Theorem isridlrng 14461

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 2229	. . . 4 |- (oppr ` R ) = (oppr ` R )
2	1	opprrng 14055	. . 3 |- ( R e. Rng -> (oppr ` R ) e. Rng )
3	1	opprsubgg 14062	. . . . 5 |- ( R e. Rng -> (SubGrp ` R ) = (SubGrp ` (oppr ` R ) ) )
4	3	eleq2d 2299	. . . 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 2229	. . . 4 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
8		eqid 2229	. . . 4 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
9	6, 7, 8	dflidl2rng 14460	. . 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 597	. 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 14053	. . . 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 2734	. 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 14049	. . . . . 6 |- ( ( R e. Rng /\ x e. B /\ y e. I ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y .x. x ) )
17	16	ad4ant134 1241	. . . . 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 2298	. . . 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 2526	. . 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 2526	. 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 1395   e. wcel 2200   A.wral 2508   ` cfv 5318  (class class class)co 6007   Basecbs 13047   .rcmulr 13126  SubGrpcsubg 13719  Rngcrng 13910  opp_rcoppr 14045  LIdealclidl 14446

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 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-lttrn 8124  ax-pre-ltadd 8126

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 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-tpos 6397  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-ndx 13050  df-slot 13051  df-base 13053  df-sets 13054  df-iress 13055  df-plusg 13138  df-mulr 13139  df-sca 13141  df-vsca 13142  df-ip 13143  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-grp 13551  df-subg 13722  df-cmn 13838  df-abl 13839  df-mgp 13899  df-rng 13911  df-oppr 14046  df-lssm 14332  df-sra 14414  df-rgmod 14415  df-lidl 14448

This theorem is referenced by:  df2idl2rng  14487