Theorem isridl 14208

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 2204	. . . 4 |- (oppr ` R ) = (oppr ` R )
2	1	opprring 13783	. . 3 |- ( R e. Ring -> (oppr ` R ) e. Ring )
3		isridl.u	. . . 4 |- U = (LIdeal ` (oppr ` R ) )
4		eqid 2204	. . . 4 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
5		eqid 2204	. . . 4 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
6	3, 4, 5	dflidl2 14192	. . 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 13788	. . . . 5 |- ( R e. Ring -> (SubGrp ` R ) = (SubGrp ` (oppr ` R ) ) )
9	8	eqcomd 2210	. . . 4 |- ( R e. Ring -> (SubGrp ` (oppr ` R ) ) = (SubGrp ` R ) )
10	9	eleq2d 2274	. . 3 |- ( R e. Ring -> ( I e. (SubGrp ` (oppr ` R ) ) <-> I e. (SubGrp ` R ) ) )
11		isridl.b	. . . . . 6 |- B = ( Base ` R )
12	1, 11	opprbasg 13779	. . . . 5 |- ( R e. Ring -> B = ( Base ` (oppr ` R ) ) )
13	12	eqcomd 2210	. . . 4 |- ( R e. Ring -> ( Base ` (oppr ` R ) ) = B )
14	12	eleq2d 2274	. . . . . 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 2774	. . . . . . . . 9 |- x e. _V
17		vex 2774	. . . . . . . . 9 |- y e. _V
18		isridl.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
19	11, 18, 1, 5	opprmulg 13775	. . . . . . . . 9 |- ( ( R e. Ring /\ x e. _V /\ y e. _V ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y .x. x ) )
20	16, 17, 19	mp3an23 1341	. . . . . . . 8 |- ( R e. Ring -> ( x ( .r ` (oppr ` R ) ) y ) = ( y .x. x ) )
21	20	eleq1d 2273	. . . . . . 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 2501	. . . . 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 2719	. . 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 1372   e. wcel 2175   A.wral 2483   _Vcvv 2771   ` cfv 5270  (class class class)co 5943   Basecbs 12774   .rcmulr 12852  SubGrpcsubg 13445   Ringcrg 13700  opp_rcoppr 13771  LIdealclidl 14171

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-tpos 6330  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-5 9097  df-6 9098  df-7 9099  df-8 9100  df-ndx 12777  df-slot 12778  df-base 12780  df-sets 12781  df-iress 12782  df-plusg 12864  df-mulr 12865  df-sca 12867  df-vsca 12868  df-ip 12869  df-0g 13032  df-mgm 13130  df-sgrp 13176  df-mnd 13191  df-grp 13277  df-minusg 13278  df-sbg 13279  df-subg 13448  df-cmn 13564  df-abl 13565  df-mgp 13625  df-rng 13637  df-ur 13664  df-ring 13702  df-oppr 13772  df-subrg 13923  df-lmod 13993  df-lssm 14057  df-sra 14139  df-rgmod 14140  df-lidl 14173

This theorem is referenced by: (None)