Theorem isridlrng 14819

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 2238	. . . 4 |- (oppr ` R ) = (oppr ` R )
2	1	opprrng 14382	. . 3 |- ( R e. Rng -> (oppr ` R ) e. Rng )
3	1	opprsubgg 14390	. . . . 5 |- ( R e. Rng -> (SubGrp ` R ) = (SubGrp ` (oppr ` R ) ) )
4	3	eleq2d 2308	. . . 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 2238	. . . 4 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
8		eqid 2238	. . . 4 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
9	6, 7, 8	dflidl2rng 14818	. . 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 603	. 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 14380	. . . 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 2755	. 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 14376	. . . . . 6 |- ( ( R e. Rng /\ x e. B /\ y e. I ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y .x. x ) )
17	16	ad4ant134 1248	. . . . 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 2307	. . . 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 2546	. . 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 2546	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5377  (class class class)co 6085   Basecbs 13352   .rcmulr 13432  SubGrpcsubg 13970  Rngcrng 14231  opp_rcoppr 14372  LIdealclidl 14804

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-tpos 6516  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-sca 13447  df-vsca 13448  df-ip 13449  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-subg 13973  df-cmn 14089  df-abl 14090  df-mgp 14218  df-rng 14232  df-oppr 14373  df-lssm 14690  df-sra 14772  df-rgmod 14773  df-lidl 14806

This theorem is used by:  df2idl2rng  14845