Theorem qus2idrng 14402

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 14404 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 2208	. 2 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
4		eqid 2207	. 2 |- ( +g ` R ) = ( +g ` R )
5		eqid 2207	. 2 |- ( .r ` R ) = ( .r ` R )
6		simp3 1002	. . 3 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> S e. (SubGrp ` R ) )
7		eqid 2207	. . . 4 |- ( Base ` R ) = ( Base ` R )
8		eqid 2207	. . . 4 |- ( R ~QG S ) = ( R ~QG S )
9	7, 8	eqger 13675	. . 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 13812	. . . . . 6 |- ( R e. Rng -> R e. Abel )
12	11	3ad2ant1 1021	. . . . 5 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> R e. Abel )
13		ablnsg 13785	. . . . 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 2287	. . 3 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> S e. (NrmSGrp ` R ) )
16	7, 8, 4	eqgcpbl 13679	. . 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 14400	. 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 1000	. 2 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> R e. Rng )
21	2, 3, 4, 5, 10, 17, 19, 20	qusrng 13835	1 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> U e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2178   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   Er wer 6640   Basecbs 12947   +g cplusg 13024   .rcmulr 13025   /._s cqus 13247  SubGrpcsubg 13618  NrmSGrpcnsg 13619   ~_QG cqg 13620   Abelcabl 13736  Rngcrng 13809  2Idealc2idl 14376

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-tpos 6354  df-er 6643  df-ec 6645  df-qs 6649  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-7 9135  df-8 9136  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-mulr 13038  df-sca 13040  df-vsca 13041  df-ip 13042  df-0g 13205  df-iimas 13249  df-qus 13250  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-sbg 13452  df-subg 13621  df-nsg 13622  df-eqg 13623  df-cmn 13737  df-abl 13738  df-mgp 13798  df-rng 13810  df-oppr 13945  df-lssm 14230  df-sra 14312  df-rgmod 14313  df-lidl 14346  df-2idl 14377

This theorem is referenced by: (None)