Theorem quscrng 14328

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 13803	. . 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 14326	. . . . 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 2284	. . 3 |- ( ( R e. CRing /\ S e. I ) -> S e. (2Ideal ` R ) )
7		quscrng.u	. . . 4 |- U = ( R /.s ( R ~QG S ) )
8		eqid 2205	. . . 4 |- (2Ideal ` R ) = (2Ideal ` R )
9	7, 8	qusring 14322	. . 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 2206	. . . . . . 7 |- ( ( R e. CRing /\ S e. I ) -> ( Base ` R ) = ( Base ` R ) )
13		eqgex 13590	. . . . . . 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 13192	. . . . . 6 |- ( ( R e. CRing /\ S e. I ) -> ( ( Base ` R ) /. ( R ~QG S ) ) = ( Base ` U ) )
16	15	eleq2d 2275	. . . . 5 |- ( ( R e. CRing /\ S e. I ) -> ( x e. ( ( Base ` R ) /. ( R ~QG S ) ) <-> x e. ( Base ` U ) ) )
17	15	eleq2d 2275	. . . . 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 2205	. . . . . 6 |- ( ( Base ` R ) /. ( R ~QG S ) ) = ( ( Base ` R ) /. ( R ~QG S ) )
20		oveq2 5954	. . . . . . 7 |- ( [ u ] ( R ~QG S ) = y -> ( x ( .r ` U ) [ u ] ( R ~QG S ) ) = ( x ( .r ` U ) y ) )
21		oveq1 5953	. . . . . . 7 |- ( [ u ] ( R ~QG S ) = y -> ( [ u ] ( R ~QG S ) ( .r ` U ) x ) = ( y ( .r ` U ) x ) )
22	20, 21	eqeq12d 2220	. . . . . 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 5953	. . . . . . . . 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 5954	. . . . . . . . 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 2220	. . . . . . . 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 2205	. . . . . . . . . . . 12 |- ( Base ` R ) = ( Base ` R )
27		eqid 2205	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
28	26, 27	crngcom 13809	. . . . . . . . . . 11 |- ( ( R e. CRing /\ u e. ( Base ` R ) /\ v e. ( Base ` R ) ) -> ( u ( .r ` R ) v ) = ( v ( .r ` R ) u ) )
29	28	ad4ant134 1220	. . . . . . . . . 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 6658	. . . . . . . . 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 13831	. . . . . . . . . . . . . 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 14281	. . . . . . . . . . . . 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 1180	. . . . . . . . . . 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 2205	. . . . . . . . . . 11 |- ( R ~QG S ) = ( R ~QG S )
41		eqid 2205	. . . . . . . . . . 11 |- ( .r ` U ) = ( .r ` U )
42	40, 7, 26, 27, 41	qusmulrng 14327	. . . . . . . . . 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 14327	. . . . . . . . . 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 2249	. . . . . . . 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 6690	. . . . . . 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 6690	. . . . 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 2587	. 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 2205	. . 3 |- ( Base ` U ) = ( Base ` U )
55	54, 41	iscrng2 13810	. 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 981   = wceq 1373   e. wcel 2176   A.wral 2484   _Vcvv 2772   ` cfv 5272  (class class class)co 5946   [cec 6620   /.cqs 6621   Basecbs 12865   .rcmulr 12943   /._s cqus 13165  SubGrpcsubg 13536   ~_QG cqg 13538  Rngcrng 13727   Ringcrg 13791   CRingccrg 13792  LIdealclidl 14262  2Idealc2idl 14294

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 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-pre-ltirr 8039  ax-pre-lttrn 8041  ax-pre-ltadd 8043

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 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-tpos 6333  df-er 6622  df-ec 6624  df-qs 6628  df-pnf 8111  df-mnf 8112  df-ltxr 8114  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-5 9100  df-6 9101  df-7 9102  df-8 9103  df-ndx 12868  df-slot 12869  df-base 12871  df-sets 12872  df-iress 12873  df-plusg 12955  df-mulr 12956  df-sca 12958  df-vsca 12959  df-ip 12960  df-0g 13123  df-iimas 13167  df-qus 13168  df-mgm 13221  df-sgrp 13267  df-mnd 13282  df-grp 13368  df-minusg 13369  df-sbg 13370  df-subg 13539  df-nsg 13540  df-eqg 13541  df-cmn 13655  df-abl 13656  df-mgp 13716  df-rng 13728  df-ur 13755  df-srg 13759  df-ring 13793  df-cring 13794  df-oppr 13863  df-subrg 14014  df-lmod 14084  df-lssm 14148  df-lsp 14182  df-sra 14230  df-rgmod 14231  df-lidl 14264  df-rsp 14265  df-2idl 14295

This theorem is referenced by:  zncrng2  14430