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	|- .x. = ( .r ` R )

Assertion

Ref	Expression
isridlrng	|- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( y .x. x ) e. I ) )

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

Allowed substitution hints:   .x. ( x, y)

Proof of Theorem isridlrng

Step	Hyp	Ref	Expression
1		eqid 2193	. . . 4 |- (oppr ` R ) = (oppr ` R )
2	1	opprrng 13576	. . 3 |- ( R e. Rng -> (oppr ` R ) e. Rng )
3	1	opprsubgg 13583	. . . . 5 |- ( R e. Rng -> (SubGrp ` R ) = (SubGrp ` (oppr ` R ) ) )
4	3	eleq2d 2263	. . . 4 |- ( R e. Rng -> ( I e. (SubGrp ` R ) <-> I e. (SubGrp ` (oppr ` R ) ) ) )
5	4	biimpa 296	. . 3 |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> I e. (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 ) e. Rng /\ I e. (SubGrp ` (oppr ` R ) ) ) -> ( I e. U <-> A. x e. ( Base ` (oppr ` R ) ) A. y e. I ( x ( .r ` (oppr ` R ) ) y ) e. I ) )
10	2, 5, 9	syl2an2r 595	. 2 |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( I e. U <-> A. x e. ( Base ` (oppr ` R ) ) A. y e. I ( x ( .r ` (oppr ` R ) ) y ) e. I ) )
11		isridlrng.b	. . . . 5 |- B = ( Base ` R )
12	1, 11	opprbasg 13574	. . . 4 |- ( R e. Rng -> B = ( Base ` (oppr ` R ) ) )
13	12	adantr 276	. . 3 |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> B = ( Base ` (oppr ` R ) ) )
14	13	raleqdv 2696	. 2 |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( A. x e. B A. y e. I ( x ( .r ` (oppr ` R ) ) y ) e. I <-> A. x e. ( Base ` (oppr ` R ) ) A. y e. I ( x ( .r ` (oppr ` R ) ) y ) e. I ) )
15		isridlrng.t	. . . . . . 7 |- .x. = ( .r ` R )
16	11, 15, 1, 8	opprmulg 13570	. . . . . 6 |- ( ( R e. Rng /\ x e. B /\ y e. I ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y .x. x ) )
17	16	ad4ant134 1219	. . . . 5 |- ( ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ x e. B ) /\ y e. I ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y .x. x ) )
18	17	eleq1d 2262	. . . 4 |- ( ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ x e. B ) /\ y e. I ) -> ( ( x ( .r ` (oppr ` R ) ) y ) e. I <-> ( y .x. x ) e. I ) )
19	18	ralbidva 2490	. . 3 |- ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ x e. B ) -> ( A. y e. I ( x ( .r ` (oppr ` R ) ) y ) e. I <-> A. y e. I ( y .x. x ) e. I ) )
20	19	ralbidva 2490	. 2 |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( A. x e. B A. y e. I ( x ( .r ` (oppr ` R ) ) y ) e. I <-> A. x e. B A. y e. I ( y .x. x ) e. I ) )
21	10, 14, 20	3bitr2d 216	1 |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( y .x. x ) e. I ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   A.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