Theorem qsidomlem2 33528

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 20217	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		prmidlidl 33519	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ∈ (LIdeal‘𝑅))
3	1, 2	sylan 581	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ∈ (LIdeal‘𝑅))
4		qsidom.1	. . . 4 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
5		eqid 2737	. . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
6	4, 5	quscrng 21273	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑄 ∈ CRing)
7	3, 6	syldan 592	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ CRing)
8	5	crng2idl 21271	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (LIdeal‘𝑅) = (2Ideal‘𝑅))
9	8	eleq2d 2823	. . . . . . 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 592	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
12		eqid 2737	. . . . . 6 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
13	4, 12	qusring 21265	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
14	1, 11, 13	syl2an2r 686	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ Ring)
15		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝑄) = (Base‘𝑄)
16		eqid 2737	. . . . . . . . 9 ⊢ (0g‘𝑄) = (0g‘𝑄)
17	15, 16	ring0cl 20239	. . . . . . . 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 4753	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g‘𝑄)} ⊆ (Base‘𝑄))
20		lidlnsg 21238	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
21	1, 20	sylan 581	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
22		eqid 2737	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
23	4, 22	qus0 19155	. . . . . . . . . . 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 21213	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
26	1, 25	sylan 581	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
27		eqid 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
28		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝑅 ~QG 𝐼) = (𝑅 ~QG 𝐼)
29	27, 28, 22	eqgid 19146	. . . . . . . . . . 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 2774	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (0g‘𝑄) = 𝐼)
32	3, 31	syldan 592	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (0g‘𝑄) = 𝐼)
33	32	sneqd 4580	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g‘𝑄)} = {𝐼})
34		eqid 2737	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
35	27, 34	isprmidlc 33522	. . . . . . . . . . 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 1144	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ≠ (Base‘𝑅))
38		ringgrp 20210	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
39	1, 38	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Grp)
40	39	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝑅 ∈ Grp)
41	1	ad2antrr 727	. . . . . . . . . . 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 585	. . . . . . . . . 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 33440	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 = (Base‘𝑅))
46	40, 43, 44, 45	syl3anc 1374	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 = (Base‘𝑅))
47	37, 46	mteqand 3024	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (Base‘𝑄) ≠ {𝐼})
48	47	necomd 2988	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {𝐼} ≠ (Base‘𝑄))
49	33, 48	eqnetrd 3000	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g‘𝑄)} ≠ (Base‘𝑄))
50		pssdifn0 4309	. . . . . 6 ⊢ (({(0g‘𝑄)} ⊆ (Base‘𝑄) ∧ {(0g‘𝑄)} ≠ (Base‘𝑄)) → ((Base‘𝑄) ∖ {(0g‘𝑄)}) ≠ ∅)
51	19, 49, 50	syl2anc 585	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ((Base‘𝑄) ∖ {(0g‘𝑄)}) ≠ ∅)
52		n0 4294	. . . . 5 ⊢ (((Base‘𝑄) ∖ {(0g‘𝑄)}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}))
53	51, 52	sylib 218	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ∃𝑥 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}))
54	16, 15	ringelnzr 20491	. . . . . 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 1935	. . . 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 1145	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)))
59	58	ad7antr 739	. . . . . . . . . 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 784	. . . . . . . . . 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 769	. . . . . . . . . 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 791	. . . . . . . . . . . 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 739	. . . . . . . . . . . 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 585	. . . . . . . . . . 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 2738	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
68	27, 28	eqger 19144	. . . . . . . . . . . . . . 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 21262	. . . . . . . . . . . . . . 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 686	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
74		simprl 771	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑒 ∈ (Base‘𝑅))
75		simprr 773	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑓 ∈ (Base‘𝑅))
76	27, 34	ringcl 20222	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅)) → (𝑒(.r‘𝑅)𝑓) ∈ (Base‘𝑅))
77	73, 74, 75, 76	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → (𝑒(.r‘𝑅)𝑓) ∈ (Base‘𝑅))
78		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑄) = (.r‘𝑄)
79	66, 67, 69, 70, 72, 77, 34, 78	qusmulval 17510	. . . . . . . . . . . . 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 1379	. . . . . . . . . . . 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 733	. . . . . . . . . . . . 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 776	. . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . 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 585	. . . . . . . . . . . . 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 2780	. . . . . . . . . . . 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 2774	. . . . . . . . . . 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 19149	. . . . . . . . . . . 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 838	. . . . . . . . . 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 3255	. . . . . . . . . . . 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 841	. . . . . . . . 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 2748	. . . . . . . . . . 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 2739	. . . . . . . . . . 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 19149	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑥 ∈ 𝐼))
99	63, 65, 98	syl2anc 585	. . . . . . . . . . 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 306	. . . . . . . . . 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 2748	. . . . . . . . . . 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 2739	. . . . . . . . . . 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 19149	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑦 ∈ 𝐼))
104	63, 65, 103	syl2anc 585	. . . . . . . . . . 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 306	. . . . . . . . . 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 919	. . . . . . . . 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 232	. . . . . . . 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 769	. . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
111		ovexd 7395	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → (𝑅 ~QG 𝐼) ∈ V)
112		id 22	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing)
113	109, 110, 111, 112	qusbas 17500	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → ((Base‘𝑅) / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
114	113	ad4antr 733	. . . . . . . . . . 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 2840	. . . . . . . . . 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 727	. . . . . . . . 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 8705	. . . . . . . . 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 3146	. . . . . . 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 776	. . . . . . . . 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 2840	. . . . . . . 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 8705	. . . . . . . 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 3146	. . . . . 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 3181	. . 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 20673	. . 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 584	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ Domn)
130		isidom 20693	. 2 ⊢ (𝑄 ∈ IDomn ↔ (𝑄 ∈ CRing ∧ 𝑄 ∈ Domn))
131	7, 129, 130	sylanbrc 584	1 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4274  {csn 4568   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360   Er wer 8633  [cec 8634   / cqs 8635  Basecbs 17170  ._rcmulr 17212  0_gc0g 17393   /_s cqus 17460  Grpcgrp 18900  SubGrpcsubg 19087  NrmSGrpcnsg 19088   ~_QG cqg 19089  Ringcrg 20205  CRingccrg 20206  NzRingcnzr 20480  Domncdomn 20660  IDomncidom 20661  LIdealclidl 21196  2Idealc2idl 21239  PrmIdealcprmidl 33510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-tpos 8169  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-ec 8638  df-qs 8642  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-inf 9349  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-0g 17395  df-imas 17463  df-qus 17464  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-minusg 18904  df-sbg 18905  df-subg 19090  df-nsg 19091  df-eqg 19092  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-cring 20208  df-oppr 20308  df-nzr 20481  df-subrg 20538  df-domn 20663  df-idom 20664  df-lmod 20848  df-lss 20918  df-lsp 20958  df-sra 21160  df-rgmod 21161  df-lidl 21198  df-rsp 21199  df-2idl 21240  df-prmidl 33511

This theorem is referenced by:  qsidom  33529