Theorem dflidl2rng 14501

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 2231	. . . . . . 7 |- ( 0g ` R ) = ( 0g ` R )
4	3	subg0cl 13774	. . . . . 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 1203	. . . 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 14500	. . . 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 2614	. 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 13766	. . . 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 2819	. . . . 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 2231	. . . . . . . . 9 |- ( +g ` R ) = ( +g ` R )
19	18	subgcl 13776	. . . . . . . 8 |- ( ( I e. (SubGrp ` R ) /\ ( x .x. y ) e. I /\ z e. I ) -> ( ( x .x. y ) ( +g ` R ) z ) e. I )
20	19	ad5ant245 1262	. . . . . . 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 2605	. . . . . 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 2601	. . . 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 14499	. . 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 1207	. 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 1004   = wceq 1397   E.wex 1540   e. wcel 2202   A.wral 2510   C_ wss 3200   ` cfv 5326  (class class class)co 6018   Basecbs 13087   +g cplusg 13165   .rcmulr 13166   0gc0g 13344  SubGrpcsubg 13759  Rngcrng 13951  LIdealclidl 14487

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-ndx 13090  df-slot 13091  df-base 13093  df-sets 13094  df-iress 13095  df-plusg 13178  df-mulr 13179  df-sca 13181  df-vsca 13182  df-ip 13183  df-0g 13346  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-grp 13591  df-subg 13762  df-abl 13879  df-mgp 13940  df-rng 13952  df-lssm 14373  df-sra 14455  df-rgmod 14456  df-lidl 14489

This theorem is referenced by:  isridlrng  14502  dflidl2  14508  df2idl2rng  14528