Theorem quscrng 14089

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 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ 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 13564	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		simpr 110	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ 𝐼)
3		quscrng.i	. . . . . 6 ⊢ 𝐼 = (LIdeal‘𝑅)
4	3	crng2idl 14087	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝐼 = (2Ideal‘𝑅))
5	4	adantr 276	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝐼 = (2Ideal‘𝑅))
6	2, 5	eleqtrd 2275	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (2Ideal‘𝑅))
7		quscrng.u	. . . 4 ⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))
8		eqid 2196	. . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
9	7, 8	qusring 14083	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (2Ideal‘𝑅)) → 𝑈 ∈ Ring)
10	1, 6, 9	syl2an2r 595	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring)
11	7	a1i 9	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆)))
12		eqidd 2197	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → (Base‘𝑅) = (Base‘𝑅))
13		eqgex 13351	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → (𝑅 ~QG 𝑆) ∈ V)
14	1	adantr 276	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Ring)
15	11, 12, 13, 14	qusbas 12970	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → ((Base‘𝑅) / (𝑅 ~QG 𝑆)) = (Base‘𝑈))
16	15	eleq2d 2266	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → (𝑥 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑆)) ↔ 𝑥 ∈ (Base‘𝑈)))
17	15	eleq2d 2266	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → (𝑦 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑆)) ↔ 𝑦 ∈ (Base‘𝑈)))
18	16, 17	anbi12d 473	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → ((𝑥 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑆)) ∧ 𝑦 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑆))) ↔ (𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈))))
19		eqid 2196	. . . . . 6 ⊢ ((Base‘𝑅) / (𝑅 ~QG 𝑆)) = ((Base‘𝑅) / (𝑅 ~QG 𝑆))
20		oveq2 5930	. . . . . . 7 ⊢ ([𝑢](𝑅 ~QG 𝑆) = 𝑦 → (𝑥(.r‘𝑈)[𝑢](𝑅 ~QG 𝑆)) = (𝑥(.r‘𝑈)𝑦))
21		oveq1 5929	. . . . . . 7 ⊢ ([𝑢](𝑅 ~QG 𝑆) = 𝑦 → ([𝑢](𝑅 ~QG 𝑆)(.r‘𝑈)𝑥) = (𝑦(.r‘𝑈)𝑥))
22	20, 21	eqeq12d 2211	. . . . . 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 5929	. . . . . . . . 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 5930	. . . . . . . . 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 2211	. . . . . . . 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 2196	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
27		eqid 2196	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
28	26, 27	crngcom 13570	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢(.r‘𝑅)𝑣) = (𝑣(.r‘𝑅)𝑢))
29	28	ad4ant134 1219	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) ∧ 𝑢 ∈ (Base‘𝑅)) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢(.r‘𝑅)𝑣) = (𝑣(.r‘𝑅)𝑢))
30	29	eceq1d 6628	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) ∧ 𝑢 ∈ (Base‘𝑅)) ∧ 𝑣 ∈ (Base‘𝑅)) → [(𝑢(.r‘𝑅)𝑣)](𝑅 ~QG 𝑆) = [(𝑣(.r‘𝑅)𝑢)](𝑅 ~QG 𝑆))
31		ringrng 13592	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Rng)
32	1, 31	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Rng)
33	32	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Rng)
34	3	lidlsubg 14042	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅))
35	1, 34	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅))
36	33, 6, 35	3jca 1179	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → (𝑅 ∈ Rng ∧ 𝑆 ∈ (2Ideal‘𝑅) ∧ 𝑆 ∈ (SubGrp‘𝑅)))
37	36	adantr 276	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) ∧ 𝑢 ∈ (Base‘𝑅)) → (𝑅 ∈ Rng ∧ 𝑆 ∈ (2Ideal‘𝑅) ∧ 𝑆 ∈ (SubGrp‘𝑅)))
