Theorem quscrng 14673

Description: The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) (Proof shortened by AV, 3-Apr-2025.)

Hypotheses

Ref	Expression
quscrng.u	|- U = ( R /.s ( R ~QG S ) )
quscrng.i	|- I = (LIdeal ` R )

Assertion

Ref	Expression
quscrng	|- ( ( R e. CRing /\ S e. I ) -> U e. CRing )

Proof of Theorem quscrng

Dummy variables u v x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngring 14144	. . 3 |- ( R e. CRing -> R e. Ring )
2		simpr 110	. . . 4 |- ( ( R e. CRing /\ S e. I ) -> S e. I )
3		quscrng.i	. . . . . 6 |- I = (LIdeal ` R )
4	3	crng2idl 14671	. . . . 5 |- ( R e. CRing -> I = (2Ideal ` R ) )
5	4	adantr 276	. . . 4 |- ( ( R e. CRing /\ S e. I ) -> I = (2Ideal ` R ) )
6	2, 5	eleqtrd 2311	. . 3 |- ( ( R e. CRing /\ S e. I ) -> S e. (2Ideal ` R ) )
7		quscrng.u	. . . 4 |- U = ( R /.s ( R ~QG S ) )
8		eqid 2232	. . . 4 |- (2Ideal ` R ) = (2Ideal ` R )
9	7, 8	qusring 14667	. . 3 |- ( ( R e. Ring /\ S e. (2Ideal ` R ) ) -> U e. Ring )
10	1, 6, 9	syl2an2r 599	. 2 |- ( ( R e. CRing /\ S e. I ) -> U e. Ring )
11	7	a1i 9	. . . . . . 7 |- ( ( R e. CRing /\ S e. I ) -> U = ( R /.s ( R ~QG S ) ) )
12		eqidd 2233	. . . . . . 7 |- ( ( R e. CRing /\ S e. I ) -> ( Base ` R ) = ( Base ` R ) )
13		eqgex 13930	. . . . . . 7 |- ( ( R e. CRing /\ S e. I ) -> ( R ~QG S ) e. _V )
14	1	adantr 276	. . . . . . 7 |- ( ( R e. CRing /\ S e. I ) -> R e. Ring )
15	11, 12, 13, 14	qusbas 13532	. . . . . 6 |- ( ( R e. CRing /\ S e. I ) -> ( ( Base ` R ) /. ( R ~QG S ) ) = ( Base ` U ) )
16	15	eleq2d 2302	. . . . 5 |- ( ( R e. CRing /\ S e. I ) -> ( x e. ( ( Base ` R ) /. ( R ~QG S ) ) <-> x e. ( Base ` U ) ) )
17	15	eleq2d 2302	. . . . 5 |- ( ( R e. CRing /\ S e. I ) -> ( y e. ( ( Base ` R ) /. ( R ~QG S ) ) <-> y e. ( Base ` U ) ) )
18	16, 17	anbi12d 473	. . . 4 |- ( ( R e. CRing /\ S e. I ) -> ( ( x e. ( ( Base ` R ) /. ( R ~QG S ) ) /\ y e. ( ( Base ` R ) /. ( R ~QG S ) ) ) <-> ( x e. ( Base ` U ) /\ y e. ( Base ` U ) ) ) )
19		eqid 2232	. . . . . 6 |- ( ( Base ` R ) /. ( R ~QG S ) ) = ( ( Base ` R ) /. ( R ~QG S ) )
20		oveq2 6057	. . . . . . 7 |- ( [ u ] ( R ~QG S ) = y -> ( x ( .r ` U ) [ u ] ( R ~QG S ) ) = ( x ( .r ` U ) y ) )
21		oveq1 6056	. . . . . . 7 |- ( [ u ] ( R ~QG S ) = y -> ( [ u ] ( R ~QG S ) ( .r ` U ) x ) = ( y ( .r ` U ) x ) )
22	20, 21	eqeq12d 2247	. . . . . 6 |- ( [ u ] ( R ~QG S ) = y -> ( ( x ( .r ` U ) [ u ] ( R ~QG S ) ) = ( [ u ] ( R ~QG S ) ( .r ` U ) x ) <-> ( x ( .r ` U ) y ) = ( y ( .r ` U ) x ) ) )
23		oveq1 6056	. . . . . . . . 9 |- ( [ v ] ( R ~QG S ) = x -> ( [ v ] ( R ~QG S ) ( .r ` U ) [ u ] ( R ~QG S ) ) = ( x ( .r ` U ) [ u ] ( R ~QG S ) ) )
24		oveq2 6057	. . . . . . . . 9 |- ( [ v ] ( R ~QG S ) = x -> ( [ u ] ( R ~QG S ) ( .r ` U ) [ v ] ( R ~QG S ) ) = ( [ u ] ( R ~QG S ) ( .r ` U ) x ) )
25	23, 24	eqeq12d 2247	. . . . . . . 8 |- ( [ v ] ( R ~QG S ) = x -> ( ( [ v ] ( R ~QG S ) ( .r ` U ) [ u ] ( R ~QG S ) ) = ( [ u ] ( R ~QG S ) ( .r ` U ) [ v ] ( R ~QG S ) ) <-> ( x ( .r ` U ) [ u ] ( R ~QG S ) ) = ( [ u ] ( R ~QG S ) ( .r ` U ) x ) ) )
