Theorem rnglidlmsgrp 14817

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 14816	. 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 14218	. . . . . . . . 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 14797	. . . . . . . . . . . . 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 14207	. . . . . . . . . 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 13706	. . . . . . 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 14204	. . . . . . . . 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 6096	. . . . . . 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 6100	. . . . . 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 6096	. . . . . . 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 6100	. . . . . 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 13482	. . . . . . . . . 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 6096	. . . . . . . . 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 6100	. . . . . . . 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 6096	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( b ( .r ` I ) c ) = ( b ( .r ` R ) c ) )
50	44, 48, 49	oveq123d 6100	. . . . . . . 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 13402	. . . . . . 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 14207	. . . . 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 14204	. . . . . . . . 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 6096	. . . . . . . 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 6100	. . . . . . 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 6096	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( b ( .r ` I ) c ) = ( b ( +g ` (mulGrp ` I ) ) c ) )
71	65, 69, 70	oveq123d 6100	. . . . . . 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 13701	. 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 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   ` cfv 5375  (class class class)co 6079   Basecbs 13335   ↾_s cress 13336   +g cplusg 13414   .rcmulr 13415   0gc0g 13593  Mgmcmgm 13657  Smgrpcsgrp 13699  mulGrpcmgp 14200  Rngcrng 14214  LIdealclidl 14787

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 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 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-sca 13430  df-vsca 13431  df-ip 13432  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-abl 14073  df-mgp 14201  df-rng 14215  df-lssm 14673  df-sra 14755  df-rgmod 14756  df-lidl 14789

This theorem is referenced by:  rnglidlrng  14818