38		simpr 110	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) ∧ 𝑢 ∈ (Base‘𝑅)) → 𝑢 ∈ (Base‘𝑅))
39	38	anim1i 340	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) ∧ 𝑢 ∈ (Base‘𝑅)) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)))
40		eqid 2196	. . . . . . . . . . 11 ⊢ (𝑅 ~QG 𝑆) = (𝑅 ~QG 𝑆)
41		eqid 2196	. . . . . . . . . . 11 ⊢ (.r‘𝑈) = (.r‘𝑈)
42	40, 7, 26, 27, 41	qusmulrng 14088	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ (2Ideal‘𝑅) ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (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 ∈ CRing ∧ 𝑆 ∈ 𝐼) ∧ 𝑢 ∈ (Base‘𝑅)) ∧ 𝑣 ∈ (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 ∈ CRing ∧ 𝑆 ∈ 𝐼) ∧ 𝑢 ∈ (Base‘𝑅)) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑣 ∈ (Base‘𝑅) ∧ 𝑢 ∈ (Base‘𝑅)))
45	40, 7, 26, 27, 41	qusmulrng 14088	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ (2Ideal‘𝑅) ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝑣 ∈ (Base‘𝑅) ∧ 𝑢 ∈ (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 ∈ CRing ∧ 𝑆 ∈ 𝐼) ∧ 𝑢 ∈ (Base‘𝑅)) ∧ 𝑣 ∈ (Base‘𝑅)) → ([𝑣](𝑅 ~QG 𝑆)(.r‘𝑈)[𝑢](𝑅 ~QG 𝑆)) = [(𝑣(.r‘𝑅)𝑢)](𝑅 ~QG 𝑆))
47	30, 43, 46	3eqtr4rd 2240	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) ∧ 𝑢 ∈ (Base‘𝑅)) ∧ 𝑣 ∈ (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 6660	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) ∧ 𝑢 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ ((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 ∈ CRing ∧ 𝑆 ∈ 𝐼) ∧ 𝑥 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑆))) ∧ 𝑢 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑈)[𝑢](𝑅 ~QG 𝑆)) = ([𝑢](𝑅 ~QG 𝑆)(.r‘𝑈)𝑥))
50	19, 22, 49	ectocld 6660	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) ∧ 𝑥 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑆))) ∧ 𝑦 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑆))) → (𝑥(.r‘𝑈)𝑦) = (𝑦(.r‘𝑈)𝑥))
51	50	expl 378	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → ((𝑥 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑆)) ∧ 𝑦 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑆))) → (𝑥(.r‘𝑈)𝑦) = (𝑦(.r‘𝑈)𝑥)))
52	18, 51	sylbird 170	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → ((𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) → (𝑥(.r‘𝑈)𝑦) = (𝑦(.r‘𝑈)𝑥)))
53	52	ralrimivv 2578	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(.r‘𝑈)𝑦) = (𝑦(.r‘𝑈)𝑥))
54		eqid 2196	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
55	54, 41	iscrng2 13571	. 2 ⊢ (𝑈 ∈ CRing ↔ (𝑈 ∈ Ring ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(.r‘𝑈)𝑦) = (𝑦(.r‘𝑈)𝑥)))
56	10, 53, 55	sylanbrc 417	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ CRing)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763  ‘cfv 5258  (class class class)co 5922  [cec 6590   / cqs 6591  Basecbs 12678  ._rcmulr 12756   /_s cqus 12943  SubGrpcsubg 13297   ~_QG cqg 13299  Rngcrng 13488  Ringcrg 13552  CRingccrg 13553  LIdealclidl 14023  2Idealc2idl 14055

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-tpos 6303  df-er 6592  df-ec 6594  df-qs 6598  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-sca 12771  df-vsca 12772  df-ip 12773  df-0g 12929  df-iimas 12945  df-qus 12946  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-sbg 13137  df-subg 13300  df-nsg 13301  df-eqg 13302  df-cmn 13416  df-abl 13417  df-mgp 13477  df-rng 13489  df-ur 13516  df-srg 13520  df-ring 13554  df-cring 13555  df-oppr 13624  df-subrg 13775  df-lmod 13845  df-lssm 13909  df-lsp 13943  df-sra 13991  df-rgmod 13992  df-lidl 14025  df-rsp 14026  df-2idl 14056

This theorem is referenced by:  zncrng2  14191