Theorem rnglidlrng 21541

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 20383	. . . 4 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	3ad2ant1 1151	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Abel)
3		simp3 1156	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ (SubGrp‘𝑅))
4		rnglidlabl.i	. . . 4 ⊢ 𝐼 = (𝑅 ↾s 𝑈)
5	4	subgabl 20056	. . 3 ⊢ ((𝑅 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Abel)
6	2, 3, 5	syl2anc 596	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Abel)
7		eqid 2760	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
8	7	subg0cl 19350	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝑅) → (0g‘𝑅) ∈ 𝑈)
9		rnglidlabl.l	. . . 4 ⊢ 𝐿 = (LIdeal‘𝑅)
10	9, 4, 7	rnglidlmsgrp 21540	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ (0g‘𝑅) ∈ 𝑈) → (mulGrp‘𝐼) ∈ Smgrp)
11	8, 10	syl3an3 1183	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (mulGrp‘𝐼) ∈ Smgrp)
12		simpl1 1210	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → 𝑅 ∈ Rng)
13	9, 4	lidlssbas 21498	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅))
14	13	sseld 3907	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅)))
15	13	sseld 3907	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑏 ∈ (Base‘𝐼) → 𝑏 ∈ (Base‘𝑅)))
16	13	sseld 3907	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑐 ∈ (Base‘𝐼) → 𝑐 ∈ (Base‘𝑅)))
17	14, 15, 16	3anim123d 1471	. . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → ((𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼)) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))))
18	17	3ad2ant2 1152	. . . . . 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 412	. . . . 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 2760	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
21		eqid 2760	. . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅)
22		eqid 2760	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
23	20, 21, 22	rngdi 20388	. . . . 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 596	. . . 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 20389	. . . . 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 596	. . . 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 17484	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝑅) = (.r‘𝐼))
28	27	eqcomd 2766	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝐼) = (.r‘𝑅))
29		eqidd 2761	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → 𝑎 = 𝑎)
30	4, 21	ressplusg 17468	. . . . . . . . . . 11 ⊢ (𝑈 ∈ 𝐿 → (+g‘𝑅) = (+g‘𝐼))
31	30	eqcomd 2766	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐿 → (+g‘𝐼) = (+g‘𝑅))
32	31	oveqd 7438	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (𝑏(+g‘𝐼)𝑐) = (𝑏(+g‘𝑅)𝑐))
33	28, 29, 32	oveq123d 7442	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = (𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)))
34	28	oveqd 7438	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (𝑎(.r‘𝐼)𝑏) = (𝑎(.r‘𝑅)𝑏))
35	28	oveqd 7438	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (𝑎(.r‘𝐼)𝑐) = (𝑎(.r‘𝑅)𝑐))
36	31, 34, 35	oveq123d 7442	. . . . . . . 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 2776	. . . . . . 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 7438	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (𝑎(+g‘𝐼)𝑏) = (𝑎(+g‘𝑅)𝑏))
39		eqidd 2761	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → 𝑐 = 𝑐)
40	28, 38, 39	oveq123d 7442	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐))
41	28	oveqd 7438	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (𝑏(.r‘𝐼)𝑐) = (𝑏(.r‘𝑅)𝑐))
42	31, 35, 41	oveq123d 7442	. . . . . . . 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 2776	. . . . . . 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 644	. . . . . 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 1152	. . . . 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 486	. . . 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 726	. . 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 3208	. 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 2760	. . 3 ⊢ (Base‘𝐼) = (Base‘𝐼)
50		eqid 2760	. . 3 ⊢ (mulGrp‘𝐼) = (mulGrp‘𝐼)
51		eqid 2760	. . 3 ⊢ (+g‘𝐼) = (+g‘𝐼)
52		eqid 2760	. . 3 ⊢ (.r‘𝐼) = (.r‘𝐼)
53	49, 50, 51, 52	isrng 20382	. 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 1362	1 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Rng)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ‘cfv 6539  (class class class)co 7421  Basecbs 17393   ↾_s cress 17414  +_gcplusg 17434  ._rcmulr 17435  0_gc0g 17616  Smgrpcsgrp 18913  SubGrpcsubg 19336  Abelcabl 20001  mulGrpcmgp 20366  Rngcrng 20380  LIdealclidl 21490

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-er 8721  df-en 8978  df-dom 8979  df-sdom 8980  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-nn 12340  df-2 12409  df-3 12410  df-4 12411  df-5 12412  df-6 12413  df-7 12414  df-8 12415  df-sets 17348  df-slot 17366  df-ndx 17378  df-base 17394  df-ress 17415  df-plusg 17447  df-mulr 17448  df-sca 17450  df-vsca 17451  df-ip 17452  df-0g 17618  df-mgm 18822  df-sgrp 18914  df-mnd 18930  df-grp 19153  df-subg 19339  df-cmn 20002  df-abl 20003  df-mgp 20367  df-rng 20381  df-lss 21213  df-sra 21454  df-rgmod 21455  df-lidl 21492

This theorem is used by:  rng2idlsubgsubrng  21568  lidlrng  49424