26		eqid 2232	. . . . . . . . . . . 12 |- ( Base ` R ) = ( Base ` R )
27		eqid 2232	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
28	26, 27	crngcom 14150	. . . . . . . . . . 11 |- ( ( R e. CRing /\ u e. ( Base ` R ) /\ v e. ( Base ` R ) ) -> ( u ( .r ` R ) v ) = ( v ( .r ` R ) u ) )
29	28	ad4ant134 1244	. . . . . . . . . 10 |- ( ( ( ( R e. CRing /\ S e. I ) /\ u e. ( Base ` R ) ) /\ v e. ( Base ` R ) ) -> ( u ( .r ` R ) v ) = ( v ( .r ` R ) u ) )
30	29	eceq1d 6802	. . . . . . . . 9 |- ( ( ( ( R e. CRing /\ S e. I ) /\ u e. ( Base ` R ) ) /\ v e. ( Base ` R ) ) -> [ ( u ( .r ` R ) v ) ] ( R ~QG S ) = [ ( v ( .r ` R ) u ) ] ( R ~QG S ) )
31		ringrng 14172	. . . . . . . . . . . . . 14 |- ( R e. Ring -> R e. Rng )
32	1, 31	syl 14	. . . . . . . . . . . . 13 |- ( R e. CRing -> R e. Rng )
33	32	adantr 276	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ S e. I ) -> R e. Rng )
34	3	lidlsubg 14626	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ S e. I ) -> S e. (SubGrp ` R ) )
35	1, 34	sylan 283	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ S e. I ) -> S e. (SubGrp ` R ) )
36	33, 6, 35	3jca 1204	. . . . . . . . . . 11 |- ( ( R e. CRing /\ S e. I ) -> ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) )
37	36	adantr 276	. . . . . . . . . 10 |- ( ( ( R e. CRing /\ S e. I ) /\ u e. ( Base ` R ) ) -> ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) )
38		simpr 110	. . . . . . . . . . 11 |- ( ( ( R e. CRing /\ S e. I ) /\ u e. ( Base ` R ) ) -> u e. ( Base ` R ) )
39	38	anim1i 340	. . . . . . . . . 10 |- ( ( ( ( R e. CRing /\ S e. I ) /\ u e. ( Base ` R ) ) /\ v e. ( Base ` R ) ) -> ( u e. ( Base ` R ) /\ v e. ( Base ` R ) ) )
40		eqid 2232	. . . . . . . . . . 11 |- ( R ~QG S ) = ( R ~QG S )
41		eqid 2232	. . . . . . . . . . 11 |- ( .r ` U ) = ( .r ` U )
42	40, 7, 26, 27, 41	qusmulrng 14672	. . . . . . . . . 10 |- ( ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) /\ ( u e. ( Base ` R ) /\ v e. ( Base ` R ) ) ) -> ( [ u ] ( R ~QG S ) ( .r ` U ) [ v ] ( R ~QG S ) ) = [ ( u ( .r ` R ) v ) ] ( R ~QG S ) )
43	37, 39, 42	syl2an2r 599	. . . . . . . . 9 |- ( ( ( ( R e. CRing /\ S e. I ) /\ u e. ( Base ` R ) ) /\ v e. ( Base ` R ) ) -> ( [ u ] ( R ~QG S ) ( .r ` U ) [ v ] ( R ~QG S ) ) = [ ( u ( .r ` R ) v ) ] ( R ~QG S ) )
44	39	ancomd 267	. . . . . . . . . 10 |- ( ( ( ( R e. CRing /\ S e. I ) /\ u e. ( Base ` R ) ) /\ v e. ( Base ` R ) ) -> ( v e. ( Base ` R ) /\ u e. ( Base ` R ) ) )
45	40, 7, 26, 27, 41	qusmulrng 14672	. . . . . . . . . 10 |- ( ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) /\ ( v e. ( Base ` R ) /\ u e. ( Base ` R ) ) ) -> ( [ v ] ( R ~QG S ) ( .r ` U ) [ u ] ( R ~QG S ) ) = [ ( v ( .r ` R ) u ) ] ( R ~QG S ) )
46	37, 44, 45	syl2an2r 599	. . . . . . . . 9 |- ( ( ( ( R e. CRing /\ S e. I ) /\ u e. ( Base ` R ) ) /\ v e. ( Base ` R ) ) -> ( [ v ] ( R ~QG S ) ( .r ` U ) [ u ] ( R ~QG S ) ) = [ ( v ( .r ` R ) u ) ] ( R ~QG S ) )
