Theorem rnglidlmmgm 14128

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 ∈ 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 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ 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 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 𝑅 ∈ Rng)
2		rnglidlabl.l	. . . . . . . . 9 ⊢ 𝐿 = (LIdeal‘𝑅)
3		rnglidlabl.i	. . . . . . . . 9 ⊢ 𝐼 = (𝑅 ↾s 𝑈)
4	2, 3	lidlbas 14110	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) = 𝑈)
5		eleq1a 2268	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → ((Base‘𝐼) = 𝑈 → (Base‘𝐼) ∈ 𝐿))
6	4, 5	mpd 13	. . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ∈ 𝐿)
7	6	3ad2ant2 1021	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (Base‘𝐼) ∈ 𝐿)
8	4	eqcomd 2202	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → 𝑈 = (Base‘𝐼))
9	8	eleq2d 2266	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → ( 0 ∈ 𝑈 ↔ 0 ∈ (Base‘𝐼)))
10	9	biimpa 296	. . . . . . 7 ⊢ ((𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 0 ∈ (Base‘𝐼))
11	10	3adant1 1017	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 0 ∈ (Base‘𝐼))
12	1, 7, 11	3jca 1179	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑅 ∈ Rng ∧ (Base‘𝐼) ∈ 𝐿 ∧ 0 ∈ (Base‘𝐼)))
13	2, 3	lidlssbas 14109	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅))
14	13	sseld 3183	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅)))
15	14	3ad2ant2 1021	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅)))
16	15	anim1d 336	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ((𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼)) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝐼))))
17	16	imp 124	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝐼)))
18		rnglidlabl.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
19		eqid 2196	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
20		eqid 2196	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
21	18, 19, 20, 2	rnglidlmcl 14112	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ (Base‘𝐼) ∈ 𝐿 ∧ 0 ∈ (Base‘𝐼)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼))
22	12, 17, 21	syl2an2r 595	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼))
23		simp2 1000	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 𝑈 ∈ 𝐿)
24	3, 20	ressmulrg 12847	. . . . . . . . 9 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑅 ∈ Rng) → (.r‘𝑅) = (.r‘𝐼))
25	23, 1, 24	syl2anc 411	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (.r‘𝑅) = (.r‘𝐼))
26	25	eqcomd 2202	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (.r‘𝐼) = (.r‘𝑅))
27	26	oveqd 5942	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑎(.r‘𝐼)𝑏) = (𝑎(.r‘𝑅)𝑏))
28	27	eleq1d 2265	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ((𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼) ↔ (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼)))
29	28	adantr 276	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → ((𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼) ↔ (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼)))
30	22, 29	mpbird 167	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼))
31	30	ralrimivva 2579	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼))
32		ressex 12768	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿) → (𝑅 ↾s 𝑈) ∈ V)
33	3, 32	eqeltrid 2283	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ V)
34	1, 23, 33	syl2anc 411	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 𝐼 ∈ V)
35		eqid 2196	. . . . 5 ⊢ (mulGrp‘𝐼) = (mulGrp‘𝐼)
36	35	mgpex 13557	. . . 4 ⊢ (𝐼 ∈ V → (mulGrp‘𝐼) ∈ V)
37		eqid 2196	. . . . 5 ⊢ (Base‘(mulGrp‘𝐼)) = (Base‘(mulGrp‘𝐼))
38		eqid 2196	. . . . 5 ⊢ (+g‘(mulGrp‘𝐼)) = (+g‘(mulGrp‘𝐼))
39	37, 38	ismgm 13059	. . . 4 ⊢ ((mulGrp‘𝐼) ∈ V → ((mulGrp‘𝐼) ∈ Mgm ↔ ∀𝑎 ∈ (Base‘(mulGrp‘𝐼))∀𝑏 ∈ (Base‘(mulGrp‘𝐼))(𝑎(+g‘(mulGrp‘𝐼))𝑏) ∈ (Base‘(mulGrp‘𝐼))))
40	34, 36, 39	3syl 17	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ((mulGrp‘𝐼) ∈ Mgm ↔ ∀𝑎 ∈ (Base‘(mulGrp‘𝐼))∀𝑏 ∈ (Base‘(mulGrp‘𝐼))(𝑎(+g‘(mulGrp‘𝐼))𝑏) ∈ (Base‘(mulGrp‘𝐼))))
41		eqid 2196	. . . . . 6 ⊢ (Base‘𝐼) = (Base‘𝐼)
42	35, 41	mgpbasg 13558	. . . . 5 ⊢ (𝐼 ∈ V → (Base‘𝐼) = (Base‘(mulGrp‘𝐼)))
43	34, 42	syl 14	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (Base‘𝐼) = (Base‘(mulGrp‘𝐼)))
44		eqid 2196	. . . . . . . . 9 ⊢ (.r‘𝐼) = (.r‘𝐼)
45	35, 44	mgpplusgg 13556	. . . . . . . 8 ⊢ (𝐼 ∈ V → (.r‘𝐼) = (+g‘(mulGrp‘𝐼)))
46	34, 45	syl 14	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (.r‘𝐼) = (+g‘(mulGrp‘𝐼)))
47	46	oveqd 5942	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑎(.r‘𝐼)𝑏) = (𝑎(+g‘(mulGrp‘𝐼))𝑏))
48	47, 43	eleq12d 2267	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ((𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼) ↔ (𝑎(+g‘(mulGrp‘𝐼))𝑏) ∈ (Base‘(mulGrp‘𝐼))))
49	43, 48	raleqbidv 2709	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼) ↔ ∀𝑏 ∈ (Base‘(mulGrp‘𝐼))(𝑎(+g‘(mulGrp‘𝐼))𝑏) ∈ (Base‘(mulGrp‘𝐼))))
50	43, 49	raleqbidv 2709	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼) ↔ ∀𝑎 ∈ (Base‘(mulGrp‘𝐼))∀𝑏 ∈ (Base‘(mulGrp‘𝐼))(𝑎(+g‘(mulGrp‘𝐼))𝑏) ∈ (Base‘(mulGrp‘𝐼))))
51	40, 50	bitr4d 191	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ((mulGrp‘𝐼) ∈ Mgm ↔ ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼)))
52	31, 51	mpbird 167	1 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Mgm)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763  ‘cfv 5259  (class class class)co 5925  Basecbs 12703   ↾_s cress 12704  +_gcplusg 12780  ._rcmulr 12781  0_gc0g 12958  Mgmcmgm 13056  mulGrpcmgp 13552  Rngcrng 13564  LIdealclidl 14099

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-sca 12796  df-vsca 12797  df-ip 12798  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-abl 13493  df-mgp 13553  df-rng 13565  df-lssm 13985  df-sra 14067  df-rgmod 14068  df-lidl 14101

This theorem is referenced by:  rnglidlmsgrp  14129