Theorem isridl 14701

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 2234	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
2	1	opprring 14244	. . 3 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
3		isridl.u	. . . 4 ⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))
4		eqid 2234	. . . 4 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
5		eqid 2234	. . . 4 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
6	3, 4, 5	dflidl2 14685	. . 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 14250	. . . . 5 ⊢ (𝑅 ∈ Ring → (SubGrp‘𝑅) = (SubGrp‘(oppr‘𝑅)))
9	8	eqcomd 2240	. . . 4 ⊢ (𝑅 ∈ Ring → (SubGrp‘(oppr‘𝑅)) = (SubGrp‘𝑅))
10	9	eleq2d 2304	. . 3 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ (SubGrp‘(oppr‘𝑅)) ↔ 𝐼 ∈ (SubGrp‘𝑅)))
11		isridl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
12	1, 11	opprbasg 14240	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝐵 = (Base‘(oppr‘𝑅)))
13	12	eqcomd 2240	. . . 4 ⊢ (𝑅 ∈ Ring → (Base‘(oppr‘𝑅)) = 𝐵)
14	12	eleq2d 2304	. . . . . 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 2818	. . . . . . . . 9 ⊢ 𝑥 ∈ V
17		vex 2818	. . . . . . . . 9 ⊢ 𝑦 ∈ V
18		isridl.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
19	11, 18, 1, 5	opprmulg 14236	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦 · 𝑥))
20	16, 17, 19	mp3an23 1366	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦 · 𝑥))
21	20	eleq1d 2303	. . . . . . 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 2540	. . . . 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 2761	. . 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 1398   ∈ wcel 2205  ∀wral 2522  Vcvv 2815  ‘cfv 5354  (class class class)co 6052  Basecbs 13233  ._rcmulr 13312  SubGrpcsubg 13905  Ringcrg 14161  opp_rcoppr 14232  LIdealclidl 14664

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-pre-ltirr 8244  ax-pre-lttrn 8246  ax-pre-ltadd 8248

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-tpos 6478  df-pnf 8315  df-mnf 8316  df-ltxr 8318  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-ndx 13236  df-slot 13237  df-base 13239  df-sets 13240  df-iress 13241  df-plusg 13324  df-mulr 13325  df-sca 13327  df-vsca 13328  df-ip 13329  df-0g 13492  df-mgm 13590  df-sgrp 13636  df-mnd 13651  df-grp 13737  df-minusg 13738  df-sbg 13739  df-subg 13908  df-cmn 14024  df-abl 14025  df-mgp 14086  df-rng 14098  df-ur 14125  df-ring 14163  df-oppr 14233  df-subrg 14387  df-lmod 14486  df-lssm 14550  df-sra 14632  df-rgmod 14633  df-lidl 14666

This theorem is referenced by: (None)