Theorem qus2idrng 14834

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 14836 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 2239	. 2 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
4		eqid 2238	. 2 |- ( +g ` R ) = ( +g ` R )
5		eqid 2238	. 2 |- ( .r ` R ) = ( .r ` R )
6		simp3 1030	. . 3 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> S e. (SubGrp ` R ) )
7		eqid 2238	. . . 4 |- ( Base ` R ) = ( Base ` R )
8		eqid 2238	. . . 4 |- ( R ~QG S ) = ( R ~QG S )
9	7, 8	eqger 14004	. . 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 14209	. . . . . 6 |- ( R e. Rng -> R e. Abel )
12	11	3ad2ant1 1049	. . . . 5 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> R e. Abel )
13		ablnsg 14115	. . . . 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 2318	. . 3 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> S e. (NrmSGrp ` R ) )
16	7, 8, 4	eqgcpbl 14008	. . 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 14832	. 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 1028	. 2 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> R e. Rng )
21	2, 3, 4, 5, 10, 17, 19, 20	qusrng 14232	1 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> U e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   Er wer 6794   Basecbs 13330   +g cplusg 13408   .rcmulr 13409   /._s cqus 13600  SubGrpcsubg 13947  NrmSGrpcnsg 13948   ~_QG cqg 13949   Abelcabl 14065  Rngcrng 14206  2Idealc2idl 14808

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-tp 3713  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-tpos 6506  df-er 6797  df-ec 6799  df-qs 6803  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-mulr 13422  df-sca 13424  df-vsca 13425  df-ip 13426  df-0g 13589  df-iimas 13601  df-qus 13602  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-sbg 13787  df-subg 13950  df-nsg 13951  df-eqg 13952  df-cmn 14066  df-abl 14067  df-mgp 14195  df-rng 14207  df-oppr 14346  df-lssm 14662  df-sra 14744  df-rgmod 14745  df-lidl 14778  df-2idl 14809

This theorem is referenced by: (None)