Theorem isridl 14537

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 2231	. . . 4 |- (oppr ` R ) = (oppr ` R )
2	1	opprring 14111	. . 3 |- ( R e. Ring -> (oppr ` R ) e. Ring )
3		isridl.u	. . . 4 |- U = (LIdeal ` (oppr ` R ) )
4		eqid 2231	. . . 4 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
5		eqid 2231	. . . 4 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
6	3, 4, 5	dflidl2 14521	. . 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 14116	. . . . 5 |- ( R e. Ring -> (SubGrp ` R ) = (SubGrp ` (oppr ` R ) ) )
9	8	eqcomd 2237	. . . 4 |- ( R e. Ring -> (SubGrp ` (oppr ` R ) ) = (SubGrp ` R ) )
10	9	eleq2d 2301	. . 3 |- ( R e. Ring -> ( I e. (SubGrp ` (oppr ` R ) ) <-> I e. (SubGrp ` R ) ) )
11		isridl.b	. . . . . 6 |- B = ( Base ` R )
12	1, 11	opprbasg 14107	. . . . 5 |- ( R e. Ring -> B = ( Base ` (oppr ` R ) ) )
13	12	eqcomd 2237	. . . 4 |- ( R e. Ring -> ( Base ` (oppr ` R ) ) = B )
14	12	eleq2d 2301	. . . . . 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 2805	. . . . . . . . 9 |- x e. _V
17		vex 2805	. . . . . . . . 9 |- y e. _V
18		isridl.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
19	11, 18, 1, 5	opprmulg 14103	. . . . . . . . 9 |- ( ( R e. Ring /\ x e. _V /\ y e. _V ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y .x. x ) )
20	16, 17, 19	mp3an23 1365	. . . . . . . 8 |- ( R e. Ring -> ( x ( .r ` (oppr ` R ) ) y ) = ( y .x. x ) )
21	20	eleq1d 2300	. . . . . . 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 2528	. . . . 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 2748	. . 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 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   ` cfv 5326  (class class class)co 6018   Basecbs 13100   .rcmulr 13179  SubGrpcsubg 13772   Ringcrg 14028  opp_rcoppr 14099  LIdealclidl 14500

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-tpos 6411  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-ndx 13103  df-slot 13104  df-base 13106  df-sets 13107  df-iress 13108  df-plusg 13191  df-mulr 13192  df-sca 13194  df-vsca 13195  df-ip 13196  df-0g 13359  df-mgm 13457  df-sgrp 13503  df-mnd 13518  df-grp 13604  df-minusg 13605  df-sbg 13606  df-subg 13775  df-cmn 13891  df-abl 13892  df-mgp 13953  df-rng 13965  df-ur 13992  df-ring 14030  df-oppr 14100  df-subrg 14252  df-lmod 14322  df-lssm 14386  df-sra 14468  df-rgmod 14469  df-lidl 14502

This theorem is referenced by: (None)