Theorem rnglidlmmgm 14805

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 1028	. . . . . 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 14787	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) = 𝑈)
5		eleq1a 2310	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → ((Base‘𝐼) = 𝑈 → (Base‘𝐼) ∈ 𝐿))
6	4, 5	mpd 13	. . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ∈ 𝐿)
7	6	3ad2ant2 1050	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (Base‘𝐼) ∈ 𝐿)
8	4	eqcomd 2244	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → 𝑈 = (Base‘𝐼))
9	8	eleq2d 2308	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → ( 0 ∈ 𝑈 ↔ 0 ∈ (Base‘𝐼)))
10	9	biimpa 296	. . . . . . 7 ⊢ ((𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 0 ∈ (Base‘𝐼))
11	10	3adant1 1046	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 0 ∈ (Base‘𝐼))
12	1, 7, 11	3jca 1208	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑅 ∈ Rng ∧ (Base‘𝐼) ∈ 𝐿 ∧ 0 ∈ (Base‘𝐼)))
13	2, 3	lidlssbas 14786	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅))
14	13	sseld 3247	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅)))
15	14	3ad2ant2 1050	. . . . . . 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 2238	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
20		eqid 2238	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
21	18, 19, 20, 2	rnglidlmcl 14789	. . . . 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 603	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼))
23		simp2 1029	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 𝑈 ∈ 𝐿)
24	3, 20	ressmulrg 13476	. . . . . . . . 9 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑅 ∈ Rng) → (.r‘𝑅) = (.r‘𝐼))
25	23, 1, 24	syl2anc 415	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (.r‘𝑅) = (.r‘𝐼))
26	25	eqcomd 2244	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (.r‘𝐼) = (.r‘𝑅))
27	26	oveqd 6092	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑎(.r‘𝐼)𝑏) = (𝑎(.r‘𝑅)𝑏))
28	27	eleq1d 2307	. . . . 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 2632	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼))
32		ressex 13396	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿) → (𝑅 ↾s 𝑈) ∈ V)
33	3, 32	eqeltrid 2325	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ V)
34	1, 23, 33	syl2anc 415	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 𝐼 ∈ V)
35		eqid 2238	. . . . 5 ⊢ (mulGrp‘𝐼) = (mulGrp‘𝐼)
36	35	mgpex 14199	. . . 4 ⊢ (𝐼 ∈ V → (mulGrp‘𝐼) ∈ V)
37		eqid 2238	. . . . 5 ⊢ (Base‘(mulGrp‘𝐼)) = (Base‘(mulGrp‘𝐼))
38		eqid 2238	. . . . 5 ⊢ (+g‘(mulGrp‘𝐼)) = (+g‘(mulGrp‘𝐼))
39	37, 38	ismgm 13654	. . . 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 2238	. . . . . 6 ⊢ (Base‘𝐼) = (Base‘𝐼)
42	35, 41	mgpbasg 14200	. . . . 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 2238	. . . . . . . . 9 ⊢ (.r‘𝐼) = (.r‘𝐼)
45	35, 44	mgpplusgg 14198	. . . . . . . 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 6092	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑎(.r‘𝐼)𝑏) = (𝑎(+g‘(mulGrp‘𝐼))𝑏))
48	47, 43	eleq12d 2309	. . . . 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 2765	. . . 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 2765	. . 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 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821  ‘cfv 5372  (class class class)co 6075  Basecbs 13330   ↾_s cress 13331  +_gcplusg 13408  ._rcmulr 13409  0_gc0g 13587  Mgmcmgm 13651  mulGrpcmgp 14194  Rngcrng 14206  LIdealclidl 14776

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-mulr 13422  df-sca 13424  df-vsca 13425  df-ip 13426  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-abl 14067  df-mgp 14195  df-rng 14207  df-lssm 14662  df-sra 14744  df-rgmod 14745  df-lidl 14778

This theorem is referenced by:  rnglidlmsgrp  14806