Theorem rnglidlrng 46748

Description: A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 𝑈 ∈ (SubGrp‘𝑅) 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)

Assertion

Ref	Expression
rnglidlrng	⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Rng)

Proof of Theorem rnglidlrng

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngabl 46641	. . . 4 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	3ad2ant1 1133	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Abel)
3		simp3 1138	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ (SubGrp‘𝑅))
4		rnglidlabl.i	. . . 4 ⊢ 𝐼 = (𝑅 ↾s 𝑈)
5	4	subgabl 19703	. . 3 ⊢ ((𝑅 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Abel)
6	2, 3, 5	syl2anc 584	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Abel)
7		eqid 2732	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
8	7	subg0cl 19013	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝑅) → (0g‘𝑅) ∈ 𝑈)
9		rnglidlabl.l	. . . 4 ⊢ 𝐿 = (LIdeal‘𝑅)
10	9, 4, 7	rnglidlmsgrp 46747	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ (0g‘𝑅) ∈ 𝑈) → (mulGrp‘𝐼) ∈ Smgrp)
11	8, 10	syl3an3 1165	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (mulGrp‘𝐼) ∈ Smgrp)
12		simpl1 1191	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → 𝑅 ∈ Rng)
13	9, 4	lidlssbas 46744	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅))
14	13	sseld 3981	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅)))
15	13	sseld 3981	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑏 ∈ (Base‘𝐼) → 𝑏 ∈ (Base‘𝑅)))
16	13	sseld 3981	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑐 ∈ (Base‘𝐼) → 𝑐 ∈ (Base‘𝑅)))
17	14, 15, 16	3anim123d 1443	. . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → ((𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼)) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))))
18	17	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → ((𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼)) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))))
19	18	imp 407	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅)))
20		eqid 2732	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
21		eqid 2732	. . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅)
22		eqid 2732	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
23	20, 21, 22	rngdi 46649	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))) → (𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)) = ((𝑎(.r‘𝑅)𝑏)(+g‘𝑅)(𝑎(.r‘𝑅)𝑐)))
24	12, 19, 23	syl2anc 584	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)) = ((𝑎(.r‘𝑅)𝑏)(+g‘𝑅)(𝑎(.r‘𝑅)𝑐)))
25	20, 21, 22	rngdir 46650	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))) → ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐) = ((𝑎(.r‘𝑅)𝑐)(+g‘𝑅)(𝑏(.r‘𝑅)𝑐)))
26	12, 19, 25	syl2anc 584	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐) = ((𝑎(.r‘𝑅)𝑐)(+g‘𝑅)(𝑏(.r‘𝑅)𝑐)))
27	4, 22	ressmulr 17251	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝑅) = (.r‘𝐼))
28	27	eqcomd 2738	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝐼) = (.r‘𝑅))
29		eqidd 2733	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → 𝑎 = 𝑎)
30	4, 21	ressplusg 17234	. . . . . . . . . . 11 ⊢ (𝑈 ∈ 𝐿 → (+g‘𝑅) = (+g‘𝐼))
31	30	eqcomd 2738	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐿 → (+g‘𝐼) = (+g‘𝑅))
32	31	oveqd 7425	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (𝑏(+g‘𝐼)𝑐) = (𝑏(+g‘𝑅)𝑐))
33	28, 29, 32	oveq123d 7429	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = (𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)))
34	28	oveqd 7425	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (𝑎(.r‘𝐼)𝑏) = (𝑎(.r‘𝑅)𝑏))
