Theorem isridl 14341

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 2206	. . . 4 |- (oppr ` R ) = (oppr ` R )
2	1	opprring 13916	. . 3 |- ( R e. Ring -> (oppr ` R ) e. Ring )
3		isridl.u	. . . 4 |- U = (LIdeal ` (oppr ` R ) )
4		eqid 2206	. . . 4 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
5		eqid 2206	. . . 4 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
6	3, 4, 5	dflidl2 14325	. . 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 13921	. . . . 5 |- ( R e. Ring -> (SubGrp ` R ) = (SubGrp ` (oppr ` R ) ) )
9	8	eqcomd 2212	. . . 4 |- ( R e. Ring -> (SubGrp ` (oppr ` R ) ) = (SubGrp ` R ) )
10	9	eleq2d 2276	. . 3 |- ( R e. Ring -> ( I e. (SubGrp ` (oppr ` R ) ) <-> I e. (SubGrp ` R ) ) )
11		isridl.b	. . . . . 6 |- B = ( Base ` R )
12	1, 11	opprbasg 13912	. . . . 5 |- ( R e. Ring -> B = ( Base ` (oppr ` R ) ) )
13	12	eqcomd 2212	. . . 4 |- ( R e. Ring -> ( Base ` (oppr ` R ) ) = B )
14	12	eleq2d 2276	. . . . . 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 2776	. . . . . . . . 9 |- x e. _V
17		vex 2776	. . . . . . . . 9 |- y e. _V
18		isridl.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
19	11, 18, 1, 5	opprmulg 13908	. . . . . . . . 9 |- ( ( R e. Ring /\ x e. _V /\ y e. _V ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y .x. x ) )
20	16, 17, 19	mp3an23 1342	. . . . . . . 8 |- ( R e. Ring -> ( x ( .r ` (oppr ` R ) ) y ) = ( y .x. x ) )
21	20	eleq1d 2275	. . . . . . 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 2503	. . . . 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 2721	. . 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 1373   e. wcel 2177   A.wral 2485   _Vcvv 2773   ` cfv 5280  (class class class)co 5957   Basecbs 12907   .rcmulr 12985  SubGrpcsubg 13578   Ringcrg 13833  opp_rcoppr 13904  LIdealclidl 14304

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-tpos 6344  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-5 9118  df-6 9119  df-7 9120  df-8 9121  df-ndx 12910  df-slot 12911  df-base 12913  df-sets 12914  df-iress 12915  df-plusg 12997  df-mulr 12998  df-sca 13000  df-vsca 13001  df-ip 13002  df-0g 13165  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411  df-sbg 13412  df-subg 13581  df-cmn 13697  df-abl 13698  df-mgp 13758  df-rng 13770  df-ur 13797  df-ring 13835  df-oppr 13905  df-subrg 14056  df-lmod 14126  df-lssm 14190  df-sra 14272  df-rgmod 14273  df-lidl 14306

This theorem is referenced by: (None)