Theorem isridl 14890

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 2238	. . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
2	1	opprring 14433	. . 3 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
3		isridl.u	. . . 4 ⊢ 𝑈 = (LIdeal‘(oppr‘𝑅))
4		eqid 2238	. . . 4 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
5		eqid 2238	. . . 4 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
6	3, 4, 5	dflidl2 14874	. . 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 14439	. . . . 5 ⊢ (𝑅 ∈ Ring → (SubGrp‘𝑅) = (SubGrp‘(oppr‘𝑅)))
9	8	eqcomd 2244	. . . 4 ⊢ (𝑅 ∈ Ring → (SubGrp‘(oppr‘𝑅)) = (SubGrp‘𝑅))
10	9	eleq2d 2308	. . 3 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ (SubGrp‘(oppr‘𝑅)) ↔ 𝐼 ∈ (SubGrp‘𝑅)))
11		isridl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
12	1, 11	opprbasg 14429	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝐵 = (Base‘(oppr‘𝑅)))
13	12	eqcomd 2244	. . . 4 ⊢ (𝑅 ∈ Ring → (Base‘(oppr‘𝑅)) = 𝐵)
14	12	eleq2d 2308	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘(oppr‘𝑅))))
15	14	pm5.32i 458	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ↔ (𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘(oppr‘𝑅))))
16		vex 2824	. . . . . . . . 9 ⊢ 𝑥 ∈ V
17		vex 2824	. . . . . . . . 9 ⊢ 𝑦 ∈ V
18		isridl.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
19	11, 18, 1, 5	opprmulg 14425	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦 · 𝑥))
20	16, 17, 19	mp3an23 1370	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦 · 𝑥))
21	20	eleq1d 2307	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ((𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼 ↔ (𝑦 · 𝑥) ∈ 𝐼))
22	21	ad2antrr 492	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐼) → ((𝑥(.r‘(oppr‘𝑅))𝑦) ∈ 𝐼 ↔ (𝑦 · 𝑥) ∈ 𝐼))
23	22	ralbidva 2546	. . . . 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 2767	. . 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 477	. 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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821  ‘cfv 5377  (class class class)co 6085  Basecbs 13401  ._rcmulr 13481  SubGrpcsubg 14019  Ringcrg 14349  opp_rcoppr 14421  LIdealclidl 14853

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-tpos 6516  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-5 9368  df-6 9369  df-7 9370  df-8 9371  df-ndx 13404  df-slot 13405  df-base 13407  df-sets 13408  df-iress 13409  df-plusg 13493  df-mulr 13494  df-sca 13496  df-vsca 13497  df-ip 13498  df-0g 13661  df-mgm 13725  df-sgrp 13766  df-mnd 13779  df-grp 13857  df-minusg 13858  df-sbg 13859  df-subg 14022  df-cmn 14138  df-abl 14139  df-mgp 14267  df-rng 14281  df-ur 14312  df-ring 14351  df-oppr 14422  df-subrg 14576  df-lmod 14674  df-lssm 14739  df-sra 14821  df-rgmod 14822  df-lidl 14855

This theorem is used by: (None)