Theorem isridlrng 13981

Description: A right ideal is a left ideal of the opposite non-unital ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)

Hypotheses

Ref	Expression
isridlrng.u	⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))
isridlrng.b	⊢ 𝐵 = (Base‘𝑅)
isridlrng.t	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
isridlrng	⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐼,𝑦   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦

Allowed substitution hints:   · (𝑥,𝑦)

Proof of Theorem isridlrng

Step	Hyp	Ref	Expression
1		eqid 2193	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
2	1	opprrng 13576	. . 3 ⊢ (𝑅 ∈ Rng → (oppr‘𝑅) ∈ Rng)
3	1	opprsubgg 13583	. . . . 5 ⊢ (𝑅 ∈ Rng → (SubGrp‘𝑅) = (SubGrp‘(oppr‘𝑅)))
4	3	eleq2d 2263	. . . 4 ⊢ (𝑅 ∈ Rng → (𝐼 ∈ (SubGrp‘𝑅) ↔ 𝐼 ∈ (SubGrp‘(oppr‘𝑅))))
5	4	biimpa 296	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ (SubGrp‘(oppr‘𝑅)))
6		isridlrng.u	. . . 4 ⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))
7		eqid 2193	. . . 4 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
8		eqid 2193	. . . 4 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
9	6, 7, 8	dflidl2rng 13980	. . 3 ⊢ (((oppr‘𝑅) ∈ Rng ∧ 𝐼 ∈ (SubGrp‘(oppr‘𝑅))) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ (Base‘(oppr‘𝑅))∀𝑦 ∈ 𝐼 (𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼))
10	2, 5, 9	syl2an2r 595	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ (Base‘(oppr‘𝑅))∀𝑦 ∈ 𝐼 (𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼))
11		isridlrng.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
12	1, 11	opprbasg 13574	. . . 4 ⊢ (𝑅 ∈ Rng → 𝐵 = (Base‘(oppr‘𝑅)))
13	12	adantr 276	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → 𝐵 = (Base‘(oppr‘𝑅)))
14	13	raleqdv 2696	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼 ↔ ∀𝑥 ∈ (Base‘(oppr‘𝑅))∀𝑦 ∈ 𝐼 (𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼))
15		isridlrng.t	. . . . . . 7 ⊢ · = (.r‘𝑅)
16	11, 15, 1, 8	opprmulg 13570	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐼) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦 · 𝑥))
17	16	ad4ant134 1219	. . . . 5 ⊢ ((((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐼) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦 · 𝑥))
18	17	eleq1d 2262	. . . 4 ⊢ ((((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐼) → ((𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼 ↔ (𝑦 · 𝑥) ∈ 𝐼))
19	18	ralbidva 2490	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐼 (𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼 ↔ ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼))
20	19	ralbidva 2490	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼))
21	10, 14, 20	3bitr2d 216	1 ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ‘cfv 5255  (class class class)co 5919  Basecbs 12621  ._rcmulr 12699  SubGrpcsubg 13240  Rngcrng 13431  opp_rcoppr 13566  LIdealclidl 13966

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-tpos 6300  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-mulr 12712  df-sca 12714  df-vsca 12715  df-ip 12716  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-subg 13243  df-cmn 13359  df-abl 13360  df-mgp 13420  df-rng 13432  df-oppr 13567  df-lssm 13852  df-sra 13934  df-rgmod 13935  df-lidl 13968

This theorem is referenced by:  df2idl2rng  14007