Theorem rnglidlmsgrp 14836

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 14835	. 2 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Mgm )
5		eqid 2238	. . . . . . . . . 10 |- (mulGrp ` R ) = (mulGrp ` R )
6	5	rngmgp 14237	. . . . . . . . 9 |- ( R e. Rng -> (mulGrp ` R ) e. Smgrp )
7	6	3ad2ant1 1049	. . . . . . . 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 14816	. . . . . . . . . . . . 13 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
10	9	sseld 3247	. . . . . . . . . . . 12 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
11	9	sseld 3247	. . . . . . . . . . . 12 |- ( U e. L -> ( b e. ( Base ` I ) -> b e. ( Base ` R ) ) )
12	9	sseld 3247	. . . . . . . . . . . 12 |- ( U e. L -> ( c e. ( Base ` I ) -> c e. ( Base ` R ) ) )
13	10, 11, 12	3anim123d 1360	. . . . . . . . . . 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 1050	. . . . . . . . . 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 1040	. . . . . . . 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 2238	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
18	5, 17	mgpbasg 14226	. . . . . . . . . 10 |- ( R e. Rng -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
19	18	3ad2ant1 1049	. . . . . . . . 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 2317	. . . . . . 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 1041	. . . . . . . 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 2317	. . . . . . 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 1042	. . . . . . . 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 2317	. . . . . . 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 2238	. . . . . . . 8 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
27		eqid 2238	. . . . . . . 8 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
28	26, 27	sgrpass 13725	. . . . . . 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 1280	. . . . . 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 2238	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
31	5, 30	mgpplusgg 14223	. . . . . . . . 9 |- ( R e. Rng -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
32	31	3ad2ant1 1049	. . . . . . . 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 6102	. . . . . . 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 2239	. . . . . . 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 6106	. . . . . 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 2239	. . . . . . 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 6102	. . . . . . 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 6106	. . . . . 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 2281	. . . . 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 1029	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> U e. L )
42		simp1 1028	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> R e. Rng )
43	2, 30	ressmulrg 13501	. . . . . . . . . 10 |- ( ( U e. L /\ R e. Rng ) -> ( .r ` R ) = ( .r ` I ) )
44	43	eqcomd 2244	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( .r ` I ) = ( .r ` R ) )
45	44	oveqd 6102	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
46		eqidd 2239	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> c = c )
47	44, 45, 46	oveq123d 6106	. . . . . . . 8 |- ( ( U e. L /\ R e. Rng ) -> ( ( a ( .r ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` R ) b ) ( .r ` R ) c ) )
48		eqidd 2239	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> a = a )
49	44	oveqd 6102	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( b ( .r ` I ) c ) = ( b ( .r ` R ) c ) )
50	44, 48, 49	oveq123d 6106	. . . . . . . 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 2253	. . . . . . 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 415	. . . . . 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 2633	. . 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 13421	. . . . . . 7 |- ( ( R e. Rng /\ U e. L ) -> ( R ↾s U ) e. _V )
57	42, 41, 56	syl2anc 415	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( R ↾s U ) e. _V )
58	2, 57	eqeltrid 2325	. . . . 5 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> I e. _V )
59		eqid 2238	. . . . . 6 |- (mulGrp ` I ) = (mulGrp ` I )
60		eqid 2238	. . . . . 6 |- ( Base ` I ) = ( Base ` I )
61	59, 60	mgpbasg 14226	. . . . 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 2238	. . . . . . . . . 10 |- ( .r ` I ) = ( .r ` I )
64	59, 63	mgpplusgg 14223	. . . . . . . . 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 6102	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a ( .r ` I ) b ) = ( a ( +g ` (mulGrp ` I ) ) b ) )
67		eqidd 2239	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> c = c )
68	65, 66, 67	oveq123d 6106	. . . . . . 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 2239	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> a = a )
70	65	oveqd 6102	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( b ( .r ` I ) c ) = ( b ( +g ` (mulGrp ` I ) ) c ) )
71	65, 69, 70	oveq123d 6106	. . . . . . 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 2253	. . . . . 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 2765	. . . . 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 2765	. . . 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 2765	. . 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 2238	. . 3 |- ( Base ` (mulGrp ` I ) ) = ( Base ` (mulGrp ` I ) )
78		eqid 2238	. . 3 |- ( +g ` (mulGrp ` I ) ) = ( +g ` (mulGrp ` I ) )
79	77, 78	issgrp 13720	. 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 421	1 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Smgrp )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   ` cfv 5377  (class class class)co 6085   Basecbs 13354   ↾_s cress 13355   +g cplusg 13433   .rcmulr 13434   0gc0g 13612  Mgmcmgm 13676  Smgrpcsgrp 13718  mulGrpcmgp 14219  Rngcrng 14233  LIdealclidl 14806

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-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-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-iress 13362  df-plusg 13446  df-mulr 13447  df-sca 13449  df-vsca 13450  df-ip 13451  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-abl 14092  df-mgp 14220  df-rng 14234  df-lssm 14692  df-sra 14774  df-rgmod 14775  df-lidl 14808

This theorem is used by:  rnglidlrng  14837