Theorem quscrng 14370

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 13845	. . 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 14368	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝐼 = (2Ideal‘𝑅))
5	4	adantr 276	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝐼 = (2Ideal‘𝑅))
6	2, 5	eleqtrd 2285	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (2Ideal‘𝑅))
7		quscrng.u	. . . 4 ⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))
8		eqid 2206	. . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
9	7, 8	qusring 14364	. . 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 2207	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → (Base‘𝑅) = (Base‘𝑅))
13		eqgex 13632	. . . . . . 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 13234	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → ((Base‘𝑅) / (𝑅 ~QG 𝑆)) = (Base‘𝑈))
16	15	eleq2d 2276	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → (𝑥 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑆)) ↔ 𝑥 ∈ (Base‘𝑈)))
17	15	eleq2d 2276	. . . . 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 2206	. . . . . 6 ⊢ ((Base‘𝑅) / (𝑅 ~QG 𝑆)) = ((Base‘𝑅) / (𝑅 ~QG 𝑆))
20		oveq2 5965	. . . . . . 7 ⊢ ([𝑢](𝑅 ~QG 𝑆) = 𝑦 → (𝑥(.r‘𝑈)[𝑢](𝑅 ~QG 𝑆)) = (𝑥(.r‘𝑈)𝑦))
21		oveq1 5964	. . . . . . 7 ⊢ ([𝑢](𝑅 ~QG 𝑆) = 𝑦 → ([𝑢](𝑅 ~QG 𝑆)(.r‘𝑈)𝑥) = (𝑦(.r‘𝑈)𝑥))
22	20, 21	eqeq12d 2221	. . . . . 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 5964	. . . . . . . . 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 5965	. . . . . . . . 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 2221	. . . . . . . 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 2206	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
27		eqid 2206	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
28	26, 27	crngcom 13851	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢(.r‘𝑅)𝑣) = (𝑣(.r‘𝑅)𝑢))
29	28	ad4ant134 1220	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) ∧ 𝑢 ∈ (Base‘𝑅)) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢(.r‘𝑅)𝑣) = (𝑣(.r‘𝑅)𝑢))
30	29	eceq1d 6669	. . . . . . . . 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 13873	. . . . . . . . . . . . . 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 14323	. . . . . . . . . . . . 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 1180	. . . . . . . . . . 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 2206	. . . . . . . . . . 11 ⊢ (𝑅 ~QG 𝑆) = (𝑅 ~QG 𝑆)
41		eqid 2206	. . . . . . . . . . 11 ⊢ (.r‘𝑈) = (.r‘𝑈)
42	40, 7, 26, 27, 41	qusmulrng 14369	. . . . . . . . . 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 14369	. . . . . . . . . 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 2250	. . . . . . . 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 6701	. . . . . . 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 6701	. . . . 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 2588	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(.r‘𝑈)𝑦) = (𝑦(.r‘𝑈)𝑥))
54		eqid 2206	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
55	54, 41	iscrng2 13852	. 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 981   = wceq 1373   ∈ wcel 2177  ∀wral 2485  Vcvv 2773  ‘cfv 5280  (class class class)co 5957  [cec 6631   / cqs 6632  Basecbs 12907  ._rcmulr 12985   /_s cqus 13207  SubGrpcsubg 13578   ~_QG cqg 13580  Rngcrng 13769  Ringcrg 13833  CRingccrg 13834  LIdealclidl 14304  2Idealc2idl 14336

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-tp 3646  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-tpos 6344  df-er 6633  df-ec 6635  df-qs 6639  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-5 9118  df-6 9119  df-7 9120  df-8 9121  df-ndx 12910  df-slot 12911  df-base 12913  df-sets 12914  df-iress 12915  df-plusg 12997  df-mulr 12998  df-sca 13000  df-vsca 13001  df-ip 13002  df-0g 13165  df-iimas 13209  df-qus 13210  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411  df-sbg 13412  df-subg 13581  df-nsg 13582  df-eqg 13583  df-cmn 13697  df-abl 13698  df-mgp 13758  df-rng 13770  df-ur 13797  df-srg 13801  df-ring 13835  df-cring 13836  df-oppr 13905  df-subrg 14056  df-lmod 14126  df-lssm 14190  df-lsp 14224  df-sra 14272  df-rgmod 14273  df-lidl 14306  df-rsp 14307  df-2idl 14337

This theorem is referenced by:  zncrng2  14472