Theorem rnglidlmmgm 14833

Description: The multiplicative group of a (left) ideal of a non-unital ring is a magma. (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
rnglidlmmgm	|- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Mgm )

Proof of Theorem rnglidlmmgm

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

Step	Hyp	Ref	Expression
1		simp1 1028	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> R e. Rng )
2		rnglidlabl.l	. . . . . . . . 9 |- L = (LIdeal ` R )
3		rnglidlabl.i	. . . . . . . . 9 |- I = ( R ↾s U )
4	2, 3	lidlbas 14815	. . . . . . . 8 |- ( U e. L -> ( Base ` I ) = U )
5		eleq1a 2310	. . . . . . . 8 |- ( U e. L -> ( ( Base ` I ) = U -> ( Base ` I ) e. L ) )
6	4, 5	mpd 13	. . . . . . 7 |- ( U e. L -> ( Base ` I ) e. L )
7	6	3ad2ant2 1050	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( Base ` I ) e. L )
8	4	eqcomd 2244	. . . . . . . . 9 |- ( U e. L -> U = ( Base ` I ) )
9	8	eleq2d 2308	. . . . . . . 8 |- ( U e. L -> ( .0. e. U <-> .0. e. ( Base ` I ) ) )
10	9	biimpa 296	. . . . . . 7 |- ( ( U e. L /\ .0. e. U ) -> .0. e. ( Base ` I ) )
11	10	3adant1 1046	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> .0. e. ( Base ` I ) )
12	1, 7, 11	3jca 1208	. . . . 5 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( R e. Rng /\ ( Base ` I ) e. L /\ .0. e. ( Base ` I ) ) )
13	2, 3	lidlssbas 14814	. . . . . . . . 9 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
14	13	sseld 3247	. . . . . . . 8 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
15	14	3ad2ant2 1050	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
16	15	anim1d 336	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( ( a e. ( Base ` I ) /\ b e. ( Base ` I ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` I ) ) ) )
17	16	imp 124	. . . . 5 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` I ) ) )
18		rnglidlabl.z	. . . . . 6 |- .0. = ( 0g ` R )
19		eqid 2238	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
20		eqid 2238	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
21	18, 19, 20, 2	rnglidlmcl 14817	. . . . 5 |- ( ( ( R e. Rng /\ ( Base ` I ) e. L /\ .0. e. ( Base ` I ) ) /\ ( a e. ( Base ` R ) /\ b e. ( Base ` I ) ) ) -> ( a ( .r ` R ) b ) e. ( Base ` I ) )
22	12, 17, 21	syl2an2r 603	. . . 4 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) ) ) -> ( a ( .r ` R ) b ) e. ( Base ` I ) )
23		simp2 1029	. . . . . . . . 9 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> U e. L )
24	3, 20	ressmulrg 13499	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( .r ` R ) = ( .r ` I ) )
25	23, 1, 24	syl2anc 415	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( .r ` R ) = ( .r ` I ) )
26	25	eqcomd 2244	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( .r ` I ) = ( .r ` R ) )
27	26	oveqd 6102	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
28	27	eleq1d 2307	. . . . 5 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( ( a ( .r ` I ) b ) e. ( Base ` I ) <-> ( a ( .r ` R ) b ) e. ( Base ` I ) ) )
29	28	adantr 276	. . . 4 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) ) ) -> ( ( a ( .r ` I ) b ) e. ( Base ` I ) <-> ( a ( .r ` R ) b ) e. ( Base ` I ) ) )
30	22, 29	mpbird 167	. . 3 |- ( ( ( R e. Rng /\ U e. L /\ .0. e. U ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) ) ) -> ( a ( .r ` I ) b ) e. ( Base ` I ) )
