . 12 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
11	9	sseld 3227	. . . . . . . . . . . 12 |- ( U e. L -> ( b e. ( Base ` I ) -> b e. ( Base ` R ) ) )
12	9	sseld 3227	. . . . . . . . . . . 12 |- ( U e. L -> ( c e. ( Base ` I ) -> c e. ( Base ` R ) ) )
13	10, 11, 12	3anim123d 1356	. . . . . . . . . . 11 |- ( U e. L -> ( ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) )
14	13	3ad2ant2 1046	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) )
15	14	imp 124	. . . . . . . . 9 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) )
16	15	simp1d 1036	. . . . . . . 8 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> a e. ( Base ` R ) )
17		eqid 2231	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
18	5, 17	mgpbasg 14001	. . . . . . . . . 10 |- ( R e. Rng -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
19	18	3ad2ant1 1045	. . . . . . . . 9 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
20	19	adantr 276	. . . . . . . 8 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
21	16, 20	eleqtrd 2310	. . . . . . 7 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> a e. ( Base ` (mulGrp ` R ) ) )
22	15	simp2d 1037	. . . . . . . 8 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> b e. ( Base ` R ) )
23	22, 20	eleqtrd 2310	. . . . . . 7 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> b e. ( Base ` (mulGrp ` R ) ) )
24	15	simp3d 1038	. . . . . . . 8 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> c e. ( Base ` R ) )
25	24, 20	eleqtrd 2310	. . . . . . 7 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> c e. ( Base ` (mulGrp ` R ) ) )
26		eqid 2231	. . . . . . . 8 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
27		eqid 2231	. . . . . . . 8 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
28	26, 27	sgrpass 13552	. . . . . . 7 |- ( ( (mulGrp ` R ) e. Smgrp /\ ( a e. ( Base ` (mulGrp ` R ) ) /\ b e. ( Base ` (mulGrp ` R ) ) /\ c e. ( Base ` (mulGrp ` R ) ) ) ) -> ( ( a ( +g ` (mulGrp ` R ) ) b ) ( +g ` (mulGrp ` R ) ) c ) = ( a ( +g ` (mulGrp ` R ) ) ( b ( +g ` (mulGrp ` R ) ) c ) ) )
29	8, 21, 23, 25, 28	syl13anc 1276	. . . . . 6 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( ( a ( +g ` (mulGrp ` R ) ) b ) ( +g ` (mulGrp ` R ) ) c ) = ( a ( +g ` (mulGrp ` R ) ) ( b ( +g ` (mulGrp ` R ) ) c ) ) )
30		eqid 2231	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
31	5, 30	mgpplusgg 13999	. . . . . . . . 9 |- ( R e. Rng -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
32	31	3ad2ant1 1045	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
33	32	adantr 276	. . . . . . 7 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
34	33	oveqd 6045	. . . . . . 7 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( a ( .r ` R ) b ) = ( a ( +g ` (mulGrp ` R ) ) b ) )
35		eqidd 2232	. . . . . . 7 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> c = c )
36	33, 34, 35	oveq123d 6049	. . . . . 6 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( ( a ( .r ` R ) b ) ( .r ` R ) c ) = ( ( a ( +g ` (mulGrp ` R ) ) b ) ( +g ` (mulGrp ` R ) ) c ) )
37		eqidd 2232	. . . . . . 7 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> a = a )
38	33	oveqd 6045	. . . . . . 7 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( b ( .r ` R ) c ) = ( b ( +g ` (mulGrp ` R ) ) c ) )
39	33, 37, 38	oveq123d 6049	. . . . . 6 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( a ( .r ` R ) ( b ( .r ` R ) c ) ) = ( a ( +g ` (mulGrp ` R ) ) ( b ( +g ` (mulGrp ` R ) ) c ) ) )
