Theorem rnglidlmmgm 14516

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 1023	. . . . . 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 14498	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) = 𝑈)
5		eleq1a 2303	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → ((Base‘𝐼) = 𝑈 → (Base‘𝐼) ∈ 𝐿))
6	4, 5	mpd 13	. . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ∈ 𝐿)
7	6	3ad2ant2 1045	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (Base‘𝐼) ∈ 𝐿)
8	4	eqcomd 2237	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → 𝑈 = (Base‘𝐼))
9	8	eleq2d 2301	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → ( 0 ∈ 𝑈 ↔ 0 ∈ (Base‘𝐼)))
10	9	biimpa 296	. . . . . . 7 ⊢ ((𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 0 ∈ (Base‘𝐼))
11	10	3adant1 1041	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 0 ∈ (Base‘𝐼))
12	1, 7, 11	3jca 1203	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑅 ∈ Rng ∧ (Base‘𝐼) ∈ 𝐿 ∧ 0 ∈ (Base‘𝐼)))
13	2, 3	lidlssbas 14497	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅))
14	13	sseld 3226	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅)))
15	14	3ad2ant2 1045	. . . . . . 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 2231	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
20		eqid 2231	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
21	18, 19, 20, 2	rnglidlmcl 14500	. . . . 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 599	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼))
23		simp2 1024	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 𝑈 ∈ 𝐿)
24	3, 20	ressmulrg 13233	. . . . . . . . 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 2237	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (.r‘𝐼) = (.r‘𝑅))
27	26	oveqd 6035	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑎(.r‘𝐼)𝑏) = (𝑎(.r‘𝑅)𝑏))
28	27	eleq1d 2300	. . . . 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 2614	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼))
32		ressex 13153	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿) → (𝑅 ↾s 𝑈) ∈ V)
33	3, 32	eqeltrid 2318	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ V)
34	1, 23, 33	syl2anc 411	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 𝐼 ∈ V)
35		eqid 2231	. . . . 5 ⊢ (mulGrp‘𝐼) = (mulGrp‘𝐼)
36	35	mgpex 13944	. . . 4 ⊢ (𝐼 ∈ V → (mulGrp‘𝐼) ∈ 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 13445	. . . 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 2231	. . . . . 6 ⊢ (Base‘𝐼) = (Base‘𝐼)
42	35, 41	mgpbasg 13945	. . . . 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 2231	. . . . . . . . 9 ⊢ (.r‘𝐼) = (.r‘𝐼)
45	35, 44	mgpplusgg 13943	. . . . . . . 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 6035	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑎(.r‘𝐼)𝑏) = (𝑎(+g‘(mulGrp‘𝐼))𝑏))
48	47, 43	eleq12d 2302	. . . . 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 2746	. . . 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 2746	. . 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 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  Vcvv 2802  ‘cfv 5326  (class class class)co 6018  Basecbs 13087   ↾_s cress 13088  +_gcplusg 13165  ._rcmulr 13166  0_gc0g 13344  Mgmcmgm 13442  mulGrpcmgp 13939  Rngcrng 13951  LIdealclidl 14487

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 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

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 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-ndx 13090  df-slot 13091  df-base 13093  df-sets 13094  df-iress 13095  df-plusg 13178  df-mulr 13179  df-sca 13181  df-vsca 13182  df-ip 13183  df-0g 13346  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-grp 13591  df-abl 13879  df-mgp 13940  df-rng 13952  df-lssm 14373  df-sra 14455  df-rgmod 14456  df-lidl 14489

This theorem is referenced by:  rnglidlmsgrp  14517