Theorem isridl 13816

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

Assertion

Ref	Expression
isridl	|- ( R e. Ring -> ( I e. U <-> ( I e. (SubGrp ` R ) /\ 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 isridl

Step	Hyp	Ref	Expression
1		eqid 2189	. . . 4 |- (oppr ` R ) = (oppr ` R )
2	1	opprring 13426	. . 3 |- ( R e. Ring -> (oppr ` R ) e. Ring )
3		isridl.u	. . . 4 |- U = (LIdeal ` (oppr ` R ) )
4		eqid 2189	. . . 4 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
5		eqid 2189	. . . 4 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
6	3, 4, 5	dflidl2 13801	. . 3 |- ( (oppr ` R ) e. Ring -> ( I e. U <-> ( I e. (SubGrp ` (oppr ` R ) ) /\ A. x e. ( Base ` (oppr ` R ) ) A. y e. I ( x ( .r ` (oppr ` R ) ) y ) e. I ) ) )
7	2, 6	syl 14	. 2 |- ( R e. Ring -> ( I e. U <-> ( I e. (SubGrp ` (oppr ` R ) ) /\ A. x e. ( Base ` (oppr ` R ) ) A. y e. I ( x ( .r ` (oppr ` R ) ) y ) e. I ) ) )
8	1	opprsubgg 13431	. . . . 5 |- ( R e. Ring -> (SubGrp ` R ) = (SubGrp ` (oppr ` R ) ) )
9	8	eqcomd 2195	. . . 4 |- ( R e. Ring -> (SubGrp ` (oppr ` R ) ) = (SubGrp ` R ) )
10	9	eleq2d 2259	. . 3 |- ( R e. Ring -> ( I e. (SubGrp ` (oppr ` R ) ) <-> I e. (SubGrp ` R ) ) )
11		isridl.b	. . . . . 6 |- B = ( Base ` R )
12	1, 11	opprbasg 13422	. . . . 5 |- ( R e. Ring -> B = ( Base ` (oppr ` R ) ) )
13	12	eqcomd 2195	. . . 4 |- ( R e. Ring -> ( Base ` (oppr ` R ) ) = B )
14	12	eleq2d 2259	. . . . . 6 |- ( R e. Ring -> ( x e. B <-> x e. ( Base ` (oppr ` R ) ) ) )
15	14	pm5.32i 454	. . . . 5 |- ( ( R e. Ring /\ x e. B ) <-> ( R e. Ring /\ x e. ( Base ` (oppr ` R ) ) ) )
16		vex 2755	. . . . . . . . 9 |- x e. _V
17		vex 2755	. . . . . . . . 9 |- y e. _V
18		isridl.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
19	11, 18, 1, 5	opprmulg 13418	. . . . . . . . 9 |- ( ( R e. Ring /\ x e. _V /\ y e. _V ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y .x. x ) )
20	16, 17, 19	mp3an23 1340	. . . . . . . 8 |- ( R e. Ring -> ( x ( .r ` (oppr ` R ) ) y ) = ( y .x. x ) )
21	20	eleq1d 2258	. . . . . . 7 |- ( R e. Ring -> ( ( x ( .r ` (oppr ` R ) ) y ) e. I <-> ( y .x. x ) e. I ) )
22	21	ad2antrr 488	. . . . . 6 |- ( ( ( R e. Ring /\ x e. B ) /\ y e. I ) -> ( ( x ( .r ` (oppr ` R ) ) y ) e. I <-> ( y .x. x ) e. I ) )
23	22	ralbidva 2486	. . . . 5 |- ( ( R e. Ring /\ x e. B ) -> ( A. y e. I ( x ( .r ` (oppr ` R ) ) y ) e. I <-> A. y e. I ( y .x. x ) e. I ) )
24	15, 23	sylbir 135	. . . 4 |- ( ( R e. Ring /\ x e. ( Base ` (oppr ` R ) ) ) -> ( A. y e. I ( x ( .r ` (oppr ` R ) ) y ) e. I <-> A. y e. I ( y .x. x ) e. I ) )
25	13, 24	raleqbidva 2700	. . 3 |- ( R e. Ring -> ( A. x e. ( Base ` (oppr ` R ) ) A. y e. I ( x ( .r ` (oppr ` R ) ) y ) e. I <-> A. x e. B A. y e. I ( y .x. x ) e. I ) )
26	10, 25	anbi12d 473	. 2 |- ( R e. Ring -> ( ( I e. (SubGrp ` (oppr ` R ) ) /\ A. x e. ( Base ` (oppr ` R ) ) A. y e. I ( x ( .r ` (oppr ` R ) ) y ) e. I ) <-> ( I e. (SubGrp ` R ) /\ A. x e. B A. y e. I ( y .x. x ) e. I ) ) )
27	7, 26	bitrd 188	1 |- ( R e. Ring -> ( I e. U <-> ( I e. (SubGrp ` R ) /\ 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 2160   A.wral 2468   _Vcvv 2752   ` cfv 5235  (class class class)co 5895   Basecbs 12511   .rcmulr 12587  SubGrpcsubg 13103   Ringcrg 13347  opp_rcoppr 13414  LIdealclidl 13780

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-pre-ltirr 7952  ax-pre-lttrn 7954  ax-pre-ltadd 7956

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-tpos 6269  df-pnf 8023  df-mnf 8024  df-ltxr 8026  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-5 9010  df-6 9011  df-7 9012  df-8 9013  df-ndx 12514  df-slot 12515  df-base 12517  df-sets 12518  df-iress 12519  df-plusg 12599  df-mulr 12600  df-sca 12602  df-vsca 12603  df-ip 12604  df-0g 12760  df-mgm 12829  df-sgrp 12862  df-mnd 12875  df-grp 12945  df-minusg 12946  df-sbg 12947  df-subg 13106  df-cmn 13222  df-abl 13223  df-mgp 13272  df-rng 13284  df-ur 13311  df-ring 13349  df-oppr 13415  df-subrg 13563  df-lmod 13602  df-lssm 13666  df-sra 13748  df-rgmod 13749  df-lidl 13782

This theorem is referenced by: (None)