40	29, 36, 39	3eqtr4d 2274	. . . . 5 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( ( a ( .r ` R ) b ) ( .r ` R ) c ) = ( a ( .r ` R ) ( b ( .r ` R ) c ) ) )
41		simp2 1025	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> U e. L )
42		simp1 1024	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> R e. Rng )
43	2, 30	ressmulrg 13289	. . . . . . . . . 10 |- ( ( U e. L /\ R e. Rng ) -> ( .r ` R ) = ( .r ` I ) )
44	43	eqcomd 2237	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( .r ` I ) = ( .r ` R ) )
45	44	oveqd 6045	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
46		eqidd 2232	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> c = c )
47	44, 45, 46	oveq123d 6049	. . . . . . . 8 |- ( ( U e. L /\ R e. Rng ) -> ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` R ) b ) ( .r ` R ) c ) )
48		eqidd 2232	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> a = a )
49	44	oveqd 6045	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( b ( .r ` I ) c ) = ( b ( .r ` R ) c ) )
50	44, 48, 49	oveq123d 6049	. . . . . . . 8 |- ( ( U e. L /\ R e. Rng ) -> ( a ( .r ` I ) ( b ( .r ` I ) c ) ) = ( a ( .r ` R ) ( b ( .r ` R ) c ) ) )
51	47, 50	eqeq12d 2246	. . . . . . 7 |- ( ( U e. L /\ R e. Rng ) -> ( ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( a ( .r ` I ) ( b ( .r ` I ) c ) ) <-> ( ( a ( .r ` R ) b ) ( .r ` R ) c ) = ( a ( .r ` R ) ( b ( .r ` R ) c ) ) ) )
52	41, 42, 51	syl2anc 411	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( a ( .r ` I ) ( b ( .r ` I ) c ) ) <-> ( ( a ( .r ` R ) b ) ( .r ` R ) c ) = ( a ( .r ` R ) ( b ( .r ` R ) c ) ) ) )
53	52	adantr 276	. . . . 5 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( a ( .r ` I ) ( b ( .r ` I ) c ) ) <-> ( ( a ( .r ` R ) b ) ( .r ` R ) c ) = ( a ( .r ` R ) ( b ( .r ` R ) c ) ) ) )
54	40, 53	mpbird 167	. . . 4 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( a ( .r ` I ) ( b ( .r ` I ) c ) ) )
55	54	ralrimivvva 2616	. . 3 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> A. a e. ( Base ` I ) A. b e. ( Base ` I ) A. c e. ( Base ` I ) ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( a ( .r ` I ) ( b ( .r ` I ) c ) ) )
56		ressex 13209	. . . . . . 7 |- ( ( R e. Rng /\ U e. L ) -> ( R ↾s U ) e. _V )
57	42, 41, 56	syl2anc 411	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( R ↾s U ) e. _V )
58	2, 57	eqeltrid 2318	. . . . 5 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> I e. _V )
59		eqid 2231	. . . . . 6 |- (mulGrp ` I ) = (mulGrp ` I )
60		eqid 2231	. . . . . 6 |- ( Base ` I ) = ( Base ` I )
61	59, 60	mgpbasg 14001	. . . . 5 |- ( I e. _V -> ( Base ` I ) = ( Base ` (mulGrp ` I ) ) )
62	58, 61	syl 14	. . . 4 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( Base ` I ) = ( Base ` (mulGrp ` I ) ) )
63		eqid 2231	. . . . . . . . . 10 |- ( .r ` I ) = ( .r ` I )
64	59, 63	mgpplusgg 13999	. . . . . . . . 9 |- ( I e. _V -> ( .r ` I ) = ( +g ` (mulGrp ` I ) ) )
65	58, 64	syl 14	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( .r ` I ) = ( +g ` (mulGrp ` I ) ) )
66	65	oveqd 6045	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a ( .r ` I ) b ) = ( a ( +g ` (mulGrp ` I ) ) b ) )
67		eqidd 2232	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> c = c )
68	65, 66, 67	oveq123d 6049	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( ( a ( +g ` (mulGrp ` I ) ) b ) ( +g ` (mulGrp ` I ) ) c ) )
