Theorem isridl 14136

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 2196	. . . 4 |- (oppr ` R ) = (oppr ` R )
2	1	opprring 13711	. . 3 |- ( R e. Ring -> (oppr ` R ) e. Ring )
3		isridl.u	. . . 4 |- U = (LIdeal ` (oppr ` R ) )
4		eqid 2196	. . . 4 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
5		eqid 2196	. . . 4 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
6	3, 4, 5	dflidl2 14120	. . 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 13716	. . . . 5 |- ( R e. Ring -> (SubGrp ` R ) = (SubGrp ` (oppr ` R ) ) )
9	8	eqcomd 2202	. . . 4 |- ( R e. Ring -> (SubGrp ` (oppr ` R ) ) = (SubGrp ` R ) )
10	9	eleq2d 2266	. . 3 |- ( R e. Ring -> ( I e. (SubGrp ` (oppr ` R ) ) <-> I e. (SubGrp ` R ) ) )
11		isridl.b	. . . . . 6 |- B = ( Base ` R )
12	1, 11	opprbasg 13707	. . . . 5 |- ( R e. Ring -> B = ( Base ` (oppr ` R ) ) )
13	12	eqcomd 2202	. . . 4 |- ( R e. Ring -> ( Base ` (oppr ` R ) ) = B )
14	12	eleq2d 2266	. . . . . 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 2766	. . . . . . . . 9 |- x e. _V
17		vex 2766	. . . . . . . . 9 |- y e. _V
18		isridl.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
19	11, 18, 1, 5	opprmulg 13703	. . . . . . . . 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 2265	. . . . . . 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 2493	. . . . 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 2711	. . 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 2167   A.wral 2475   _Vcvv 2763   ` cfv 5259  (class class class)co 5925   Basecbs 12703   .rcmulr 12781  SubGrpcsubg 13373   Ringcrg 13628  opp_rcoppr 13699  LIdealclidl 14099

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 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-tpos 6312  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-sca 12796  df-vsca 12797  df-ip 12798  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-sbg 13207  df-subg 13376  df-cmn 13492  df-abl 13493  df-mgp 13553  df-rng 13565  df-ur 13592  df-ring 13630  df-oppr 13700  df-subrg 13851  df-lmod 13921  df-lssm 13985  df-sra 14067  df-rgmod 14068  df-lidl 14101

This theorem is referenced by: (None)