Theorem rnglidlmmgm 13992

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 999	. . . . . 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 13974	. . . . . . . 8 |- ( U e. L -> ( Base ` I ) = U )
5		eleq1a 2265	. . . . . . . 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 1021	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( Base ` I ) e. L )
8	4	eqcomd 2199	. . . . . . . . 9 |- ( U e. L -> U = ( Base ` I ) )
9	8	eleq2d 2263	. . . . . . . 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 1017	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> .0. e. ( Base ` I ) )
12	1, 7, 11	3jca 1179	. . . . 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 13973	. . . . . . . . 9 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
14	13	sseld 3178	. . . . . . . 8 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
15	14	3ad2ant2 1021	. . . . . . 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 2193	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
20		eqid 2193	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
21	18, 19, 20, 2	rnglidlmcl 13976	. . . . 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 595	. . . 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 1000	. . . . . . . . 9 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> U e. L )
24	3, 20	ressmulrg 12762	. . . . . . . . 9 |- ( ( U e. L /\ R e. Rng ) -> ( .r ` R ) = ( .r ` I ) )
25	23, 1, 24	syl2anc 411	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( .r ` R ) = ( .r ` I ) )
26	25	eqcomd 2199	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( .r ` I ) = ( .r ` R ) )
27	26	oveqd 5935	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
28	27	eleq1d 2262	. . . . 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 2576	. 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 12683	. . . . . 6 |- ( ( R e. Rng /\ U e. L ) -> ( R ↾s U ) e. _V )
33	3, 32	eqeltrid 2280	. . . . 5 |- ( ( R e. Rng /\ U e. L ) -> I e. _V )
34	1, 23, 33	syl2anc 411	. . . 4 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> I e. _V )
35		eqid 2193	. . . . 5 |- (mulGrp ` I ) = (mulGrp ` I )
36	35	mgpex 13421	. . . 4 |- ( I e. _V -> (mulGrp ` I ) e. _V )
37		eqid 2193	. . . . 5 |- ( Base ` (mulGrp ` I ) ) = ( Base ` (mulGrp ` I ) )
38		eqid 2193	. . . . 5 |- ( +g ` (mulGrp ` I ) ) = ( +g ` (mulGrp ` I ) )
39	37, 38	ismgm 12940	. . . 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 2193	. . . . . 6 |- ( Base ` I ) = ( Base ` I )
42	35, 41	mgpbasg 13422	. . . . 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 2193	. . . . . . . . 9 |- ( .r ` I ) = ( .r ` I )
45	35, 44	mgpplusgg 13420	. . . . . . . 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 5935	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a ( .r ` I ) b ) = ( a ( +g ` (mulGrp ` I ) ) b ) )
48	47, 43	eleq12d 2264	. . . . 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 2706	. . . 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 2706	. . 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 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   _Vcvv 2760   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619   +g cplusg 12695   .rcmulr 12696   0gc0g 12867  Mgmcmgm 12937  mulGrpcmgp 13416  Rngcrng 13428  LIdealclidl 13963

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-ip 12713  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-abl 13357  df-mgp 13417  df-rng 13429  df-lssm 13849  df-sra 13931  df-rgmod 13932  df-lidl 13965

This theorem is referenced by:  rnglidlmsgrp  13993