69		eqidd 2232	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> a = a )
70	65	oveqd 6045	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( b ( .r ` I ) c ) = ( b ( +g ` (mulGrp ` I ) ) c ) )
71	65, 69, 70	oveq123d 6049	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a ( .r ` I ) ( b ( .r ` I ) c ) ) = ( a ( +g ` (mulGrp ` I ) ) ( b ( +g ` (mulGrp ` I ) ) c ) ) )
72	68, 71	eqeq12d 2246	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( a ( .r ` I ) ( b ( .r ` I ) c ) ) <-> ( ( a ( +g ` (mulGrp ` I ) ) b ) ( +g ` (mulGrp ` I ) ) c ) = ( a ( +g ` (mulGrp ` I ) ) ( b ( +g ` (mulGrp ` I ) ) c ) ) ) )
73	62, 72	raleqbidv 2747	. . . . 5 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( A. c e. ( Base ` I ) ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( a ( .r ` I ) ( b ( .r ` I ) c ) ) <-> A. c e. ( Base ` (mulGrp ` I ) ) ( ( a ( +g ` (mulGrp ` I ) ) b ) ( +g ` (mulGrp ` I ) ) c ) = ( a ( +g ` (mulGrp ` I ) ) ( b ( +g ` (mulGrp ` I ) ) c ) ) ) )
74	62, 73	raleqbidv 2747	. . . 4 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( A. b e. ( Base ` I ) A. c e. ( Base ` I ) ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( a ( .r ` I ) ( b ( .r ` I ) c ) ) <-> A. b e. ( Base ` (mulGrp ` I ) ) A. c e. ( Base ` (mulGrp ` I ) ) ( ( a ( +g ` (mulGrp ` I ) ) b ) ( +g ` (mulGrp ` I ) ) c ) = ( a ( +g ` (mulGrp ` I ) ) ( b ( +g ` (mulGrp ` I ) ) c ) ) ) )
75	62, 74	raleqbidv 2747	. . 3 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( A. a e. ( Base ` I ) A. b e. ( Base ` I ) A. c e. ( Base ` I ) ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( a ( .r ` I ) ( b ( .r ` I ) c ) ) <-> A. a e. ( Base ` (mulGrp ` I ) ) A. b e. ( Base ` (mulGrp ` I ) ) A. c e. ( Base ` (mulGrp ` I ) ) ( ( a ( +g ` (mulGrp ` I ) ) b ) ( +g ` (mulGrp ` I ) ) c ) = ( a ( +g ` (mulGrp ` I ) ) ( b ( +g ` (mulGrp ` I ) ) c ) ) ) )
76	55, 75	mpbid 147	. 2 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> A. a e. ( Base ` (mulGrp ` I ) ) A. b e. ( Base ` (mulGrp ` I ) ) A. c e. ( Base ` (mulGrp ` I ) ) ( ( a ( +g ` (mulGrp ` I ) ) b ) ( +g ` (mulGrp ` I ) ) c ) = ( a ( +g ` (mulGrp ` I ) ) ( b ( +g ` (mulGrp ` I ) ) c ) ) )
77		eqid 2231	. . 3 |- ( Base ` (mulGrp ` I ) ) = ( Base ` (mulGrp ` I ) )
78		eqid 2231	. . 3 |- ( +g ` (mulGrp ` I ) ) = ( +g ` (mulGrp ` I ) )
79	77, 78	issgrp 13547	. 2 |- ( (mulGrp ` I ) e. Smgrp <-> ( (mulGrp ` I ) e. Mgm /\ A. a e. ( Base ` (mulGrp ` I ) ) A. b e. ( Base ` (mulGrp ` I ) ) A. c e. ( Base ` (mulGrp ` I ) ) ( ( a ( +g ` (mulGrp ` I ) ) b ) ( +g ` (mulGrp ` I ) ) c ) = ( a ( +g ` (mulGrp ` I ) ) ( b ( +g ` (mulGrp ` I ) ) c ) ) ) )
80	4, 76, 79	sylanbrc 417	1 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Smgrp )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   ` cfv 5333  (class class class)co 6028   Basecbs 13143   ↾_s cress 13144   +g cplusg 13221   .rcmulr 13222   0gc0g 13400  Mgmcmgm 13498  Smgrpcsgrp 13545  mulGrpcmgp 13995  Rngcrng 14007  LIdealclidl 14543

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8259  df-mnf 8260  df-ltxr 8262  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-ndx 13146  df-slot 13147  df-base 13149  df-sets 13150  df-iress 13151  df-plusg 13234  df-mulr 13235  df-sca 13237  df-vsca 13238  df-ip 13239  df-0g 13402  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-grp 13647  df-abl 13935  df-mgp 13996  df-rng 14008  df-lssm 14429  df-sra 14511  df-rgmod 14512  df-lidl 14545

This theorem is referenced by:  rnglidlrng  14574