Theorem rnglidlrng 14345

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 13782	. . . 4 |- ( R e. Rng -> R e. Abel )
2	1	3ad2ant1 1021	. . 3 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> R e. Abel )
3		simp3 1002	. . 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 13753	. . 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 2206	. . . 4 |- ( 0g ` R ) = ( 0g ` R )
8	7	subg0cl 13603	. . 3 |- ( U e. (SubGrp ` R ) -> ( 0g ` R ) e. U )
9		rnglidlabl.l	. . . 4 |- L = (LIdeal ` R )
10	9, 4, 7	rnglidlmsgrp 14344	. . 3 |- ( ( R e. Rng /\ U e. L /\ ( 0g ` R ) e. U ) -> (mulGrp ` I ) e. Smgrp )
11	8, 10	syl3an3 1285	. 2 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> (mulGrp ` I ) e. Smgrp )
12		simpl1 1003	. . . . 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 14324	. . . . . . . . 9 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
14	13	sseld 3196	. . . . . . . 8 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
15	13	sseld 3196	. . . . . . . 8 |- ( U e. L -> ( b e. ( Base ` I ) -> b e. ( Base ` R ) ) )
16	13	sseld 3196	. . . . . . . 8 |- ( U e. L -> ( c e. ( Base ` I ) -> c e. ( Base ` R ) ) )
17	14, 15, 16	3anim123d 1332	. . . . . . 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 1022	. . . . . 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 2206	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
21		eqid 2206	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
22		eqid 2206	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
23	20, 21, 22	rngdi 13787	. . . . 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 13788	. . . . 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 1001	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> U e. L )
28		simp1 1000	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> R e. Rng )
29	4, 22	ressmulrg 13062	. . . . . . . . . 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 2212	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( .r ` I ) = ( .r ` R ) )
32		eqidd 2207	. . . . . . . 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 2207	. . . . . . . . . . 11 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` R ) = ( +g ` R ) )
35	33, 34, 27, 28	ressplusgd 13046	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` R ) = ( +g ` I ) )
36	35	eqcomd 2212	. . . . . . . . 9 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` I ) = ( +g ` R ) )
37	36	oveqd 5979	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( b ( +g ` I ) c ) = ( b ( +g ` R ) c ) )
38	31, 32, 37	oveq123d 5983	. . . . . . 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 5979	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
40	31	oveqd 5979	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( .r ` I ) c ) = ( a ( .r ` R ) c ) )
41	36, 39, 40	oveq123d 5983	. . . . . . 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 2221	. . . . . 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 5979	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( +g ` I ) b ) = ( a ( +g ` R ) b ) )
44		eqidd 2207	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> c = c )
45	31, 43, 44	oveq123d 5983	. . . . . . 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 5979	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( b ( .r ` I ) c ) = ( b ( .r ` R ) c ) )
47	36, 40, 46	oveq123d 5983	. . . . . . 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 2221	. . . . . 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 947	. . 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 2590	. 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 2206	. . 3 |- ( Base ` I ) = ( Base ` I )
54		eqid 2206	. . 3 |- (mulGrp ` I ) = (mulGrp ` I )
55		eqid 2206	. . 3 |- ( +g ` I ) = ( +g ` I )
56		eqid 2206	. . 3 |- ( .r ` I ) = ( .r ` I )
57	53, 54, 55, 56	isrng 13781	. 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 1184	1 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2177   A.wral 2485   ` cfv 5285  (class class class)co 5962   Basecbs 12917   ↾_s cress 12918   +g cplusg 12994   .rcmulr 12995   0gc0g 13173  Smgrpcsgrp 13318  SubGrpcsubg 13588   Abelcabl 13706  mulGrpcmgp 13767  Rngcrng 13779  LIdealclidl 14314

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-i2m1 8060  ax-0lt1 8061  ax-0id 8063  ax-rnegex 8064  ax-pre-ltirr 8067  ax-pre-lttrn 8069  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-pnf 8139  df-mnf 8140  df-ltxr 8142  df-inn 9067  df-2 9125  df-3 9126  df-4 9127  df-5 9128  df-6 9129  df-7 9130  df-8 9131  df-ndx 12920  df-slot 12921  df-base 12923  df-sets 12924  df-iress 12925  df-plusg 13007  df-mulr 13008  df-sca 13010  df-vsca 13011  df-ip 13012  df-0g 13175  df-mgm 13273  df-sgrp 13319  df-mnd 13334  df-grp 13420  df-subg 13591  df-cmn 13707  df-abl 13708  df-mgp 13768  df-rng 13780  df-lssm 14200  df-sra 14282  df-rgmod 14283  df-lidl 14316

This theorem is referenced by:  rng2idlsubgsubrng  14367