47	30, 43, 46	3eqtr4rd 2276	. . . . . . . 8 |- ( ( ( ( R e. CRing /\ S e. I ) /\ u e. ( Base ` R ) ) /\ v e. ( Base ` R ) ) -> ( [ v ] ( R ~QG S ) ( .r ` U ) [ u ] ( R ~QG S ) ) = ( [ u ] ( R ~QG S ) ( .r ` U ) [ v ] ( R ~QG S ) ) )
48	19, 25, 47	ectocld 6834	. . . . . . 7 |- ( ( ( ( R e. CRing /\ S e. I ) /\ u e. ( Base ` R ) ) /\ x e. ( ( Base ` R ) /. ( R ~QG S ) ) ) -> ( x ( .r ` U ) [ u ] ( R ~QG S ) ) = ( [ u ] ( R ~QG S ) ( .r ` U ) x ) )
49	48	an32s 570	. . . . . 6 |- ( ( ( ( R e. CRing /\ S e. I ) /\ x e. ( ( Base ` R ) /. ( R ~QG S ) ) ) /\ u e. ( Base ` R ) ) -> ( x ( .r ` U ) [ u ] ( R ~QG S ) ) = ( [ u ] ( R ~QG S ) ( .r ` U ) x ) )
50	19, 22, 49	ectocld 6834	. . . . 5 |- ( ( ( ( R e. CRing /\ S e. I ) /\ x e. ( ( Base ` R ) /. ( R ~QG S ) ) ) /\ y e. ( ( Base ` R ) /. ( R ~QG S ) ) ) -> ( x ( .r ` U ) y ) = ( y ( .r ` U ) x ) )
51	50	expl 378	. . . 4 |- ( ( R e. CRing /\ S e. I ) -> ( ( x e. ( ( Base ` R ) /. ( R ~QG S ) ) /\ y e. ( ( Base ` R ) /. ( R ~QG S ) ) ) -> ( x ( .r ` U ) y ) = ( y ( .r ` U ) x ) ) )
52	18, 51	sylbird 170	. . 3 |- ( ( R e. CRing /\ S e. I ) -> ( ( x e. ( Base ` U ) /\ y e. ( Base ` U ) ) -> ( x ( .r ` U ) y ) = ( y ( .r ` U ) x ) ) )
53	52	ralrimivv 2623	. 2 |- ( ( R e. CRing /\ S e. I ) -> A. x e. ( Base ` U ) A. y e. ( Base ` U ) ( x ( .r ` U ) y ) = ( y ( .r ` U ) x ) )
54		eqid 2232	. . 3 |- ( Base ` U ) = ( Base ` U )
55	54, 41	iscrng2 14151	. 2 |- ( U e. CRing <-> ( U e. Ring /\ A. x e. ( Base ` U ) A. y e. ( Base ` U ) ( x ( .r ` U ) y ) = ( y ( .r ` U ) x ) ) )
56	10, 53, 55	sylanbrc 417	1 |- ( ( R e. CRing /\ S e. I ) -> U e. CRing )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   _Vcvv 2812   ` cfv 5351  (class class class)co 6049   [cec 6764   /.cqs 6765   Basecbs 13204   .rcmulr 13283   /._s cqus 13505  SubGrpcsubg 13876   ~_QG cqg 13878  Rngcrng 14068   Ringcrg 14132   CRingccrg 14133  LIdealclidl 14607  2Idealc2idl 14639

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-pre-ltirr 8238  ax-pre-lttrn 8240  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-tp 3696  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-tpos 6475  df-er 6766  df-ec 6768  df-qs 6772  df-pnf 8309  df-mnf 8310  df-ltxr 8312  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-5 9298  df-6 9299  df-7 9300  df-8 9301  df-ndx 13207  df-slot 13208  df-base 13210  df-sets 13211  df-iress 13212  df-plusg 13295  df-mulr 13296  df-sca 13298  df-vsca 13299  df-ip 13300  df-0g 13463  df-iimas 13507  df-qus 13508  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-grp 13708  df-minusg 13709  df-sbg 13710  df-subg 13879  df-nsg 13880  df-eqg 13881  df-cmn 13995  df-abl 13996  df-mgp 14057  df-rng 14069  df-ur 14096  df-srg 14100  df-ring 14134  df-cring 14135  df-oppr 14204  df-subrg 14356  df-lmod 14429  df-lssm 14493  df-lsp 14527  df-sra 14575  df-rgmod 14576  df-lidl 14609  df-rsp 14610  df-2idl 14640

This theorem is referenced by:  zncrng2  14775