Theorem 2idlcpblrng 14561

Description: The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) Generalization for non-unital rings and two-sided ideals which are subgroups of the additive group of the non-unital ring. (Revised by AV, 23-Feb-2025.)

Hypotheses

Ref	Expression
2idlcpblrng.x	|- X = ( Base ` R )
2idlcpblrng.r	|- E = ( R ~QG S )
2idlcpblrng.i	|- I = (2Ideal ` R )
2idlcpblrng.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
2idlcpblrng	|- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> ( ( A E C /\ B E D ) -> ( A .x. B ) E ( C .x. D ) ) )

Proof of Theorem 2idlcpblrng

Step	Hyp	Ref	Expression
1		simpl1 1026	. . . 4 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> R e. Rng )
2		simpl3 1028	. . . . . . . 8 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> S e. (SubGrp ` R ) )
3		2idlcpblrng.x	. . . . . . . . 9 |- X = ( Base ` R )
4		2idlcpblrng.r	. . . . . . . . 9 |- E = ( R ~QG S )
5	3, 4	eqger 13834	. . . . . . . 8 |- ( S e. (SubGrp ` R ) -> E Er X )
6	2, 5	syl 14	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> E Er X )
7		simprl 531	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> A E C )
8	6, 7	ersym 6719	. . . . . 6 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> C E A )
9		rngabl 13972	. . . . . . . 8 |- ( R e. Rng -> R e. Abel )
10	9	3ad2ant1 1044	. . . . . . 7 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> R e. Abel )
11		eqid 2230	. . . . . . . . . . . 12 |- (LIdeal ` R ) = (LIdeal ` R )
12		eqid 2230	. . . . . . . . . . . 12 |- (oppr ` R ) = (oppr ` R )
13		eqid 2230	. . . . . . . . . . . 12 |- (LIdeal ` (oppr ` R ) ) = (LIdeal ` (oppr ` R ) )
14		2idlcpblrng.i	. . . . . . . . . . . 12 |- I = (2Ideal ` R )
15	11, 12, 13, 14	2idlelb 14543	. . . . . . . . . . 11 |- ( S e. I <-> ( S e. (LIdeal ` R ) /\ S e. (LIdeal ` (oppr ` R ) ) ) )
16	15	simplbi 274	. . . . . . . . . 10 |- ( S e. I -> S e. (LIdeal ` R ) )
17	16	3ad2ant2 1045	. . . . . . . . 9 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> S e. (LIdeal ` R ) )
18	17	adantr 276	. . . . . . . 8 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> S e. (LIdeal ` R ) )
19	3, 11	lidlss 14514	. . . . . . . 8 |- ( S e. (LIdeal ` R ) -> S C_ X )
20	18, 19	syl 14	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> S C_ X )
21		eqid 2230	. . . . . . . 8 |- ( -g ` R ) = ( -g ` R )
22	3, 21, 4	eqgabl 13940	. . . . . . 7 |- ( ( R e. Abel /\ S C_ X ) -> ( C E A <-> ( C e. X /\ A e. X /\ ( A ( -g ` R ) C ) e. S ) ) )
23	10, 20, 22	syl2an2r 599	. . . . . 6 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( C E A <-> ( C e. X /\ A e. X /\ ( A ( -g ` R ) C ) e. S ) ) )
24	8, 23	mpbid 147	. . . . 5 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( C e. X /\ A e. X /\ ( A ( -g ` R ) C ) e. S ) )
25	24	simp2d 1036	. . . 4 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> A e. X )
26		simprr 533	. . . . . 6 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> B E D )
27	3, 21, 4	eqgabl 13940	. . . . . . 7 |- ( ( R e. Abel /\ S C_ X ) -> ( B E D <-> ( B e. X /\ D e. X /\ ( D ( -g ` R ) B ) e. S ) ) )
28	10, 20, 27	syl2an2r 599	. . . . . 6 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( B E D <-> ( B e. X /\ D e. X /\ ( D ( -g ` R ) B ) e. S ) ) )
29	26, 28	mpbid 147	. . . . 5 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( B e. X /\ D e. X /\ ( D ( -g ` R ) B ) e. S ) )
30	29	simp1d 1035	. . . 4 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> B e. X )
31		2idlcpblrng.t	. . . . 5 |- .x. = ( .r ` R )
32	3, 31	rngcl 13981	. . . 4 |- ( ( R e. Rng /\ A e. X /\ B e. X ) -> ( A .x. B ) e. X )
33	1, 25, 30, 32	syl3anc 1273	. . 3 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( A .x. B ) e. X )
34	24	simp1d 1035	. . . 4 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> C e. X )
35	29	simp2d 1036	. . . 4 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> D e. X )
