Theorem qsidomlem2 31531

Description: A quotient by a prime ideal is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)

Hypothesis

Ref	Expression
qsidom.1	⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))

Assertion

Ref	Expression
qsidomlem2	⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn)

Proof of Theorem qsidomlem2

Dummy variables 𝑎 𝑦 𝑏 𝑒 𝑓 𝑥 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngring 19710	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		prmidlidl 31521	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ∈ (LIdeal‘𝑅))
3	1, 2	sylan 579	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ∈ (LIdeal‘𝑅))
4		qsidom.1	. . . 4 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
5		eqid 2738	. . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
6	4, 5	quscrng 20424	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑄 ∈ CRing)
7	3, 6	syldan 590	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ CRing)
8	5	crng2idl 20423	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (LIdeal‘𝑅) = (2Ideal‘𝑅))
9	8	eleq2d 2824	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (𝐼 ∈ (LIdeal‘𝑅) ↔ 𝐼 ∈ (2Ideal‘𝑅)))
10	9	biimpa 476	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
11	3, 10	syldan 590	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
12		eqid 2738	. . . . . 6 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
13	4, 12	qusring 20420	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
14	1, 11, 13	syl2an2r 681	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ Ring)
15		eqid 2738	. . . . . . . . 9 ⊢ (Base‘𝑄) = (Base‘𝑄)
16		eqid 2738	. . . . . . . . 9 ⊢ (0g‘𝑄) = (0g‘𝑄)
17	15, 16	ring0cl 19723	. . . . . . . 8 ⊢ (𝑄 ∈ Ring → (0g‘𝑄) ∈ (Base‘𝑄))
18	14, 17	syl 17	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (0g‘𝑄) ∈ (Base‘𝑄))
19	18	snssd 4739	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g‘𝑄)} ⊆ (Base‘𝑄))
20		lidlnsg 31523	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
21	1, 20	sylan 579	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
22		eqid 2738	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
23	4, 22	qus0 18729	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → [(0g‘𝑅)](𝑅 ~QG 𝐼) = (0g‘𝑄))
24	21, 23	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → [(0g‘𝑅)](𝑅 ~QG 𝐼) = (0g‘𝑄))
25	5	lidlsubg 20399	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
26	1, 25	sylan 579	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
27		eqid 2738	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
28		eqid 2738	. . . . . . . . . . . 12 ⊢ (𝑅 ~QG 𝐼) = (𝑅 ~QG 𝐼)
29	27, 28, 22	eqgid 18723	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → [(0g‘𝑅)](𝑅 ~QG 𝐼) = 𝐼)
30	26, 29	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → [(0g‘𝑅)](𝑅 ~QG 𝐼) = 𝐼)
31	24, 30	eqtr3d 2780	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (0g‘𝑄) = 𝐼)
32	3, 31	syldan 590	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (0g‘𝑄) = 𝐼)
33	32	sneqd 4570	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g‘𝑄)} = {𝐼})
34		eqid 2738	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
35	27, 34	isprmidlc 31525	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → (𝐼 ∈ (PrmIdeal‘𝑅) ↔ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)))))
36	35	biimpa 476	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼))))
37	36	simp2d 1141	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ≠ (Base‘𝑅))
38		ringgrp 19703	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
39	1, 38	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Grp)
40	39	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝑅 ∈ Grp)
