Theorem qus2idrng 14558

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 14560 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 2232	. 2 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
4		eqid 2231	. 2 |- ( +g ` R ) = ( +g ` R )
5		eqid 2231	. 2 |- ( .r ` R ) = ( .r ` R )
6		simp3 1025	. . 3 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> S e. (SubGrp ` R ) )
7		eqid 2231	. . . 4 |- ( Base ` R ) = ( Base ` R )
8		eqid 2231	. . . 4 |- ( R ~QG S ) = ( R ~QG S )
9	7, 8	eqger 13829	. . 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 13967	. . . . . 6 |- ( R e. Rng -> R e. Abel )
12	11	3ad2ant1 1044	. . . . 5 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> R e. Abel )
13		ablnsg 13939	. . . . 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 2311	. . 3 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> S e. (NrmSGrp ` R ) )
16	7, 8, 4	eqgcpbl 13833	. . 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 14556	. 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 1023	. 2 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> R e. Rng )
21	2, 3, 4, 5, 10, 17, 19, 20	qusrng 13990	1 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> U e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   Er wer 6699   Basecbs 13100   +g cplusg 13178   .rcmulr 13179   /._s cqus 13401  SubGrpcsubg 13772  NrmSGrpcnsg 13773   ~_QG cqg 13774   Abelcabl 13890  Rngcrng 13964  2Idealc2idl 14532

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-tpos 6411  df-er 6702  df-ec 6704  df-qs 6708  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-ndx 13103  df-slot 13104  df-base 13106  df-sets 13107  df-iress 13108  df-plusg 13191  df-mulr 13192  df-sca 13194  df-vsca 13195  df-ip 13196  df-0g 13359  df-iimas 13403  df-qus 13404  df-mgm 13457  df-sgrp 13503  df-mnd 13518  df-grp 13604  df-minusg 13605  df-sbg 13606  df-subg 13775  df-nsg 13776  df-eqg 13777  df-cmn 13891  df-abl 13892  df-mgp 13953  df-rng 13965  df-oppr 14100  df-lssm 14386  df-sra 14468  df-rgmod 14469  df-lidl 14502  df-2idl 14533

This theorem is referenced by: (None)