Theorem rnglidlmsgrp 14482

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 14481	. 2 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Mgm )
5		eqid 2229	. . . . . . . . . 10 |- (mulGrp ` R ) = (mulGrp ` R )
6	5	rngmgp 13920	. . . . . . . . 9 |- ( R e. Rng -> (mulGrp ` R ) e. Smgrp )
7	6	3ad2ant1 1042	. . . . . . . 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 14462	. . . . . . . . . . . . 13 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
10	9	sseld 3223	. . . . . . . . . . . 12 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
11	9	sseld 3223	. . . . . . . . . . . 12 |- ( U e. L -> ( b e. ( Base ` I ) -> b e. ( Base ` R ) ) )
12	9	sseld 3223	. . . . . . . . . . . 12 |- ( U e. L -> ( c e. ( Base ` I ) -> c e. ( Base ` R ) ) )
13	10, 11, 12	3anim123d 1353	. . . . . . . . . . 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 1043	. . . . . . . . . 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 1033	. . . . . . . 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 2229	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
18	5, 17	mgpbasg 13910	. . . . . . . . . 10 |- ( R e. Rng -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
19	18	3ad2ant1 1042	. . . . . . . . 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 2308	. . . . . . 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 1034	. . . . . . . 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 2308	. . . . . . 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 1035	. . . . . . . 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 2308	. . . . . . 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 2229	. . . . . . . 8 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
27		eqid 2229	. . . . . . . 8 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
28	26, 27	sgrpass 13462	. . . . . . 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 1273	. . . . . 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 2229	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
31	5, 30	mgpplusgg 13908	. . . . . . . . 9 |- ( R e. Rng -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
32	31	3ad2ant1 1042	. . . . . . . 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 6027	. . . . . . 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 2230	. . . . . . 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 6031	. . . . . 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 2230	. . . . . . 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 6027	. . . . . . 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 6031	. . . . . 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 2272	. . . . 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 1022	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> U e. L )
42		simp1 1021	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> R e. Rng )
43	2, 30	ressmulrg 13199	. . . . . . . . . 10 |- ( ( U e. L /\ R e. Rng ) -> ( .r ` R ) = ( .r ` I ) )
44	43	eqcomd 2235	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( .r ` I ) = ( .r ` R ) )
45	44	oveqd 6027	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
46		eqidd 2230	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> c = c )
47	44, 45, 46	oveq123d 6031	. . . . . . . 8 |- ( ( U e. L /\ R e. Rng ) -> ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` R ) b ) ( .r ` R ) c ) )
48		eqidd 2230	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> a = a )
49	44	oveqd 6027	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( b ( .r ` I ) c ) = ( b ( .r ` R ) c ) )
50	44, 48, 49	oveq123d 6031	. . . . . . . 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 2244	. . . . . . 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 2613	. . 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 13119	. . . . . . 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 2316	. . . . 5 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> I e. _V )
59		eqid 2229	. . . . . 6 |- (mulGrp ` I ) = (mulGrp ` I )
60		eqid 2229	. . . . . 6 |- ( Base ` I ) = ( Base ` I )
61	59, 60	mgpbasg 13910	. . . . 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 2229	. . . . . . . . . 10 |- ( .r ` I ) = ( .r ` I )
64	59, 63	mgpplusgg 13908	. . . . . . . . 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 6027	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a ( .r ` I ) b ) = ( a ( +g ` (mulGrp ` I ) ) b ) )
67		eqidd 2230	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> c = c )
68	65, 66, 67	oveq123d 6031	. . . . . . 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 2230	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> a = a )
70	65	oveqd 6027	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( b ( .r ` I ) c ) = ( b ( +g ` (mulGrp ` I ) ) c ) )
71	65, 69, 70	oveq123d 6031	. . . . . . 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 2244	. . . . . 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 2744	. . . . 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 2744	. . . 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 2744	. . 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 2229	. . 3 |- ( Base ` (mulGrp ` I ) ) = ( Base ` (mulGrp ` I ) )
78		eqid 2229	. . 3 |- ( +g ` (mulGrp ` I ) ) = ( +g ` (mulGrp ` I ) )
79	77, 78	issgrp 13457	. 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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   ` cfv 5321  (class class class)co 6010   Basecbs 13053   ↾_s cress 13054   +g cplusg 13131   .rcmulr 13132   0gc0g 13310  Mgmcmgm 13408  Smgrpcsgrp 13455  mulGrpcmgp 13904  Rngcrng 13916  LIdealclidl 14452

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-pre-ltirr 8127  ax-pre-lttrn 8129  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-pnf 8199  df-mnf 8200  df-ltxr 8202  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-ndx 13056  df-slot 13057  df-base 13059  df-sets 13060  df-iress 13061  df-plusg 13144  df-mulr 13145  df-sca 13147  df-vsca 13148  df-ip 13149  df-0g 13312  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-grp 13557  df-abl 13845  df-mgp 13905  df-rng 13917  df-lssm 14338  df-sra 14420  df-rgmod 14421  df-lidl 14454

This theorem is referenced by:  rnglidlrng  14483