Theorem isridlrng 14244

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 2205	. . . 4 |- (oppr ` R ) = (oppr ` R )
2	1	opprrng 13839	. . 3 |- ( R e. Rng -> (oppr ` R ) e. Rng )
3	1	opprsubgg 13846	. . . . 5 |- ( R e. Rng -> (SubGrp ` R ) = (SubGrp ` (oppr ` R ) ) )
4	3	eleq2d 2275	. . . 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 2205	. . . 4 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
8		eqid 2205	. . . 4 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
9	6, 7, 8	dflidl2rng 14243	. . 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 595	. 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 13837	. . . 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 2708	. 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 13833	. . . . . 6 |- ( ( R e. Rng /\ x e. B /\ y e. I ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y .x. x ) )
17	16	ad4ant134 1220	. . . . 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 2274	. . . 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 2502	. . 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 2502	. 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 1373   e. wcel 2176   A.wral 2484   ` cfv 5271  (class class class)co 5944   Basecbs 12832   .rcmulr 12910  SubGrpcsubg 13503  Rngcrng 13694  opp_rcoppr 13829  LIdealclidl 14229

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-tpos 6331  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-7 9100  df-8 9101  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-mulr 12923  df-sca 12925  df-vsca 12926  df-ip 12927  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-subg 13506  df-cmn 13622  df-abl 13623  df-mgp 13683  df-rng 13695  df-oppr 13830  df-lssm 14115  df-sra 14197  df-rgmod 14198  df-lidl 14231

This theorem is referenced by:  df2idl2rng  14270