Theorem rnglidlrng 14130

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 13567	. . . 4 |- ( R e. Rng -> R e. Abel )
2	1	3ad2ant1 1020	. . 3 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> R e. Abel )
3		simp3 1001	. . 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 13538	. . 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 2196	. . . 4 |- ( 0g ` R ) = ( 0g ` R )
8	7	subg0cl 13388	. . 3 |- ( U e. (SubGrp ` R ) -> ( 0g ` R ) e. U )
9		rnglidlabl.l	. . . 4 |- L = (LIdeal ` R )
10	9, 4, 7	rnglidlmsgrp 14129	. . 3 |- ( ( R e. Rng /\ U e. L /\ ( 0g ` R ) e. U ) -> (mulGrp ` I ) e. Smgrp )
11	8, 10	syl3an3 1284	. 2 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> (mulGrp ` I ) e. Smgrp )
12		simpl1 1002	. . . . 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 14109	. . . . . . . . 9 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
14	13	sseld 3183	. . . . . . . 8 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
15	13	sseld 3183	. . . . . . . 8 |- ( U e. L -> ( b e. ( Base ` I ) -> b e. ( Base ` R ) ) )
16	13	sseld 3183	. . . . . . . 8 |- ( U e. L -> ( c e. ( Base ` I ) -> c e. ( Base ` R ) ) )
17	14, 15, 16	3anim123d 1330	. . . . . . 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 1021	. . . . . 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 2196	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
21		eqid 2196	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
22		eqid 2196	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
23	20, 21, 22	rngdi 13572	. . . . 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 13573	. . . . 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 1000	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> U e. L )
28		simp1 999	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> R e. Rng )
29	4, 22	ressmulrg 12847	. . . . . . . . . 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 2202	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( .r ` I ) = ( .r ` R ) )
32		eqidd 2197	. . . . . . . 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 2197	. . . . . . . . . . 11 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` R ) = ( +g ` R ) )
35	33, 34, 27, 28	ressplusgd 12831	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` R ) = ( +g ` I ) )
36	35	eqcomd 2202	. . . . . . . . 9 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` I ) = ( +g ` R ) )
37	36	oveqd 5942	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( b ( +g ` I ) c ) = ( b ( +g ` R ) c ) )
38	31, 32, 37	oveq123d 5946	. . . . . . 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 5942	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
40	31	oveqd 5942	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( .r ` I ) c ) = ( a ( .r ` R ) c ) )
41	36, 39, 40	oveq123d 5946	. . . . . . 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 2211	. . . . . 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 5942	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( +g ` I ) b ) = ( a ( +g ` R ) b ) )
44		eqidd 2197	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> c = c )
45	31, 43, 44	oveq123d 5946	. . . . . . 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 5942	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( b ( .r ` I ) c ) = ( b ( .r ` R ) c ) )
47	36, 40, 46	oveq123d 5946	. . . . . . 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 2211	. . . . . 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 946	. . 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 2580	. 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 2196	. . 3 |- ( Base ` I ) = ( Base ` I )
54		eqid 2196	. . 3 |- (mulGrp ` I ) = (mulGrp ` I )
55		eqid 2196	. . 3 |- ( +g ` I ) = ( +g ` I )
56		eqid 2196	. . 3 |- ( .r ` I ) = ( .r ` I )
57	53, 54, 55, 56	isrng 13566	. 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 1183	1 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   A.wral 2475   ` cfv 5259  (class class class)co 5925   Basecbs 12703   ↾_s cress 12704   +g cplusg 12780   .rcmulr 12781   0gc0g 12958  Smgrpcsgrp 13103  SubGrpcsubg 13373   Abelcabl 13491  mulGrpcmgp 13552  Rngcrng 13564  LIdealclidl 14099

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-sca 12796  df-vsca 12797  df-ip 12798  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-subg 13376  df-cmn 13492  df-abl 13493  df-mgp 13553  df-rng 13565  df-lssm 13985  df-sra 14067  df-rgmod 14068  df-lidl 14101

This theorem is referenced by:  rng2idlsubgsubrng  14152