36	3, 31	rngcl 13981	. . . 4 |- ( ( R e. Rng /\ C e. X /\ D e. X ) -> ( C .x. D ) e. X )
37	1, 34, 35, 36	syl3anc 1273	. . 3 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( C .x. D ) e. X )
38		rnggrp 13975	. . . . . . 7 |- ( R e. Rng -> R e. Grp )
39	38	3ad2ant1 1044	. . . . . 6 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> R e. Grp )
40	39	adantr 276	. . . . 5 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> R e. Grp )
41	3, 31	rngcl 13981	. . . . . 6 |- ( ( R e. Rng /\ C e. X /\ B e. X ) -> ( C .x. B ) e. X )
42	1, 34, 30, 41	syl3anc 1273	. . . . 5 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( C .x. B ) e. X )
43	3, 21	grpnnncan2 13703	. . . . 5 |- ( ( R e. Grp /\ ( ( C .x. D ) e. X /\ ( A .x. B ) e. X /\ ( C .x. B ) e. X ) ) -> ( ( ( C .x. D ) ( -g ` R ) ( C .x. B ) ) ( -g ` R ) ( ( A .x. B ) ( -g ` R ) ( C .x. B ) ) ) = ( ( C .x. D ) ( -g ` R ) ( A .x. B ) ) )
44	40, 37, 33, 42, 43	syl13anc 1275	. . . 4 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( ( ( C .x. D ) ( -g ` R ) ( C .x. B ) ) ( -g ` R ) ( ( A .x. B ) ( -g ` R ) ( C .x. B ) ) ) = ( ( C .x. D ) ( -g ` R ) ( A .x. B ) ) )
45	3, 31, 21, 1, 34, 35, 30	rngsubdi 13988	. . . . . 6 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( C .x. ( D ( -g ` R ) B ) ) = ( ( C .x. D ) ( -g ` R ) ( C .x. B ) ) )
46		eqid 2230	. . . . . . . . . 10 |- ( 0g ` R ) = ( 0g ` R )
47	46	subg0cl 13792	. . . . . . . . 9 |- ( S e. (SubGrp ` R ) -> ( 0g ` R ) e. S )
48	47	3ad2ant3 1046	. . . . . . . 8 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> ( 0g ` R ) e. S )
49	48	adantr 276	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( 0g ` R ) e. S )
50	29	simp3d 1037	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( D ( -g ` R ) B ) e. S )
51	46, 3, 31, 11	rnglidlmcl 14518	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. (LIdeal ` R ) /\ ( 0g ` R ) e. S ) /\ ( C e. X /\ ( D ( -g ` R ) B ) e. S ) ) -> ( C .x. ( D ( -g ` R ) B ) ) e. S )
52	1, 18, 49, 34, 50, 51	syl32anc 1281	. . . . . 6 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( C .x. ( D ( -g ` R ) B ) ) e. S )
53	45, 52	eqeltrrd 2308	. . . . 5 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( ( C .x. D ) ( -g ` R ) ( C .x. B ) ) e. S )
54	3, 21	grpsubcl 13686	. . . . . . . . 9 |- ( ( R e. Grp /\ A e. X /\ C e. X ) -> ( A ( -g ` R ) C ) e. X )
55	40, 25, 34, 54	syl3anc 1273	. . . . . . . 8 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( A ( -g ` R ) C ) e. X )
56		eqid 2230	. . . . . . . . 9 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
57	3, 31, 12, 56	opprmulg 14108	. . . . . . . 8 |- ( ( R e. Rng /\ B e. X /\ ( A ( -g ` R ) C ) e. X ) -> ( B ( .r ` (oppr ` R ) ) ( A ( -g ` R ) C ) ) = ( ( A ( -g ` R ) C ) .x. B ) )
58	1, 30, 55, 57	syl3anc 1273	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( B ( .r ` (oppr ` R ) ) ( A ( -g ` R ) C ) ) = ( ( A ( -g ` R ) C ) .x. B ) )
59	3, 31, 21, 1, 25, 34, 30	rngsubdir 13989	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( ( A ( -g ` R ) C ) .x. B ) = ( ( A .x. B ) ( -g ` R ) ( C .x. B ) ) )
60	58, 59	eqtrd 2263	. . . . . 6 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( B ( .r ` (oppr ` R ) ) ( A ( -g ` R ) C ) ) = ( ( A .x. B ) ( -g ` R ) ( C .x. B ) ) )
61	12	opprrng 14114	. . . . . . . . 9 |- ( R e. Rng -> (oppr ` R ) e. Rng )
62	61	3ad2ant1 1044	. . . . . . . 8 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> (oppr ` R ) e. Rng )
63	62	adantr 276	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> (oppr ` R ) e. Rng )
64	15	simprbi 275	. . . . . . . . 9 |- ( S e. I -> S e. (LIdeal ` (oppr ` R ) ) )
65	64	3ad2ant2 1045	. . . . . . . 8 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> S e. (LIdeal ` (oppr ` R ) ) )
66	65	adantr 276	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> S e. (LIdeal ` (oppr ` R ) ) )