31	30	ralrimivva 2632	. 2 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> A. a e. ( Base ` I ) A. b e. ( Base ` I ) ( a ( .r ` I ) b ) e. ( Base ` I ) )
32		ressex 13419	. . . . . 6 |- ( ( R e. Rng /\ U e. L ) -> ( R ↾s U ) e. _V )
33	3, 32	eqeltrid 2325	. . . . 5 |- ( ( R e. Rng /\ U e. L ) -> I e. _V )
34	1, 23, 33	syl2anc 415	. . . 4 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> I e. _V )
35		eqid 2238	. . . . 5 |- (mulGrp ` I ) = (mulGrp ` I )
36	35	mgpex 14223	. . . 4 |- ( I e. _V -> (mulGrp ` I ) e. _V )
37		eqid 2238	. . . . 5 |- ( Base ` (mulGrp ` I ) ) = ( Base ` (mulGrp ` I ) )
38		eqid 2238	. . . . 5 |- ( +g ` (mulGrp ` I ) ) = ( +g ` (mulGrp ` I ) )
39	37, 38	ismgm 13677	. . . 4 |- ( (mulGrp ` I ) e. _V -> ( (mulGrp ` I ) e. Mgm <-> A. a e. ( Base ` (mulGrp ` I ) ) A. b e. ( Base ` (mulGrp ` I ) ) ( a ( +g ` (mulGrp ` I ) ) b ) e. ( Base ` (mulGrp ` I ) ) ) )
40	34, 36, 39	3syl 17	. . 3 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( (mulGrp ` I ) e. Mgm <-> A. a e. ( Base ` (mulGrp ` I ) ) A. b e. ( Base ` (mulGrp ` I ) ) ( a ( +g ` (mulGrp ` I ) ) b ) e. ( Base ` (mulGrp ` I ) ) ) )
41		eqid 2238	. . . . . 6 |- ( Base ` I ) = ( Base ` I )
42	35, 41	mgpbasg 14224	. . . . 5 |- ( I e. _V -> ( Base ` I ) = ( Base ` (mulGrp ` I ) ) )
43	34, 42	syl 14	. . . 4 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( Base ` I ) = ( Base ` (mulGrp ` I ) ) )
44		eqid 2238	. . . . . . . . 9 |- ( .r ` I ) = ( .r ` I )
45	35, 44	mgpplusgg 14221	. . . . . . . 8 |- ( I e. _V -> ( .r ` I ) = ( +g ` (mulGrp ` I ) ) )
46	34, 45	syl 14	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( .r ` I ) = ( +g ` (mulGrp ` I ) ) )
47	46	oveqd 6102	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a ( .r ` I ) b ) = ( a ( +g ` (mulGrp ` I ) ) b ) )
48	47, 43	eleq12d 2309	. . . . 5 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( ( a ( .r ` I ) b ) e. ( Base ` I ) <-> ( a ( +g ` (mulGrp ` I ) ) b ) e. ( Base ` (mulGrp ` I ) ) ) )
49	43, 48	raleqbidv 2765	. . . 4 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( A. b e. ( Base ` I ) ( a ( .r ` I ) b ) e. ( Base ` I ) <-> A. b e. ( Base ` (mulGrp ` I ) ) ( a ( +g ` (mulGrp ` I ) ) b ) e. ( Base ` (mulGrp ` I ) ) ) )
50	43, 49	raleqbidv 2765	. . 3 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( A. a e. ( Base ` I ) A. b e. ( Base ` I ) ( a ( .r ` I ) b ) e. ( Base ` I ) <-> A. a e. ( Base ` (mulGrp ` I ) ) A. b e. ( Base ` (mulGrp ` I ) ) ( a ( +g ` (mulGrp ` I ) ) b ) e. ( Base ` (mulGrp ` I ) ) ) )
51	40, 50	bitr4d 191	. 2 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( (mulGrp ` I ) e. Mgm <-> A. a e. ( Base ` I ) A. b e. ( Base ` I ) ( a ( .r ` I ) b ) e. ( Base ` I ) ) )
52	31, 51	mpbird 167	1 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Mgm )

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 13352   ↾_s cress 13353   +g cplusg 13431   .rcmulr 13432   0gc0g 13610  Mgmcmgm 13674  mulGrpcmgp 14217  Rngcrng 14231  LIdealclidl 14804

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 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-sca 13447  df-vsca 13448  df-ip 13449  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-abl 14090  df-mgp 14218  df-rng 14232  df-lssm 14690  df-sra 14772  df-rgmod 14773  df-lidl 14806

This theorem is used by:  rnglidlmsgrp  14834