41	1	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝑅 ∈ Ring)
42	3	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 ∈ (LIdeal‘𝑅))
43	41, 42, 25	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 ∈ (SubGrp‘𝑅))
44		simpr 484	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → (Base‘𝑄) = {𝐼})
45	27, 4	qustrivr 31463	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 = (Base‘𝑅))
46	40, 43, 44, 45	syl3anc 1369	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 = (Base‘𝑅))
47	37, 46	mteqand 3047	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (Base‘𝑄) ≠ {𝐼})
48	47	necomd 2998	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {𝐼} ≠ (Base‘𝑄))
49	33, 48	eqnetrd 3010	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g‘𝑄)} ≠ (Base‘𝑄))
50		pssdifn0 4296	. . . . . 6 ⊢ (({(0g‘𝑄)} ⊆ (Base‘𝑄) ∧ {(0g‘𝑄)} ≠ (Base‘𝑄)) → ((Base‘𝑄) ∖ {(0g‘𝑄)}) ≠ ∅)
51	19, 49, 50	syl2anc 583	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ((Base‘𝑄) ∖ {(0g‘𝑄)}) ≠ ∅)
52		n0 4277	. . . . 5 ⊢ (((Base‘𝑄) ∖ {(0g‘𝑄)}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}))
53	51, 52	sylib 217	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ∃𝑥 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}))
54	16, 15	ringelnzr 20450	. . . . . 6 ⊢ ((𝑄 ∈ Ring ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → 𝑄 ∈ NzRing)
55	54	ex 412	. . . . 5 ⊢ (𝑄 ∈ Ring → (𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}) → 𝑄 ∈ NzRing))
56	55	exlimdv 1937	. . . 4 ⊢ (𝑄 ∈ Ring → (∃𝑥 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}) → 𝑄 ∈ NzRing))
57	14, 53, 56	sylc 65	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ NzRing)
58	36	simp3d 1142	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)))
59	58	ad7antr 734	. . . . . . . . . 10 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)))
60		simp-4r 780	. . . . . . . . . 10 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝑥 ∈ (Base‘𝑅))
61		simplr 765	. . . . . . . . . 10 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝑦 ∈ (Base‘𝑅))
62		simp-8l 787	. . . . . . . . . . . 12 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝑅 ∈ CRing)
63	62, 39	syl 17	. . . . . . . . . . 11 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝑅 ∈ Grp)
64	3	ad7antr 734	. . . . . . . . . . . 12 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝐼 ∈ (LIdeal‘𝑅))
65	62, 64, 26	syl2anc 583	. . . . . . . . . . 11 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝐼 ∈ (SubGrp‘𝑅))
66	4	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)))
67		eqidd 2739	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
68	27, 28	eqger 18721	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → (𝑅 ~QG 𝐼) Er (Base‘𝑅))
69	26, 68	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑅 ~QG 𝐼) Er (Base‘𝑅))
70		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
71	27, 28, 12, 34	2idlcpbl 20418	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → ((𝑔(𝑅 ~QG 𝐼)𝑒 ∧ ℎ(𝑅 ~QG 𝐼)𝑓) → (𝑔(.r‘𝑅)ℎ)(𝑅 ~QG 𝐼)(𝑒(.r‘𝑅)𝑓)))
72	1, 10, 71	syl2an2r 681	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ((𝑔(𝑅 ~QG 𝐼)𝑒 ∧ ℎ(𝑅 ~QG 𝐼)𝑓) → (𝑔(.r‘𝑅)ℎ)(𝑅 ~QG 𝐼)(𝑒(.r‘𝑅)𝑓)))
73	1	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
74		simprl 767	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑒 ∈ (Base‘𝑅))
75		simprr 769	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑓 ∈ (Base‘𝑅))
76	27, 34	ringcl 19715	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅)) → (𝑒(.r‘𝑅)𝑓) ∈ (Base‘𝑅))
77	73, 74, 75, 76	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → (𝑒(.r‘𝑅)𝑓) ∈ (Base‘𝑅))
78		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑄) = (.r‘𝑄)
