Theorem rnglidlrng 14456

Description: A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption U e. (SubGrp ` R ) is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)

Hypotheses

Ref	Expression
rnglidlabl.l	|- L = (LIdeal ` R )
rnglidlabl.i	|- I = ( R ↾s U )

Assertion

Ref	Expression
rnglidlrng	|- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I e. Rng )

Proof of Theorem rnglidlrng

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngabl 13893	. . . 4 |- ( R e. Rng -> R e. Abel )
2	1	3ad2ant1 1042	. . 3 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> R e. Abel )
3		simp3 1023	. . 3 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> U e. (SubGrp ` R ) )
4		rnglidlabl.i	. . . 4 |- I = ( R ↾s U )
5	4	subgabl 13864	. . 3 |- ( ( R e. Abel /\ U e. (SubGrp ` R ) ) -> I e. Abel )
6	2, 3, 5	syl2anc 411	. 2 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I e. Abel )
7		eqid 2229	. . . 4 |- ( 0g ` R ) = ( 0g ` R )
8	7	subg0cl 13714	. . 3 |- ( U e. (SubGrp ` R ) -> ( 0g ` R ) e. U )
9		rnglidlabl.l	. . . 4 |- L = (LIdeal ` R )
10	9, 4, 7	rnglidlmsgrp 14455	. . 3 |- ( ( R e. Rng /\ U e. L /\ ( 0g ` R ) e. U ) -> (mulGrp ` I ) e. Smgrp )
11	8, 10	syl3an3 1306	. 2 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> (mulGrp ` I ) e. Smgrp )
12		simpl1 1024	. . . . 5 |- ( ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> R e. Rng )
13	9, 4	lidlssbas 14435	. . . . . . . . 9 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
14	13	sseld 3223	. . . . . . . 8 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
15	13	sseld 3223	. . . . . . . 8 |- ( U e. L -> ( b e. ( Base ` I ) -> b e. ( Base ` R ) ) )
16	13	sseld 3223	. . . . . . . 8 |- ( U e. L -> ( c e. ( Base ` I ) -> c e. ( Base ` R ) ) )
17	14, 15, 16	3anim123d 1353	. . . . . . 7 |- ( U e. L -> ( ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) )
18	17	3ad2ant2 1043	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) )
19	18	imp 124	. . . . 5 |- ( ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) )
20		eqid 2229	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
21		eqid 2229	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
22		eqid 2229	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
23	20, 21, 22	rngdi 13898	. . . . 5 |- ( ( R e. Rng /\ ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) -> ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) )
24	12, 19, 23	syl2anc 411	. . . 4 |- ( ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) )
25	20, 21, 22	rngdir 13899	. . . . 5 |- ( ( R e. Rng /\ ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) -> ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) )
26	12, 19, 25	syl2anc 411	. . . 4 |- ( ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) )
27		simp2 1022	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> U e. L )
28		simp1 1021	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> R e. Rng )
29	4, 22	ressmulrg 13173	. . . . . . . . . 10 |- ( ( U e. L /\ R e. Rng ) -> ( .r ` R ) = ( .r ` I ) )
30	27, 28, 29	syl2anc 411	. . . . . . . . 9 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( .r ` R ) = ( .r ` I ) )
31	30	eqcomd 2235	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( .r ` I ) = ( .r ` R ) )
32		eqidd 2230	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> a = a )
33	4	a1i 9	. . . . . . . . . . 11 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I = ( R ↾s U ) )
34		eqidd 2230	. . . . . . . . . . 11 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` R ) = ( +g ` R ) )
35	33, 34, 27, 28	ressplusgd 13157	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` R ) = ( +g ` I ) )
36	35	eqcomd 2235	. . . . . . . . 9 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` I ) = ( +g ` R ) )
37	36	oveqd 6017	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( b ( +g ` I ) c ) = ( b ( +g ` R ) c ) )
38	31, 32, 37	oveq123d 6021	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( a ( .r ` R ) ( b ( +g ` R ) c ) ) )
39	31	oveqd 6017	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
40	31	oveqd 6017	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( .r ` I ) c ) = ( a ( .r ` R ) c ) )
41	36, 39, 40	oveq123d 6021	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) )
42	38, 41	eqeq12d 2244	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) <-> ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) ) )
43	36	oveqd 6017	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( +g ` I ) b ) = ( a ( +g ` R ) b ) )
44		eqidd 2230	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> c = c )
45	31, 43, 44	oveq123d 6021	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( +g ` R ) b ) ( .r ` R ) c ) )
46	31	oveqd 6017	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( b ( .r ` I ) c ) = ( b ( .r ` R ) c ) )
47	36, 40, 46	oveq123d 6021	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) )
48	45, 47	eqeq12d 2244	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) <-> ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) ) )
49	42, 48	anbi12d 473	. . . . 5 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) <-> ( ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) /\ ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) ) ) )
50	49	adantr 276	. . . 4 |- ( ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) <-> ( ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) /\ ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) ) ) )
51	24, 26, 50	mpbir2and 950	. . 3 |- ( ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) )
52	51	ralrimivvva 2613	. 2 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> A. a e. ( Base ` I ) A. b e. ( Base ` I ) A. c e. ( Base ` I ) ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) )
53		eqid 2229	. . 3 |- ( Base ` I ) = ( Base ` I )
54		eqid 2229	. . 3 |- (mulGrp ` I ) = (mulGrp ` I )
55		eqid 2229	. . 3 |- ( +g ` I ) = ( +g ` I )
56		eqid 2229	. . 3 |- ( .r ` I ) = ( .r ` I )
57	53, 54, 55, 56	isrng 13892	. 2 |- ( I e. Rng <-> ( I e. Abel /\ (mulGrp ` I ) e. Smgrp /\ A. a e. ( Base ` I ) A. b e. ( Base ` I ) A. c e. ( Base ` I ) ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) ) )
58	6, 11, 52, 57	syl3anbrc 1205	1 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   ` cfv 5317  (class class class)co 6000   Basecbs 13027   ↾_s cress 13028   +g cplusg 13105   .rcmulr 13106   0gc0g 13284  Smgrpcsgrp 13429  SubGrpcsubg 13699   Abelcabl 13817  mulGrpcmgp 13878  Rngcrng 13890  LIdealclidl 14425

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-6 9169  df-7 9170  df-8 9171  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-iress 13035  df-plusg 13118  df-mulr 13119  df-sca 13121  df-vsca 13122  df-ip 13123  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-subg 13702  df-cmn 13818  df-abl 13819  df-mgp 13879  df-rng 13891  df-lssm 14311  df-sra 14393  df-rgmod 14394  df-lidl 14427

This theorem is referenced by:  rng2idlsubgsubrng  14478