Theorem quscrng 14513

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 13987	. . 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 14511	. . . . 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 2308	. . 3 |- ( ( R e. CRing /\ S e. I ) -> S e. (2Ideal ` R ) )
7		quscrng.u	. . . 4 |- U = ( R /.s ( R ~QG S ) )
8		eqid 2229	. . . 4 |- (2Ideal ` R ) = (2Ideal ` R )
9	7, 8	qusring 14507	. . 3 |- ( ( R e. Ring /\ S e. (2Ideal ` R ) ) -> U e. Ring )
10	1, 6, 9	syl2an2r 597	. 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 2230	. . . . . . 7 |- ( ( R e. CRing /\ S e. I ) -> ( Base ` R ) = ( Base ` R ) )
13		eqgex 13774	. . . . . . 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 13376	. . . . . 6 |- ( ( R e. CRing /\ S e. I ) -> ( ( Base ` R ) /. ( R ~QG S ) ) = ( Base ` U ) )
16	15	eleq2d 2299	. . . . 5 |- ( ( R e. CRing /\ S e. I ) -> ( x e. ( ( Base ` R ) /. ( R ~QG S ) ) <-> x e. ( Base ` U ) ) )
17	15	eleq2d 2299	. . . . 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 2229	. . . . . 6 |- ( ( Base ` R ) /. ( R ~QG S ) ) = ( ( Base ` R ) /. ( R ~QG S ) )
20		oveq2 6015	. . . . . . 7 |- ( [ u ] ( R ~QG S ) = y -> ( x ( .r ` U ) [ u ] ( R ~QG S ) ) = ( x ( .r ` U ) y ) )
21		oveq1 6014	. . . . . . 7 |- ( [ u ] ( R ~QG S ) = y -> ( [ u ] ( R ~QG S ) ( .r ` U ) x ) = ( y ( .r ` U ) x ) )
22	20, 21	eqeq12d 2244	. . . . . 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 6014	. . . . . . . . 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 6015	. . . . . . . . 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 2244	. . . . . . . 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 2229	. . . . . . . . . . . 12 |- ( Base ` R ) = ( Base ` R )
27		eqid 2229	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
28	26, 27	crngcom 13993	. . . . . . . . . . 11 |- ( ( R e. CRing /\ u e. ( Base ` R ) /\ v e. ( Base ` R ) ) -> ( u ( .r ` R ) v ) = ( v ( .r ` R ) u ) )
29	28	ad4ant134 1241	. . . . . . . . . 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 6724	. . . . . . . . 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 14015	. . . . . . . . . . . . . 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 14466	. . . . . . . . . . . . 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 1201	. . . . . . . . . . 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 2229	. . . . . . . . . . 11 |- ( R ~QG S ) = ( R ~QG S )
41		eqid 2229	. . . . . . . . . . 11 |- ( .r ` U ) = ( .r ` U )
42	40, 7, 26, 27, 41	qusmulrng 14512	. . . . . . . . . 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 597	. . . . . . . . 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 14512	. . . . . . . . . 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 597	. . . . . . . . 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 2273	. . . . . . . 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 6756	. . . . . . 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 6756	. . . . 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 2611	. 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 2229	. . 3 |- ( Base ` U ) = ( Base ` U )
55	54, 41	iscrng2 13994	. 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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   ` cfv 5318  (class class class)co 6007   [cec 6686   /.cqs 6687   Basecbs 13048   .rcmulr 13127   /._s cqus 13349  SubGrpcsubg 13720   ~_QG cqg 13722  Rngcrng 13911   Ringcrg 13975   CRingccrg 13976  LIdealclidl 14447  2Idealc2idl 14479

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-lttrn 8124  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-tpos 6397  df-er 6688  df-ec 6690  df-qs 6694  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-ndx 13051  df-slot 13052  df-base 13054  df-sets 13055  df-iress 13056  df-plusg 13139  df-mulr 13140  df-sca 13142  df-vsca 13143  df-ip 13144  df-0g 13307  df-iimas 13351  df-qus 13352  df-mgm 13405  df-sgrp 13451  df-mnd 13466  df-grp 13552  df-minusg 13553  df-sbg 13554  df-subg 13723  df-nsg 13724  df-eqg 13725  df-cmn 13839  df-abl 13840  df-mgp 13900  df-rng 13912  df-ur 13939  df-srg 13943  df-ring 13977  df-cring 13978  df-oppr 14047  df-subrg 14199  df-lmod 14269  df-lssm 14333  df-lsp 14367  df-sra 14415  df-rgmod 14416  df-lidl 14449  df-rsp 14450  df-2idl 14480

This theorem is referenced by:  zncrng2  14615