79	66, 67, 69, 70, 72, 77, 34, 78	qusmulval 17183	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼)(.r‘𝑄)[𝑦](𝑅 ~QG 𝐼)) = [(𝑥(.r‘𝑅)𝑦)](𝑅 ~QG 𝐼))
80	62, 64, 60, 61, 79	syl211anc 1374	. . . . . . . . . . . 12 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → ([𝑥](𝑅 ~QG 𝐼)(.r‘𝑄)[𝑦](𝑅 ~QG 𝐼)) = [(𝑥(.r‘𝑅)𝑦)](𝑅 ~QG 𝐼))
81		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) → (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄))
82	81	ad4antr 728	. . . . . . . . . . . . 13 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄))
83		simpllr 772	. . . . . . . . . . . . . 14 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝑎 = [𝑥](𝑅 ~QG 𝐼))
84		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝑏 = [𝑦](𝑅 ~QG 𝐼))
85	83, 84	oveq12d 7273	. . . . . . . . . . . . 13 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑎(.r‘𝑄)𝑏) = ([𝑥](𝑅 ~QG 𝐼)(.r‘𝑄)[𝑦](𝑅 ~QG 𝐼)))
86	62, 64, 31	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (0g‘𝑄) = 𝐼)
87	82, 85, 86	3eqtr3d 2786	. . . . . . . . . . . 12 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → ([𝑥](𝑅 ~QG 𝐼)(.r‘𝑄)[𝑦](𝑅 ~QG 𝐼)) = 𝐼)
88	80, 87	eqtr3d 2780	. . . . . . . . . . 11 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → [(𝑥(.r‘𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼)
89	28	eqg0el 31459	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([(𝑥(.r‘𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼 ↔ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼))
90	89	biimpa 476	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ [(𝑥(.r‘𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐼)
91	63, 65, 88, 90	syl21anc 834	. . . . . . . . . 10 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐼)
92		rsp2 3136	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)) → ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼))))
93	92	impl 455	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)))
94	93	imp 406	. . . . . . . . . 10 ⊢ ((((∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼))
95	59, 60, 61, 91, 94	syl1111anc 836	. . . . . . . . 9 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼))
96	86	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑎 = (0g‘𝑄) ↔ 𝑎 = 𝐼))
97	83	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑎 = 𝐼 ↔ [𝑥](𝑅 ~QG 𝐼) = 𝐼))
98	28	eqg0el 31459	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑥 ∈ 𝐼))
99	63, 65, 98	syl2anc 583	. . . . . . . . . . 11 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑥 ∈ 𝐼))
100	96, 97, 99	3bitrrd 305	. . . . . . . . . 10 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑥 ∈ 𝐼 ↔ 𝑎 = (0g‘𝑄)))
101	86	eqeq2d 2749	. . . . . . . . . . 11 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑏 = (0g‘𝑄) ↔ 𝑏 = 𝐼))
102	84	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑏 = 𝐼 ↔ [𝑦](𝑅 ~QG 𝐼) = 𝐼))
103	28	eqg0el 31459	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑦 ∈ 𝐼))
104	63, 65, 103	syl2anc 583	. . . . . . . . . . 11 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑦 ∈ 𝐼))
105	101, 102, 104	3bitrrd 305	. . . . . . . . . 10 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑦 ∈ 𝐼 ↔ 𝑏 = (0g‘𝑄)))
106	100, 105	orbi12d 915	. . . . . . . . 9 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → ((𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼) ↔ (𝑎 = (0g‘𝑄) ∨ 𝑏 = (0g‘𝑄))))
107	95, 106	mpbid 231	. . . . . . . 8 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑎 = (0g‘𝑄) ∨ 𝑏 = (0g‘𝑄)))
108		simplr 765	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) → 𝑏 ∈ (Base‘𝑄))
109	4	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)))
