Theorem quscrng 13844

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 13359	. . 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 13842	. . . . 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 2268	. . 3 |- ( ( R e. CRing /\ S e. I ) -> S e. (2Ideal ` R ) )
7		quscrng.u	. . . 4 |- U = ( R /.s ( R ~QG S ) )
8		eqid 2189	. . . 4 |- (2Ideal ` R ) = (2Ideal ` R )
9	7, 8	qusring 13839	. . 3 |- ( ( R e. Ring /\ S e. (2Ideal ` R ) ) -> U e. Ring )
10	1, 6, 9	syl2an2r 595	. 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 2190	. . . . . . 7 |- ( ( R e. CRing /\ S e. I ) -> ( Base ` R ) = ( Base ` R ) )
13		eqgex 13157	. . . . . . 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 12801	. . . . . 6 |- ( ( R e. CRing /\ S e. I ) -> ( ( Base ` R ) /. ( R ~QG S ) ) = ( Base ` U ) )
16	15	eleq2d 2259	. . . . 5 |- ( ( R e. CRing /\ S e. I ) -> ( x e. ( ( Base ` R ) /. ( R ~QG S ) ) <-> x e. ( Base ` U ) ) )
17	15	eleq2d 2259	. . . . 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 2189	. . . . . 6 |- ( ( Base ` R ) /. ( R ~QG S ) ) = ( ( Base ` R ) /. ( R ~QG S ) )
20		oveq2 5903	. . . . . . 7 |- ( [ u ] ( R ~QG S ) = y -> ( x ( .r ` U ) [ u ] ( R ~QG S ) ) = ( x ( .r ` U ) y ) )
21		oveq1 5902	. . . . . . 7 |- ( [ u ] ( R ~QG S ) = y -> ( [ u ] ( R ~QG S ) ( .r ` U ) x ) = ( y ( .r ` U ) x ) )
22	20, 21	eqeq12d 2204	. . . . . 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 5902	. . . . . . . . 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 5903	. . . . . . . . 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 2204	. . . . . . . 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 2189	. . . . . . . . . . . 12 |- ( Base ` R ) = ( Base ` R )
27		eqid 2189	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
28	26, 27	crngcom 13365	. . . . . . . . . . 11 |- ( ( R e. CRing /\ u e. ( Base ` R ) /\ v e. ( Base ` R ) ) -> ( u ( .r ` R ) v ) = ( v ( .r ` R ) u ) )
29	28	ad4ant134 1219	. . . . . . . . . 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 6594	. . . . . . . . 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 13387	. . . . . . . . . . . . . 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 13799	. . . . . . . . . . . . 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 1179	. . . . . . . . . . 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 2189	. . . . . . . . . . 11 |- ( R ~QG S ) = ( R ~QG S )
41		eqid 2189	. . . . . . . . . . 11 |- ( .r ` U ) = ( .r ` U )
42	40, 7, 26, 27, 41	qusmulrng 13843	. . . . . . . . . 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 595	. . . . . . . . 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 13843	. . . . . . . . . 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 595	. . . . . . . . 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 2233	. . . . . . . 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 6626	. . . . . . 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 568	. . . . . 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 6626	. . . . 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 2571	. 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 2189	. . 3 |- ( Base ` U ) = ( Base ` U )
55	54, 41	iscrng2 13366	. 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 980   = wceq 1364   e. wcel 2160   A.wral 2468   _Vcvv 2752   ` cfv 5235  (class class class)co 5895   [cec 6556   /.cqs 6557   Basecbs 12511   .rcmulr 12587   /._s cqus 12774  SubGrpcsubg 13103   ~_QG cqg 13105  Rngcrng 13283   Ringcrg 13347   CRingccrg 13348  LIdealclidl 13780  2Idealc2idl 13812

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-pre-ltirr 7952  ax-pre-lttrn 7954  ax-pre-ltadd 7956

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-tp 3615  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-tpos 6269  df-er 6558  df-ec 6560  df-qs 6564  df-pnf 8023  df-mnf 8024  df-ltxr 8026  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-5 9010  df-6 9011  df-7 9012  df-8 9013  df-ndx 12514  df-slot 12515  df-base 12517  df-sets 12518  df-iress 12519  df-plusg 12599  df-mulr 12600  df-sca 12602  df-vsca 12603  df-ip 12604  df-0g 12760  df-iimas 12776  df-qus 12777  df-mgm 12829  df-sgrp 12862  df-mnd 12875  df-grp 12945  df-minusg 12946  df-sbg 12947  df-subg 13106  df-nsg 13107  df-eqg 13108  df-cmn 13222  df-abl 13223  df-mgp 13272  df-rng 13284  df-ur 13311  df-srg 13315  df-ring 13349  df-cring 13350  df-oppr 13415  df-subrg 13563  df-lmod 13602  df-lssm 13666  df-lsp 13700  df-sra 13748  df-rgmod 13749  df-lidl 13782  df-rsp 13783  df-2idl 13813

This theorem is referenced by:  zncrng2  13928