Theorem rnglidlrng 14807

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 14209	. . . 4 |- ( R e. Rng -> R e. Abel )
2	1	3ad2ant1 1049	. . 3 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> R e. Abel )
3		simp3 1030	. . 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 14113	. . 3 |- ( ( R e. Abel /\ U e. (SubGrp ` R ) ) -> I e. Abel )
6	2, 3, 5	syl2anc 415	. 2 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I e. Abel )
7		eqid 2238	. . . 4 |- ( 0g ` R ) = ( 0g ` R )
8	7	subg0cl 13962	. . 3 |- ( U e. (SubGrp ` R ) -> ( 0g ` R ) e. U )
9		rnglidlabl.l	. . . 4 |- L = (LIdeal ` R )
10	9, 4, 7	rnglidlmsgrp 14806	. . 3 |- ( ( R e. Rng /\ U e. L /\ ( 0g ` R ) e. U ) -> (mulGrp ` I ) e. Smgrp )
11	8, 10	syl3an3 1313	. 2 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> (mulGrp ` I ) e. Smgrp )
12		simpl1 1031	. . . . 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 14786	. . . . . . . . 9 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
14	13	sseld 3247	. . . . . . . 8 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
15	13	sseld 3247	. . . . . . . 8 |- ( U e. L -> ( b e. ( Base ` I ) -> b e. ( Base ` R ) ) )
16	13	sseld 3247	. . . . . . . 8 |- ( U e. L -> ( c e. ( Base ` I ) -> c e. ( Base ` R ) ) )
17	14, 15, 16	3anim123d 1360	. . . . . . 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 1050	. . . . . 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 2238	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
21		eqid 2238	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
22		eqid 2238	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
23	20, 21, 22	rngdi 14214	. . . . 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 415	. . . 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 14215	. . . . 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 415	. . . 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 1029	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> U e. L )
28		simp1 1028	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> R e. Rng )
29	4, 22	ressmulrg 13476	. . . . . . . . . 10 |- ( ( U e. L /\ R e. Rng ) -> ( .r ` R ) = ( .r ` I ) )
30	27, 28, 29	syl2anc 415	. . . . . . . . 9 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( .r ` R ) = ( .r ` I ) )
31	30	eqcomd 2244	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( .r ` I ) = ( .r ` R ) )
32		eqidd 2239	. . . . . . . 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 2239	. . . . . . . . . . 11 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` R ) = ( +g ` R ) )
35	33, 34, 27, 28	ressplusgd 13460	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` R ) = ( +g ` I ) )
36	35	eqcomd 2244	. . . . . . . . 9 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` I ) = ( +g ` R ) )
37	36	oveqd 6092	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( b ( +g ` I ) c ) = ( b ( +g ` R ) c ) )
38	31, 32, 37	oveq123d 6096	. . . . . . 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 6092	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
40	31	oveqd 6092	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( .r ` I ) c ) = ( a ( .r ` R ) c ) )
41	36, 39, 40	oveq123d 6096	. . . . . . 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 2253	. . . . . 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 6092	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( +g ` I ) b ) = ( a ( +g ` R ) b ) )
44		eqidd 2239	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> c = c )
45	31, 43, 44	oveq123d 6096	. . . . . . 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 6092	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( b ( .r ` I ) c ) = ( b ( .r ` R ) c ) )
47	36, 40, 46	oveq123d 6096	. . . . . . 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 2253	. . . . . 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 477	. . . . 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 957	. . 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 2633	. 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 2238	. . 3 |- ( Base ` I ) = ( Base ` I )
54		eqid 2238	. . 3 |- (mulGrp ` I ) = (mulGrp ` I )
55		eqid 2238	. . 3 |- ( +g ` I ) = ( +g ` I )
56		eqid 2238	. . 3 |- ( .r ` I ) = ( .r ` I )
57	53, 54, 55, 56	isrng 14208	. 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 1212	1 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5372  (class class class)co 6075   Basecbs 13330   ↾_s cress 13331   +g cplusg 13408   .rcmulr 13409   0gc0g 13587  Smgrpcsgrp 13693  SubGrpcsubg 13947   Abelcabl 14065  mulGrpcmgp 14194  Rngcrng 14206  LIdealclidl 14776

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-mulr 13422  df-sca 13424  df-vsca 13425  df-ip 13426  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-subg 13950  df-cmn 14066  df-abl 14067  df-mgp 14195  df-rng 14207  df-lssm 14662  df-sra 14744  df-rgmod 14745  df-lidl 14778

This theorem is referenced by:  rng2idlsubgsubrng  14829