Theorem isridlrng 14038

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 2196	. . . 4 |- (oppr ` R ) = (oppr ` R )
2	1	opprrng 13633	. . 3 |- ( R e. Rng -> (oppr ` R ) e. Rng )
3	1	opprsubgg 13640	. . . . 5 |- ( R e. Rng -> (SubGrp ` R ) = (SubGrp ` (oppr ` R ) ) )
4	3	eleq2d 2266	. . . 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 2196	. . . 4 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
8		eqid 2196	. . . 4 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
9	6, 7, 8	dflidl2rng 14037	. . 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 13631	. . . 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 2699	. 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 13627	. . . . . 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 2265	. . . 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 2493	. . 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 2493	. 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 2167   A.wral 2475   ` cfv 5258  (class class class)co 5922   Basecbs 12678   .rcmulr 12756  SubGrpcsubg 13297  Rngcrng 13488  opp_rcoppr 13623  LIdealclidl 14023

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-sca 12771  df-vsca 12772  df-ip 12773  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-subg 13300  df-cmn 13416  df-abl 13417  df-mgp 13477  df-rng 13489  df-oppr 13624  df-lssm 13909  df-sra 13991  df-rgmod 13992  df-lidl 14025

This theorem is referenced by:  df2idl2rng  14064