Theorem dflidl2rng 14621

Description: Alternate (the usual textbook) definition of a (left) ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)

Hypotheses

Ref	Expression
dflidl2rng.u	|- U = (LIdeal ` R )
dflidl2rng.b	|- B = ( Base ` R )
dflidl2rng.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
dflidl2rng	|- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( x .x. y ) 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 dflidl2rng

Dummy variables z j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 527	. . . . 5 |- ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ I e. U ) -> R e. Rng )
2		simpr 110	. . . . 5 |- ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ I e. U ) -> I e. U )
3		eqid 2232	. . . . . . 7 |- ( 0g ` R ) = ( 0g ` R )
4	3	subg0cl 13891	. . . . . 6 |- ( I e. (SubGrp ` R ) -> ( 0g ` R ) e. I )
5	4	ad2antlr 489	. . . . 5 |- ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ I e. U ) -> ( 0g ` R ) e. I )
6	1, 2, 5	3jca 1204	. . . 4 |- ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ I e. U ) -> ( R e. Rng /\ I e. U /\ ( 0g ` R ) e. I ) )
7		dflidl2rng.b	. . . . 5 |- B = ( Base ` R )
8		dflidl2rng.t	. . . . 5 |- .x. = ( .r ` R )
9		dflidl2rng.u	. . . . 5 |- U = (LIdeal ` R )
10	3, 7, 8, 9	rnglidlmcl 14620	. . . 4 |- ( ( ( R e. Rng /\ I e. U /\ ( 0g ` R ) e. I ) /\ ( x e. B /\ y e. I ) ) -> ( x .x. y ) e. I )
11	6, 10	sylan 283	. . 3 |- ( ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ I e. U ) /\ ( x e. B /\ y e. I ) ) -> ( x .x. y ) e. I )
12	11	ralrimivva 2624	. 2 |- ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ I e. U ) -> A. x e. B A. y e. I ( x .x. y ) e. I )
13	7	subgss 13883	. . . 4 |- ( I e. (SubGrp ` R ) -> I C_ B )
14	13	ad2antlr 489	. . 3 |- ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ A. x e. B A. y e. I ( x .x. y ) e. I ) -> I C_ B )
15		elex2 2829	. . . . 5 |- ( ( 0g ` R ) e. I -> E. j j e. I )
16	4, 15	syl 14	. . . 4 |- ( I e. (SubGrp ` R ) -> E. j j e. I )
17	16	ad2antlr 489	. . 3 |- ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ A. x e. B A. y e. I ( x .x. y ) e. I ) -> E. j j e. I )
18		eqid 2232	. . . . . . . . 9 |- ( +g ` R ) = ( +g ` R )
19	18	subgcl 13893	. . . . . . . 8 |- ( ( I e. (SubGrp ` R ) /\ ( x .x. y ) e. I /\ z e. I ) -> ( ( x .x. y ) ( +g ` R ) z ) e. I )
20	19	ad5ant245 1263	. . . . . . 7 |- ( ( ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ ( x e. B /\ y e. I ) ) /\ ( x .x. y ) e. I ) /\ z e. I ) -> ( ( x .x. y ) ( +g ` R ) z ) e. I )
21	20	ralrimiva 2615	. . . . . 6 |- ( ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ ( x e. B /\ y e. I ) ) /\ ( x .x. y ) e. I ) -> A. z e. I ( ( x .x. y ) ( +g ` R ) z ) e. I )
22	21	ex 115	. . . . 5 |- ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ ( x e. B /\ y e. I ) ) -> ( ( x .x. y ) e. I -> A. z e. I ( ( x .x. y ) ( +g ` R ) z ) e. I ) )
23	22	ralimdvva 2611	. . . 4 |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( A. x e. B A. y e. I ( x .x. y ) e. I -> A. x e. B A. y e. I A. z e. I ( ( x .x. y ) ( +g ` R ) z ) e. I ) )
24	23	imp 124	. . 3 |- ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ A. x e. B A. y e. I ( x .x. y ) e. I ) -> A. x e. B A. y e. I A. z e. I ( ( x .x. y ) ( +g ` R ) z ) e. I )
25	9, 7, 18, 8	islidlm 14619	. . 3 |- ( I e. U <-> ( I C_ B /\ E. j j e. I /\ A. x e. B A. y e. I A. z e. I ( ( x .x. y ) ( +g ` R ) z ) e. I ) )
26	14, 17, 24, 25	syl3anbrc 1208	. 2 |- ( ( ( R e. Rng /\ I e. (SubGrp ` R ) ) /\ A. x e. B A. y e. I ( x .x. y ) e. I ) -> I e. U )
27	12, 26	impbida 600	1 |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( x .x. y ) e. I ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   C_ wss 3210   ` cfv 5351  (class class class)co 6049   Basecbs 13204   +g cplusg 13282   .rcmulr 13283   0gc0g 13461  SubGrpcsubg 13876  Rngcrng 14068  LIdealclidl 14607

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-pre-ltirr 8238  ax-pre-lttrn 8240  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8309  df-mnf 8310  df-ltxr 8312  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-5 9298  df-6 9299  df-7 9300  df-8 9301  df-ndx 13207  df-slot 13208  df-base 13210  df-sets 13211  df-iress 13212  df-plusg 13295  df-mulr 13296  df-sca 13298  df-vsca 13299  df-ip 13300  df-0g 13463  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-grp 13708  df-subg 13879  df-abl 13996  df-mgp 14057  df-rng 14069  df-lssm 14493  df-sra 14575  df-rgmod 14576  df-lidl 14609

This theorem is referenced by:  isridlrng  14622  dflidl2  14628  df2idl2rng  14648