Theorem qus2idrng 14157

Description: The quotient of a non-unital ring modulo a two-sided ideal, which is a subgroup of the additive group of the non-unital ring, is a non-unital ring (qusring 14159 analog). (Contributed by AV, 23-Feb-2025.)

Hypotheses

Ref	Expression
qus2idrng.u	|- U = ( R /.s ( R ~QG S ) )
qus2idrng.i	|- I = (2Ideal ` R )

Assertion

Ref	Expression
qus2idrng	|- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> U e. Rng )

Proof of Theorem qus2idrng

Dummy variables a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qus2idrng.u	. . 3 |- U = ( R /.s ( R ~QG S ) )
2	1	a1i 9	. 2 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> U = ( R /.s ( R ~QG S ) ) )
3		eqidd 2197	. 2 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
4		eqid 2196	. 2 |- ( +g ` R ) = ( +g ` R )
5		eqid 2196	. 2 |- ( .r ` R ) = ( .r ` R )
6		simp3 1001	. . 3 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> S e. (SubGrp ` R ) )
7		eqid 2196	. . . 4 |- ( Base ` R ) = ( Base ` R )
8		eqid 2196	. . . 4 |- ( R ~QG S ) = ( R ~QG S )
9	7, 8	eqger 13430	. . 3 |- ( S e. (SubGrp ` R ) -> ( R ~QG S ) Er ( Base ` R ) )
10	6, 9	syl 14	. 2 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> ( R ~QG S ) Er ( Base ` R ) )
11		rngabl 13567	. . . . . 6 |- ( R e. Rng -> R e. Abel )
12	11	3ad2ant1 1020	. . . . 5 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> R e. Abel )
13		ablnsg 13540	. . . . 5 |- ( R e. Abel -> (NrmSGrp ` R ) = (SubGrp ` R ) )
14	12, 13	syl 14	. . . 4 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> (NrmSGrp ` R ) = (SubGrp ` R ) )
15	6, 14	eleqtrrd 2276	. . 3 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> S e. (NrmSGrp ` R ) )
16	7, 8, 4	eqgcpbl 13434	. . 3 |- ( S e. (NrmSGrp ` R ) -> ( ( a ( R ~QG S ) c /\ b ( R ~QG S ) d ) -> ( a ( +g ` R ) b ) ( R ~QG S ) ( c ( +g ` R ) d ) ) )
17	15, 16	syl 14	. 2 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> ( ( a ( R ~QG S ) c /\ b ( R ~QG S ) d ) -> ( a ( +g ` R ) b ) ( R ~QG S ) ( c ( +g ` R ) d ) ) )
18		qus2idrng.i	. . 3 |- I = (2Ideal ` R )
19	7, 8, 18, 5	2idlcpblrng 14155	. 2 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> ( ( a ( R ~QG S ) c /\ b ( R ~QG S ) d ) -> ( a ( .r ` R ) b ) ( R ~QG S ) ( c ( .r ` R ) d ) ) )
20		simp1 999	. 2 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> R e. Rng )
21	2, 3, 4, 5, 10, 17, 19, 20	qusrng 13590	1 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> U e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   Er wer 6598   Basecbs 12703   +g cplusg 12780   .rcmulr 12781   /._s cqus 13002  SubGrpcsubg 13373  NrmSGrpcnsg 13374   ~_QG cqg 13375   Abelcabl 13491  Rngcrng 13564  2Idealc2idl 14131

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-nul 4160  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-3or 981  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-tp 3631  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-1st 6207  df-2nd 6208  df-tpos 6312  df-er 6601  df-ec 6603  df-qs 6607  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-iimas 13004  df-qus 13005  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-sbg 13207  df-subg 13376  df-nsg 13377  df-eqg 13378  df-cmn 13492  df-abl 13493  df-mgp 13553  df-rng 13565  df-oppr 13700  df-lssm 13985  df-sra 14067  df-rgmod 14068  df-lidl 14101  df-2idl 14132

This theorem is referenced by: (None)