Theorem isridl 14542

Description: A right ideal is a left ideal of the opposite 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, 13-Feb-2025.)

Hypotheses

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

Assertion

Ref	Expression
isridl	⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀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 isridl

Step	Hyp	Ref	Expression
1		eqid 2230	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
2	1	opprring 14116	. . 3 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
3		isridl.u	. . . 4 ⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))
4		eqid 2230	. . . 4 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
5		eqid 2230	. . . 4 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
6	3, 4, 5	dflidl2 14526	. . 3 ⊢ ((oppr‘𝑅) ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘(oppr‘𝑅)) ∧ ∀𝑥 ∈ (Base‘(oppr‘𝑅))∀𝑦 ∈ 𝐼 (𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼)))
7	2, 6	syl 14	. 2 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘(oppr‘𝑅)) ∧ ∀𝑥 ∈ (Base‘(oppr‘𝑅))∀𝑦 ∈ 𝐼 (𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼)))
8	1	opprsubgg 14121	. . . . 5 ⊢ (𝑅 ∈ Ring → (SubGrp‘𝑅) = (SubGrp‘(oppr‘𝑅)))
9	8	eqcomd 2236	. . . 4 ⊢ (𝑅 ∈ Ring → (SubGrp‘(oppr‘𝑅)) = (SubGrp‘𝑅))
10	9	eleq2d 2300	. . 3 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ (SubGrp‘(oppr‘𝑅)) ↔ 𝐼 ∈ (SubGrp‘𝑅)))
11		isridl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
12	1, 11	opprbasg 14112	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝐵 = (Base‘(oppr‘𝑅)))
13	12	eqcomd 2236	. . . 4 ⊢ (𝑅 ∈ Ring → (Base‘(oppr‘𝑅)) = 𝐵)
14	12	eleq2d 2300	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘(oppr‘𝑅))))
15	14	pm5.32i 454	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ↔ (𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘(oppr‘𝑅))))
16		vex 2804	. . . . . . . . 9 ⊢ 𝑥 ∈ V
17		vex 2804	. . . . . . . . 9 ⊢ 𝑦 ∈ V
18		isridl.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
19	11, 18, 1, 5	opprmulg 14108	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦 · 𝑥))
20	16, 17, 19	mp3an23 1365	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦 · 𝑥))
21	20	eleq1d 2299	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ((𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼 ↔ (𝑦 · 𝑥) ∈ 𝐼))
22	21	ad2antrr 488	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐼) → ((𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼 ↔ (𝑦 · 𝑥) ∈ 𝐼))
23	22	ralbidva 2527	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐼 (𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼 ↔ ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼))
24	15, 23	sylbir 135	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘(oppr‘𝑅))) → (∀𝑦 ∈ 𝐼 (𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼 ↔ ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼))
25	13, 24	raleqbidva 2747	. . 3 ⊢ (𝑅 ∈ Ring → (∀𝑥 ∈ (Base‘(oppr‘𝑅))∀𝑦 ∈ 𝐼 (𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼))
26	10, 25	anbi12d 473	. 2 ⊢ (𝑅 ∈ Ring → ((𝐼 ∈ (SubGrp‘(oppr‘𝑅)) ∧ ∀𝑥 ∈ (Base‘(oppr‘𝑅))∀𝑦 ∈ 𝐼 (𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼) ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼)))
27	7, 26	bitrd 188	1 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2201  ∀wral 2509  Vcvv 2801  ‘cfv 5328  (class class class)co 6023  Basecbs 13105  ._rcmulr 13184  SubGrpcsubg 13777  Ringcrg 14033  opp_rcoppr 14104  LIdealclidl 14505

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 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-addass 8139  ax-i2m1 8142  ax-0lt1 8143  ax-0id 8145  ax-rnegex 8146  ax-pre-ltirr 8149  ax-pre-lttrn 8151  ax-pre-ltadd 8153

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-tpos 6416  df-pnf 8221  df-mnf 8222  df-ltxr 8224  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-ndx 13108  df-slot 13109  df-base 13111  df-sets 13112  df-iress 13113  df-plusg 13196  df-mulr 13197  df-sca 13199  df-vsca 13200  df-ip 13201  df-0g 13364  df-mgm 13462  df-sgrp 13508  df-mnd 13523  df-grp 13609  df-minusg 13610  df-sbg 13611  df-subg 13780  df-cmn 13896  df-abl 13897  df-mgp 13958  df-rng 13970  df-ur 13997  df-ring 14035  df-oppr 14105  df-subrg 14257  df-lmod 14327  df-lssm 14391  df-sra 14473  df-rgmod 14474  df-lidl 14507

This theorem is referenced by: (None)