110		eqidd 2739	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
111		ovexd 7290	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → (𝑅 ~QG 𝐼) ∈ V)
112		id 22	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing)
113	109, 110, 111, 112	qusbas 17173	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → ((Base‘𝑅) / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
114	113	ad4antr 728	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) → ((Base‘𝑅) / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
115	108, 114	eleqtrrd 2842	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) → 𝑏 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
116	115	ad2antrr 722	. . . . . . . . 9 ⊢ (((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) → 𝑏 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
117		elqsi 8517	. . . . . . . . 9 ⊢ (𝑏 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)) → ∃𝑦 ∈ (Base‘𝑅)𝑏 = [𝑦](𝑅 ~QG 𝐼))
118	116, 117	syl 17	. . . . . . . 8 ⊢ (((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) → ∃𝑦 ∈ (Base‘𝑅)𝑏 = [𝑦](𝑅 ~QG 𝐼))
119	107, 118	r19.29a 3217	. . . . . . 7 ⊢ (((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) → (𝑎 = (0g‘𝑄) ∨ 𝑏 = (0g‘𝑄)))
120		simpllr 772	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) → 𝑎 ∈ (Base‘𝑄))
121	120, 114	eleqtrrd 2842	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) → 𝑎 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
122		elqsi 8517	. . . . . . . 8 ⊢ (𝑎 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)) → ∃𝑥 ∈ (Base‘𝑅)𝑎 = [𝑥](𝑅 ~QG 𝐼))
123	121, 122	syl 17	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) → ∃𝑥 ∈ (Base‘𝑅)𝑎 = [𝑥](𝑅 ~QG 𝐼))
124	119, 123	r19.29a 3217	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r‘𝑄)𝑏) = (0g‘𝑄)) → (𝑎 = (0g‘𝑄) ∨ 𝑏 = (0g‘𝑄)))
125	124	ex 412	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) → ((𝑎(.r‘𝑄)𝑏) = (0g‘𝑄) → (𝑎 = (0g‘𝑄) ∨ 𝑏 = (0g‘𝑄))))
126	125	anasss 466	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (𝑎 ∈ (Base‘𝑄) ∧ 𝑏 ∈ (Base‘𝑄))) → ((𝑎(.r‘𝑄)𝑏) = (0g‘𝑄) → (𝑎 = (0g‘𝑄) ∨ 𝑏 = (0g‘𝑄))))
127	126	ralrimivva 3114	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ∀𝑎 ∈ (Base‘𝑄)∀𝑏 ∈ (Base‘𝑄)((𝑎(.r‘𝑄)𝑏) = (0g‘𝑄) → (𝑎 = (0g‘𝑄) ∨ 𝑏 = (0g‘𝑄))))
128	15, 78, 16	isdomn 20478	. . 3 ⊢ (𝑄 ∈ Domn ↔ (𝑄 ∈ NzRing ∧ ∀𝑎 ∈ (Base‘𝑄)∀𝑏 ∈ (Base‘𝑄)((𝑎(.r‘𝑄)𝑏) = (0g‘𝑄) → (𝑎 = (0g‘𝑄) ∨ 𝑏 = (0g‘𝑄)))))
129	57, 127, 128	sylanbrc 582	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ Domn)
130		isidom 20488	. 2 ⊢ (𝑄 ∈ IDomn ↔ (𝑄 ∈ CRing ∧ 𝑄 ∈ Domn))
131	7, 129, 130	sylanbrc 582	1 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  {csn 4558   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255   Er wer 8453  [cec 8454   / cqs 8455  Basecbs 16840  ._rcmulr 16889  0_gc0g 17067   /_s cqus 17133  Grpcgrp 18492  SubGrpcsubg 18664  NrmSGrpcnsg 18665   ~_QG cqg 18666  Ringcrg 19698  CRingccrg 19699  LIdealclidl 20347  2Idealc2idl 20415  NzRingcnzr 20441  Domncdomn 20464  IDomncidom 20465  PrmIdealcprmidl 31512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-tpos 8013  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-ec 8458  df-qs 8462  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-0g 17069  df-imas 17136  df-qus 17137  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-nsg 18668  df-eqg 18669  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-cring 19701  df-oppr 19777  df-subrg 19937  df-lmod 20040  df-lss 20109  df-lsp 20149  df-sra 20349  df-rgmod 20350  df-lidl 20351  df-rsp 20352  df-2idl 20416  df-nzr 20442  df-domn 20468  df-idom 20469  df-prmidl 31513

This theorem is referenced by:  qsidom  31532