35	28	oveqd 7425	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (𝑎(.r‘𝐼)𝑐) = (𝑎(.r‘𝑅)𝑐))
36	31, 34, 35	oveq123d 7429	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) = ((𝑎(.r‘𝑅)𝑏)(+g‘𝑅)(𝑎(.r‘𝑅)𝑐)))
37	33, 36	eqeq12d 2748	. . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → ((𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) ↔ (𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)) = ((𝑎(.r‘𝑅)𝑏)(+g‘𝑅)(𝑎(.r‘𝑅)𝑐))))
38	31	oveqd 7425	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (𝑎(+g‘𝐼)𝑏) = (𝑎(+g‘𝑅)𝑏))
39		eqidd 2733	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → 𝑐 = 𝑐)
40	28, 38, 39	oveq123d 7429	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐))
41	28	oveqd 7425	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (𝑏(.r‘𝐼)𝑐) = (𝑏(.r‘𝑅)𝑐))
42	31, 35, 41	oveq123d 7429	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐)) = ((𝑎(.r‘𝑅)𝑐)(+g‘𝑅)(𝑏(.r‘𝑅)𝑐)))
43	40, 42	eqeq12d 2748	. . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → (((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐)) ↔ ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐) = ((𝑎(.r‘𝑅)𝑐)(+g‘𝑅)(𝑏(.r‘𝑅)𝑐))))
44	37, 43	anbi12d 631	. . . . . 6 ⊢ (𝑈 ∈ 𝐿 → (((𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) ∧ ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐))) ↔ ((𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)) = ((𝑎(.r‘𝑅)𝑏)(+g‘𝑅)(𝑎(.r‘𝑅)𝑐)) ∧ ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐) = ((𝑎(.r‘𝑅)𝑐)(+g‘𝑅)(𝑏(.r‘𝑅)𝑐)))))
45	44	3ad2ant2 1134	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (((𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) ∧ ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐))) ↔ ((𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)) = ((𝑎(.r‘𝑅)𝑏)(+g‘𝑅)(𝑎(.r‘𝑅)𝑐)) ∧ ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐) = ((𝑎(.r‘𝑅)𝑐)(+g‘𝑅)(𝑏(.r‘𝑅)𝑐)))))
46	45	adantr 481	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (((𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) ∧ ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐))) ↔ ((𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)) = ((𝑎(.r‘𝑅)𝑏)(+g‘𝑅)(𝑎(.r‘𝑅)𝑐)) ∧ ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐) = ((𝑎(.r‘𝑅)𝑐)(+g‘𝑅)(𝑏(.r‘𝑅)𝑐)))))
47	24, 26, 46	mpbir2and 711	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → ((𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) ∧ ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐))))
48	47	ralrimivvva 3203	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)∀𝑐 ∈ (Base‘𝐼)((𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) ∧ ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐))))
49		eqid 2732	. . 3 ⊢ (Base‘𝐼) = (Base‘𝐼)
50		eqid 2732	. . 3 ⊢ (mulGrp‘𝐼) = (mulGrp‘𝐼)
51		eqid 2732	. . 3 ⊢ (+g‘𝐼) = (+g‘𝐼)
52		eqid 2732	. . 3 ⊢ (.r‘𝐼) = (.r‘𝐼)
53	49, 50, 51, 52	isrng 46640	. 2 ⊢ (𝐼 ∈ Rng ↔ (𝐼 ∈ Abel ∧ (mulGrp‘𝐼) ∈ Smgrp ∧ ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)∀𝑐 ∈ (Base‘𝐼)((𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) ∧ ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐)))))
54	6, 11, 48, 53	syl3anbrc 1343	1 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Rng)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ‘cfv 6543  (class class class)co 7408  Basecbs 17143   ↾_s cress 17172  +_gcplusg 17196  ._rcmulr 17197  0_gc0g 17384  Smgrpcsgrp 18608  SubGrpcsubg 18999  Abelcabl 19648  mulGrpcmgp 19986  LIdealclidl 20782  Rngcrng 46638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-subg 19002  df-cmn 19649  df-abl 19650  df-mgp 19987  df-lss 20542  df-sra 20784  df-rgmod 20785  df-lidl 20786  df-rng 46639

This theorem is referenced by:  rng2idlsubgsubrng  46753  lidlrng  46815