Theorem rnglidlmmgm 14475

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 1021	. . . . . 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 14457	. . . . . . . 8 |- ( U e. L -> ( Base ` I ) = U )
5		eleq1a 2301	. . . . . . . 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 1043	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( Base ` I ) e. L )
8	4	eqcomd 2235	. . . . . . . . 9 |- ( U e. L -> U = ( Base ` I ) )
9	8	eleq2d 2299	. . . . . . . 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 1039	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> .0. e. ( Base ` I ) )
12	1, 7, 11	3jca 1201	. . . . 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 14456	. . . . . . . . 9 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
14	13	sseld 3223	. . . . . . . 8 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
15	14	3ad2ant2 1043	. . . . . . 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 2229	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
20		eqid 2229	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
21	18, 19, 20, 2	rnglidlmcl 14459	. . . . 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 597	. . . 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 1022	. . . . . . . . 9 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> U e. L )
24	3, 20	ressmulrg 13193	. . . . . . . . 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 2235	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( .r ` I ) = ( .r ` R ) )
27	26	oveqd 6024	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
28	27	eleq1d 2298	. . . . 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 2612	. 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 13113	. . . . . 6 |- ( ( R e. Rng /\ U e. L ) -> ( R ↾s U ) e. _V )
33	3, 32	eqeltrid 2316	. . . . 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 2229	. . . . 5 |- (mulGrp ` I ) = (mulGrp ` I )
36	35	mgpex 13903	. . . 4 |- ( I e. _V -> (mulGrp ` I ) e. _V )
37		eqid 2229	. . . . 5 |- ( Base ` (mulGrp ` I ) ) = ( Base ` (mulGrp ` I ) )
38		eqid 2229	. . . . 5 |- ( +g ` (mulGrp ` I ) ) = ( +g ` (mulGrp ` I ) )
39	37, 38	ismgm 13405	. . . 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 2229	. . . . . 6 |- ( Base ` I ) = ( Base ` I )
42	35, 41	mgpbasg 13904	. . . . 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 2229	. . . . . . . . 9 |- ( .r ` I ) = ( .r ` I )
45	35, 44	mgpplusgg 13902	. . . . . . . 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 6024	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a ( .r ` I ) b ) = ( a ( +g ` (mulGrp ` I ) ) b ) )
48	47, 43	eleq12d 2300	. . . . 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 2744	. . . 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 2744	. . 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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   ` cfv 5318  (class class class)co 6007   Basecbs 13047   ↾_s cress 13048   +g cplusg 13125   .rcmulr 13126   0gc0g 13304  Mgmcmgm 13402  mulGrpcmgp 13898  Rngcrng 13910  LIdealclidl 14446

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-lttrn 8124  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-ndx 13050  df-slot 13051  df-base 13053  df-sets 13054  df-iress 13055  df-plusg 13138  df-mulr 13139  df-sca 13141  df-vsca 13142  df-ip 13143  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-grp 13551  df-abl 13839  df-mgp 13899  df-rng 13911  df-lssm 14332  df-sra 14414  df-rgmod 14415  df-lidl 14448

This theorem is referenced by:  rnglidlmsgrp  14476