Theorem 2idlcpblrng 13838

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 1002	. . . 4 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> R e. Rng )
2		simpl3 1004	. . . . . . . 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 13163	. . . . . . . 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 529	. . . . . . 7 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> A E C )
8	6, 7	ersym 6571	. . . . . 6 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> C E A )
9		rngabl 13289	. . . . . . . 8 |- ( R e. Rng -> R e. Abel )
10	9	3ad2ant1 1020	. . . . . . 7 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> R e. Abel )
11		eqid 2189	. . . . . . . . . . . 12 |- (LIdeal ` R ) = (LIdeal ` R )
12		eqid 2189	. . . . . . . . . . . 12 |- (oppr ` R ) = (oppr ` R )
13		eqid 2189	. . . . . . . . . . . 12 |- (LIdeal ` (oppr ` R ) ) = (LIdeal ` (oppr ` R ) )
14		2idlcpblrng.i	. . . . . . . . . . . 12 |- I = (2Ideal ` R )
15	11, 12, 13, 14	2idlelb 13820	. . . . . . . . . . 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 1021	. . . . . . . . 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 13792	. . . . . . . 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 2189	. . . . . . . 8 |- ( -g ` R ) = ( -g ` R )
22	3, 21, 4	eqgabl 13267	. . . . . . 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 595	. . . . . 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 1012	. . . 4 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> A e. X )
26		simprr 531	. . . . . 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 13267	. . . . . . 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 595	. . . . . 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 1011	. . . 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 13298	. . . 4 |- ( ( R e. Rng /\ A e. X /\ B e. X ) -> ( A .x. B ) e. X )
33	1, 25, 30, 32	syl3anc 1249	. . 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 1011	. . . 4 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> C e. X )
35	29	simp2d 1012	. . . 4 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> D e. X )
36	3, 31	rngcl 13298	. . . 4 |- ( ( R e. Rng /\ C e. X /\ D e. X ) -> ( C .x. D ) e. X )
37	1, 34, 35, 36	syl3anc 1249	. . 3 |- ( ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) /\ ( A E C /\ B E D ) ) -> ( C .x. D ) e. X )
38		rnggrp 13292	. . . . . . 7 |- ( R e. Rng -> R e. Grp )
39	38	3ad2ant1 1020	. . . . . 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 13298	. . . . . 6 |- ( ( R e. Rng /\ C e. X /\ B e. X ) -> ( C .x. B ) e. X )
42	1, 34, 30, 41	syl3anc 1249	. . . . 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 13041	. . . . 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 1251	. . . 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 13305	. . . . . 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 2189	. . . . . . . . . 10 |- ( 0g ` R ) = ( 0g ` R )
47	46	subg0cl 13121	. . . . . . . . 9 |- ( S e. (SubGrp ` R ) -> ( 0g ` R ) e. S )
48	47	3ad2ant3 1022	. . . . . . . 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 1013	. . . . . . 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 13796	. . . . . . 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 1257	. . . . . 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 2267	. . . . 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 13024	. . . . . . . . 9 |- ( ( R e. Grp /\ A e. X /\ C e. X ) -> ( A ( -g ` R ) C ) e. X )
55	40, 25, 34, 54	syl3anc 1249	. . . . . . . 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 2189	. . . . . . . . 9 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
57	3, 31, 12, 56	opprmulg 13421	. . . . . . . 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 1249	. . . . . . 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 13306	. . . . . . 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 2222	. . . . . 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 13427	. . . . . . . . 9 |- ( R e. Rng -> (oppr ` R ) e. Rng )
62	61	3ad2ant1 1020	. . . . . . . 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 1021	. . . . . . . 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 13431	. . . . . . . . 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 2267	. . . . . . 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 13425	. . . . . . . . 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 2268	. . . . . . 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 1013	. . . . . . 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 2189	. . . . . . . 8 |- ( 0g ` (oppr ` R ) ) = ( 0g ` (oppr ` R ) )
75		eqid 2189	. . . . . . . 8 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
76	74, 75, 56, 13	rnglidlmcl 13796	. . . . . . 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 1257	. . . . . 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 2267	. . . . 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 13124	. . . . 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 1249	. . . 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 2267	. . 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 13267	. . . 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 595	. . 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 1182	. 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 980   = wceq 1364   e. wcel 2160   C_ wss 3144   class class class wbr 4018   ` cfv 5235  (class class class)co 5896   Er wer 6556   Basecbs 12512   .rcmulr 12590   0gc0g 12761   Grpcgrp 12945   -gcsg 12947  SubGrpcsubg 13106   ~_QG cqg 13108   Abelcabl 13224  Rngcrng 13286  opp_rcoppr 13417  LIdealclidl 13783  2Idealc2idl 13815

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-i2m1 7946  ax-0lt1 7947  ax-0id 7949  ax-rnegex 7950  ax-pre-ltirr 7953  ax-pre-lttrn 7955  ax-pre-ltadd 7957

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-tpos 6270  df-er 6559  df-pnf 8024  df-mnf 8025  df-ltxr 8027  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-5 9011  df-6 9012  df-7 9013  df-8 9014  df-ndx 12515  df-slot 12516  df-base 12518  df-sets 12519  df-iress 12520  df-plusg 12602  df-mulr 12603  df-sca 12605  df-vsca 12606  df-ip 12607  df-0g 12763  df-mgm 12832  df-sgrp 12865  df-mnd 12878  df-grp 12948  df-minusg 12949  df-sbg 12950  df-subg 13109  df-eqg 13111  df-cmn 13225  df-abl 13226  df-mgp 13275  df-rng 13287  df-oppr 13418  df-lssm 13669  df-sra 13751  df-rgmod 13752  df-lidl 13785  df-2idl 13816

This theorem is referenced by:  2idlcpbl  13839  qus2idrng  13840  qusmulrng  13846