Theorem isridl 14489

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 2229	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
2	1	opprring 14063	. . 3 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
3		isridl.u	. . . 4 ⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))
4		eqid 2229	. . . 4 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
5		eqid 2229	. . . 4 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
6	3, 4, 5	dflidl2 14473	. . 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 14068	. . . . 5 ⊢ (𝑅 ∈ Ring → (SubGrp‘𝑅) = (SubGrp‘(oppr‘𝑅)))
9	8	eqcomd 2235	. . . 4 ⊢ (𝑅 ∈ Ring → (SubGrp‘(oppr‘𝑅)) = (SubGrp‘𝑅))
10	9	eleq2d 2299	. . 3 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ (SubGrp‘(oppr‘𝑅)) ↔ 𝐼 ∈ (SubGrp‘𝑅)))
11		isridl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
12	1, 11	opprbasg 14059	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝐵 = (Base‘(oppr‘𝑅)))
13	12	eqcomd 2235	. . . 4 ⊢ (𝑅 ∈ Ring → (Base‘(oppr‘𝑅)) = 𝐵)
14	12	eleq2d 2299	. . . . . 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 2802	. . . . . . . . 9 ⊢ 𝑥 ∈ V
17		vex 2802	. . . . . . . . 9 ⊢ 𝑦 ∈ V
18		isridl.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
19	11, 18, 1, 5	opprmulg 14055	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦 · 𝑥))
20	16, 17, 19	mp3an23 1363	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦 · 𝑥))
21	20	eleq1d 2298	. . . . . . 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 2526	. . . . 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 2746	. . 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 1395   ∈ wcel 2200  ∀wral 2508  Vcvv 2799  ‘cfv 5321  (class class class)co 6010  Basecbs 13053  ._rcmulr 13132  SubGrpcsubg 13725  Ringcrg 13980  opp_rcoppr 14051  LIdealclidl 14452

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-pre-ltirr 8127  ax-pre-lttrn 8129  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-tpos 6402  df-pnf 8199  df-mnf 8200  df-ltxr 8202  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-ndx 13056  df-slot 13057  df-base 13059  df-sets 13060  df-iress 13061  df-plusg 13144  df-mulr 13145  df-sca 13147  df-vsca 13148  df-ip 13149  df-0g 13312  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-grp 13557  df-minusg 13558  df-sbg 13559  df-subg 13728  df-cmn 13844  df-abl 13845  df-mgp 13905  df-rng 13917  df-ur 13944  df-ring 13982  df-oppr 14052  df-subrg 14204  df-lmod 14274  df-lssm 14338  df-sra 14420  df-rgmod 14421  df-lidl 14454

This theorem is referenced by: (None)