Theorem rnglidlmsgrp 14303

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 14302	. 2 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Mgm )
5		eqid 2206	. . . . . . . . . 10 |- (mulGrp ` R ) = (mulGrp ` R )
6	5	rngmgp 13742	. . . . . . . . 9 |- ( R e. Rng -> (mulGrp ` R ) e. Smgrp )
7	6	3ad2ant1 1021	. . . . . . . 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 14283	. . . . . . . . . . . . 13 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
10	9	sseld 3193	. . . . . . . . . . . 12 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
11	9	sseld 3193	. . . . . . . . . . . 12 |- ( U e. L -> ( b e. ( Base ` I ) -> b e. ( Base ` R ) ) )
12	9	sseld 3193	. . . . . . . . . . . 12 |- ( U e. L -> ( c e. ( Base ` I ) -> c e. ( Base ` R ) ) )
13	10, 11, 12	3anim123d 1332	. . . . . . . . . . 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 1022	. . . . . . . . . 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 1012	. . . . . . . 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 2206	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
18	5, 17	mgpbasg 13732	. . . . . . . . . 10 |- ( R e. Rng -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
19	18	3ad2ant1 1021	. . . . . . . . 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 2285	. . . . . . 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 1013	. . . . . . . 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 2285	. . . . . . 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 1014	. . . . . . . 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 2285	. . . . . . 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 2206	. . . . . . . 8 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
27		eqid 2206	. . . . . . . 8 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
28	26, 27	sgrpass 13284	. . . . . . 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 1252	. . . . . 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 2206	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
31	5, 30	mgpplusgg 13730	. . . . . . . . 9 |- ( R e. Rng -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
32	31	3ad2ant1 1021	. . . . . . . 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 5968	. . . . . . 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 2207	. . . . . . 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 5972	. . . . . 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 2207	. . . . . . 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 5968	. . . . . . 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 5972	. . . . . 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 2249	. . . . 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 1001	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> U e. L )
42		simp1 1000	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> R e. Rng )
43	2, 30	ressmulrg 13021	. . . . . . . . . 10 |- ( ( U e. L /\ R e. Rng ) -> ( .r ` R ) = ( .r ` I ) )
44	43	eqcomd 2212	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( .r ` I ) = ( .r ` R ) )
45	44	oveqd 5968	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
46		eqidd 2207	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> c = c )
47	44, 45, 46	oveq123d 5972	. . . . . . . 8 |- ( ( U e. L /\ R e. Rng ) -> ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` R ) b ) ( .r ` R ) c ) )
48		eqidd 2207	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> a = a )
49	44	oveqd 5968	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( b ( .r ` I ) c ) = ( b ( .r ` R ) c ) )
50	44, 48, 49	oveq123d 5972	. . . . . . . 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 2221	. . . . . . 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 2590	. . 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 12941	. . . . . . 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 2293	. . . . 5 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> I e. _V )
59		eqid 2206	. . . . . 6 |- (mulGrp ` I ) = (mulGrp ` I )
60		eqid 2206	. . . . . 6 |- ( Base ` I ) = ( Base ` I )
61	59, 60	mgpbasg 13732	. . . . 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 2206	. . . . . . . . . 10 |- ( .r ` I ) = ( .r ` I )
64	59, 63	mgpplusgg 13730	. . . . . . . . 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 5968	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a ( .r ` I ) b ) = ( a ( +g ` (mulGrp ` I ) ) b ) )
67		eqidd 2207	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> c = c )
68	65, 66, 67	oveq123d 5972	. . . . . . 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 2207	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> a = a )
70	65	oveqd 5968	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( b ( .r ` I ) c ) = ( b ( +g ` (mulGrp ` I ) ) c ) )
71	65, 69, 70	oveq123d 5972	. . . . . . 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 2221	. . . . . 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 2719	. . . . 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 2719	. . . 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 2719	. . 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 2206	. . 3 |- ( Base ` (mulGrp ` I ) ) = ( Base ` (mulGrp ` I ) )
78		eqid 2206	. . 3 |- ( +g ` (mulGrp ` I ) ) = ( +g ` (mulGrp ` I ) )
79	77, 78	issgrp 13279	. 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 981   = wceq 1373   e. wcel 2177   A.wral 2485   _Vcvv 2773   ` cfv 5276  (class class class)co 5951   Basecbs 12876   ↾_s cress 12877   +g cplusg 12953   .rcmulr 12954   0gc0g 13132  Mgmcmgm 13230  Smgrpcsgrp 13277  mulGrpcmgp 13726  Rngcrng 13738  LIdealclidl 14273

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-addass 8034  ax-i2m1 8037  ax-0lt1 8038  ax-0id 8040  ax-rnegex 8041  ax-pre-ltirr 8044  ax-pre-lttrn 8046  ax-pre-ltadd 8048

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-pnf 8116  df-mnf 8117  df-ltxr 8119  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-ndx 12879  df-slot 12880  df-base 12882  df-sets 12883  df-iress 12884  df-plusg 12966  df-mulr 12967  df-sca 12969  df-vsca 12970  df-ip 12971  df-0g 13134  df-mgm 13232  df-sgrp 13278  df-mnd 13293  df-grp 13379  df-abl 13667  df-mgp 13727  df-rng 13739  df-lssm 14159  df-sra 14241  df-rgmod 14242  df-lidl 14275

This theorem is referenced by:  rnglidlrng  14304