67	12, 46	oppr0g 14118	. . . . . . . . 9 |- ( R e. Rng -> ( 0g ` R ) = ( 0g ` (oppr ` R ) ) )
68	1, 67	syl 14	. . . . . . . 8 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( 0g ` R ) = ( 0g ` (oppr ` R ) ) )
69	68, 49	eqeltrrd 2308	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( 0g ` (oppr ` R ) ) e. S )
70	12, 3	opprbasg 14112	. . . . . . . . 9 |- ( R e. Rng -> X = ( Base ` (oppr ` R ) ) )
71	1, 70	syl 14	. . . . . . . 8 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> X = ( Base ` (oppr ` R ) ) )
72	30, 71	eleqtrd 2309	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> B e. ( Base ` (oppr ` R ) ) )
73	24	simp3d 1037	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( A ( -g ` R ) C ) e. S )
74		eqid 2230	. . . . . . . 8 |- ( 0g ` (oppr ` R ) ) = ( 0g ` (oppr ` R ) )
75		eqid 2230	. . . . . . . 8 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
76	74, 75, 56, 13	rnglidlmcl 14518	. . . . . . 7 |- ( ( ( (oppr ` R ) e. Rng /\ S e. (LIdeal ` (oppr ` R ) ) /\ ( 0g ` (oppr ` R ) ) e. S ) /\ ( B e. ( Base ` (oppr ` R ) ) /\ ( A ( -g ` R ) C ) e. S ) ) -> ( B ( .r ` (oppr ` R ) ) ( A ( -g ` R ) C ) ) e. S )
77	63, 66, 69, 72, 73, 76	syl32anc 1281	. . . . . 6 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( B ( .r ` (oppr ` R ) ) ( A ( -g ` R ) C ) ) e. S )
78	60, 77	eqeltrrd 2308	. . . . 5 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( ( A .x. B ) ( -g ` R ) ( C .x. B ) ) e. S )
79	21	subgsubcl 13795	. . . . 5 |- ( ( S e. (SubGrp ` R ) /\ ( ( C .x. D ) ( -g ` R ) ( C .x. B ) ) e. S /\ ( ( A .x. B ) ( -g ` R ) ( C .x. B ) ) e. S ) -> ( ( ( C .x. D ) ( -g ` R ) ( C .x. B ) ) ( -g ` R ) ( ( A .x. B ) ( -g ` R ) ( C .x. B ) ) ) e. S )
80	2, 53, 78, 79	syl3anc 1273	. . . 4 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( ( ( C .x. D ) ( -g ` R ) ( C .x. B ) ) ( -g ` R ) ( ( A .x. B ) ( -g ` R ) ( C .x. B ) ) ) e. S )
81	44, 80	eqeltrrd 2308	. . 3 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( ( C .x. D ) ( -g ` R ) ( A .x. B ) ) e. S )
82	3, 21, 4	eqgabl 13940	. . . 4 |- ( ( R e. Abel /\ S C_ X ) -> ( ( A .x. B ) E ( C .x. D ) <-> ( ( A .x. B ) e. X /\ ( C .x. D ) e. X /\ ( ( C .x. D ) ( -g ` R ) ( A .x. B ) ) e. S ) ) )
83	10, 20, 82	syl2an2r 599	. . 3 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( ( A .x. B ) E ( C .x. D ) <-> ( ( A .x. B ) e. X /\ ( C .x. D ) e. X /\ ( ( C .x. D ) ( -g ` R ) ( A .x. B ) ) e. S ) ) )
84	33, 37, 81, 83	mpbir3and 1206	. 2 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( A .x. B ) E ( C .x. D ) )
85	84	ex 115	1 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> ( ( A E C /\ B E D ) -> ( A .x. B ) E ( C .x. D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2201   C_ wss 3199   class class class wbr 4089   ` cfv 5328  (class class class)co 6023   Er wer 6704   Basecbs 13105   .rcmulr 13184   0gc0g 13362   Grpcgrp 13606   -gcsg 13608  SubGrpcsubg 13777   ~_QG cqg 13779   Abelcabl 13895  Rngcrng 13969  opp_rcoppr 14104  LIdealclidl 14505  2Idealc2idl 14537

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 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-addass 8139  ax-i2m1 8142  ax-0lt1 8143  ax-0id 8145  ax-rnegex 8146  ax-pre-ltirr 8149  ax-pre-lttrn 8151  ax-pre-ltadd 8153

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-tpos 6416  df-er 6707  df-pnf 8221  df-mnf 8222  df-ltxr 8224  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-ndx 13108  df-slot 13109  df-base 13111  df-sets 13112  df-iress 13113  df-plusg 13196  df-mulr 13197  df-sca 13199  df-vsca 13200  df-ip 13201  df-0g 13364  df-mgm 13462  df-sgrp 13508  df-mnd 13523  df-grp 13609  df-minusg 13610  df-sbg 13611  df-subg 13780  df-eqg 13782  df-cmn 13896  df-abl 13897  df-mgp 13958  df-rng 13970  df-oppr 14105  df-lssm 14391  df-sra 14473  df-rgmod 14474  df-lidl 14507  df-2idl 14538

This theorem is referenced by:  2idlcpbl  14562  qus2idrng  14563  qusmulrng  14570