Theorem rnglidlmmgm 14509

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 1023	. . . . . 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 14491	. . . . . . . 8 |- ( U e. L -> ( Base ` I ) = U )
5		eleq1a 2303	. . . . . . . 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 1045	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( Base ` I ) e. L )
8	4	eqcomd 2237	. . . . . . . . 9 |- ( U e. L -> U = ( Base ` I ) )
9	8	eleq2d 2301	. . . . . . . 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 1041	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> .0. e. ( Base ` I ) )
12	1, 7, 11	3jca 1203	. . . . 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 14490	. . . . . . . . 9 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
14	13	sseld 3226	. . . . . . . 8 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
15	14	3ad2ant2 1045	. . . . . . 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 2231	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
20		eqid 2231	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
21	18, 19, 20, 2	rnglidlmcl 14493	. . . . 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 599	. . . 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 1024	. . . . . . . . 9 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> U e. L )
24	3, 20	ressmulrg 13227	. . . . . . . . 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 2237	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( .r ` I ) = ( .r ` R ) )
27	26	oveqd 6034	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
28	27	eleq1d 2300	. . . . 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 2614	. 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 13147	. . . . . 6 |- ( ( R e. Rng /\ U e. L ) -> ( R ↾s U ) e. _V )
33	3, 32	eqeltrid 2318	. . . . 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 2231	. . . . 5 |- (mulGrp ` I ) = (mulGrp ` I )
36	35	mgpex 13937	. . . 4 |- ( I e. _V -> (mulGrp ` I ) e. _V )
37		eqid 2231	. . . . 5 |- ( Base ` (mulGrp ` I ) ) = ( Base ` (mulGrp ` I ) )
38		eqid 2231	. . . . 5 |- ( +g ` (mulGrp ` I ) ) = ( +g ` (mulGrp ` I ) )
39	37, 38	ismgm 13439	. . . 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 2231	. . . . . 6 |- ( Base ` I ) = ( Base ` I )
42	35, 41	mgpbasg 13938	. . . . 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 2231	. . . . . . . . 9 |- ( .r ` I ) = ( .r ` I )
45	35, 44	mgpplusgg 13936	. . . . . . . 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 6034	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> ( a ( .r ` I ) b ) = ( a ( +g ` (mulGrp ` I ) ) b ) )
48	47, 43	eleq12d 2302	. . . . 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 2746	. . . 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 2746	. . 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 1004   = wceq 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   ` cfv 5326  (class class class)co 6017   Basecbs 13081   ↾_s cress 13082   +g cplusg 13159   .rcmulr 13160   0gc0g 13338  Mgmcmgm 13436  mulGrpcmgp 13932  Rngcrng 13944  LIdealclidl 14480

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-mulr 13173  df-sca 13175  df-vsca 13176  df-ip 13177  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-abl 13873  df-mgp 13933  df-rng 13945  df-lssm 14366  df-sra 14448  df-rgmod 14449  df-lidl 14482

This theorem is referenced by:  rnglidlmsgrp  14510