Theorem qus2idrng 14287

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 14289 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 2206	. 2 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
4		eqid 2205	. 2 |- ( +g ` R ) = ( +g ` R )
5		eqid 2205	. 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 2205	. . . 4 |- ( Base ` R ) = ( Base ` R )
8		eqid 2205	. . . 4 |- ( R ~QG S ) = ( R ~QG S )
9	7, 8	eqger 13560	. . 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 13697	. . . . . 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 13670	. . . . 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 2285	. . 3 |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> S e. (NrmSGrp ` R ) )
16	7, 8, 4	eqgcpbl 13564	. . 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 14285	. 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 13720	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 2176   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   Er wer 6617   Basecbs 12832   +g cplusg 12909   .rcmulr 12910   /._s cqus 13132  SubGrpcsubg 13503  NrmSGrpcnsg 13504   ~_QG cqg 13505   Abelcabl 13621  Rngcrng 13694  2Idealc2idl 14261

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-tp 3641  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-tpos 6331  df-er 6620  df-ec 6622  df-qs 6626  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-7 9100  df-8 9101  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-mulr 12923  df-sca 12925  df-vsca 12926  df-ip 12927  df-0g 13090  df-iimas 13134  df-qus 13135  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336  df-sbg 13337  df-subg 13506  df-nsg 13507  df-eqg 13508  df-cmn 13622  df-abl 13623  df-mgp 13683  df-rng 13695  df-oppr 13830  df-lssm 14115  df-sra 14197  df-rgmod 14198  df-lidl 14231  df-2idl 14262

This theorem is referenced by: (None)