Theorem qsidomlem2 31638

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 19804	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		prmidlidl 31628	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ∈ (LIdeal‘𝑅))
3	1, 2	sylan 580	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ∈ (LIdeal‘𝑅))
4		qsidom.1	. . . 4 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
5		eqid 2739	. . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
6	4, 5	quscrng 20520	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑄 ∈ CRing)
7	3, 6	syldan 591	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ CRing)
8	5	crng2idl 20519	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (LIdeal‘𝑅) = (2Ideal‘𝑅))
9	8	eleq2d 2825	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (𝐼 ∈ (LIdeal‘𝑅) ↔ 𝐼 ∈ (2Ideal‘𝑅)))
10	9	biimpa 477	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
11	3, 10	syldan 591	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
12		eqid 2739	. . . . . 6 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
13	4, 12	qusring 20516	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
14	1, 11, 13	syl2an2r 682	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ Ring)
15		eqid 2739	. . . . . . . . 9 ⊢ (Base‘𝑄) = (Base‘𝑄)
16		eqid 2739	. . . . . . . . 9 ⊢ (0g‘𝑄) = (0g‘𝑄)
17	15, 16	ring0cl 19817	. . . . . . . 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 4743	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g‘𝑄)} ⊆ (Base‘𝑄))
20		lidlnsg 31630	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
21	1, 20	sylan 580	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
22		eqid 2739	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
23	4, 22	qus0 18823	. . . . . . . . . . 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 20495	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
26	1, 25	sylan 580	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
27		eqid 2739	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
28		eqid 2739	. . . . . . . . . . . 12 ⊢ (𝑅 ~QG 𝐼) = (𝑅 ~QG 𝐼)
29	27, 28, 22	eqgid 18817	. . . . . . . . . . 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 2781	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (0g‘𝑄) = 𝐼)
32	3, 31	syldan 591	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (0g‘𝑄) = 𝐼)
33	32	sneqd 4574	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g‘𝑄)} = {𝐼})
34		eqid 2739	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
35	27, 34	isprmidlc 31632	. . . . . . . . . . 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 477	. . . . . . . . . 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 1142	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ≠ (Base‘𝑅))
38		ringgrp 19797	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
39	1, 38	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Grp)
40	39	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝑅 ∈ Grp)
41	1	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝑅 ∈ Ring)
42	3	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 ∈ (LIdeal‘𝑅))
43	41, 42, 25	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 ∈ (SubGrp‘𝑅))
44		simpr 485	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → (Base‘𝑄) = {𝐼})
45	27, 4	qustrivr 31570	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 = (Base‘𝑅))
46	40, 43, 44, 45	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 = (Base‘𝑅))
47	37, 46	mteqand 3049	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (Base‘𝑄) ≠ {𝐼})
48	47	necomd 3000	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {𝐼} ≠ (Base‘𝑄))
49	33, 48	eqnetrd 3012	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g‘𝑄)} ≠ (Base‘𝑄))
50		pssdifn0 4300	. . . . . 6 ⊢ (({(0g‘𝑄)} ⊆ (Base‘𝑄) ∧ {(0g‘𝑄)} ≠ (Base‘𝑄)) → ((Base‘𝑄) ∖ {(0g‘𝑄)}) ≠ ∅)
51	19, 49, 50	syl2anc 584	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ((Base‘𝑄) ∖ {(0g‘𝑄)}) ≠ ∅)
52		n0 4281	. . . . 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 20546	. . . . . 6 ⊢ ((𝑄 ∈ Ring ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → 𝑄 ∈ NzRing)
55	54	ex 413	. . . . 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 1143	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)))
59	58	ad7antr 735	. . . . . . . . . 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 781	. . . . . . . . . 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 766	. . . . . . . . . 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 788	. . . . . . . . . . . 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 735	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
68	27, 28	eqger 18815	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
71	27, 28, 12, 34	2idlcpbl 20514	. . . . . . . . . . . . . . 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 682	. . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
74		simprl 768	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑒 ∈ (Base‘𝑅))
75		simprr 770	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑓 ∈ (Base‘𝑅))
76	27, 34	ringcl 19809	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅)) → (𝑒(.r‘𝑅)𝑓) ∈ (Base‘𝑅))
77	73, 74, 75, 76	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → (𝑒(.r‘𝑅)𝑓) ∈ (Base‘𝑅))
78		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑄) = (.r‘𝑄)
79	66, 67, 69, 70, 72, 77, 34, 78	qusmulval 17275	. . . . . . . . . . . . 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 1375	. . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 729	. . . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 7302	. . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 2787	. . . . . . . . . . . 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 2781	. . . . . . . . . . 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 31566	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([(𝑥(.r‘𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼 ↔ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼))
90	89	biimpa 477	. . . . . . . . . . 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 835	. . . . . . . . . 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 3139	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 407	. . . . . . . . . 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 837	. . . . . . . . 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 2750	. . . . . . . . . . 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 2741	. . . . . . . . . . 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 31566	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑥 ∈ 𝐼))
99	63, 65, 98	syl2anc 584	. . . . . . . . . . 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 2750	. . . . . . . . . . 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 2741	. . . . . . . . . . 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 31566	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑦 ∈ 𝐼))
104	63, 65, 103	syl2anc 584	. . . . . . . . . . 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 916	. . . . . . . . 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 766	. . . . . . . . . . 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 2740	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
111		ovexd 7319	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → (𝑅 ~QG 𝐼) ∈ V)
112		id 22	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing)
113	109, 110, 111, 112	qusbas 17265	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → ((Base‘𝑅) / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
114	113	ad4antr 729	. . . . . . . . . . 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 2843	. . . . . . . . . 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 723	. . . . . . . . 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 8568	. . . . . . . . 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 3219	. . . . . . 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 773	. . . . . . . . 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 2843	. . . . . . . 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 8568	. . . . . . . 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 3219	. . . . . 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 413	. . . . 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 467	. . . 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 3124	. . 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 20574	. . 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 583	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ Domn)
130		isidom 20584	. 2 ⊢ (𝑄 ∈ IDomn ↔ (𝑄 ∈ CRing ∧ 𝑄 ∈ Domn))
131	7, 129, 130	sylanbrc 583	1 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2107   ≠ wne 2944  ∀wral 3065  ∃wrex 3066  Vcvv 3433   ∖ cdif 3885   ⊆ wss 3888  ∅c0 4257  {csn 4562   class class class wbr 5075  ‘cfv 6437  (class class class)co 7284   Er wer 8504  [cec 8505   / cqs 8506  Basecbs 16921  ._rcmulr 16972  0_gc0g 17159   /_s cqus 17225  Grpcgrp 18586  SubGrpcsubg 18758  NrmSGrpcnsg 18759   ~_QG cqg 18760  Ringcrg 19792  CRingccrg 19793  LIdealclidl 20441  2Idealc2idl 20511  NzRingcnzr 20537  Domncdomn 20560  IDomncidom 20561  PrmIdealcprmidl 31619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-cnex 10936  ax-resscn 10937  ax-1cn 10938  ax-icn 10939  ax-addcl 10940  ax-addrcl 10941  ax-mulcl 10942  ax-mulrcl 10943  ax-mulcom 10944  ax-addass 10945  ax-mulass 10946  ax-distr 10947  ax-i2m1 10948  ax-1ne0 10949  ax-1rid 10950  ax-rnegex 10951  ax-rrecex 10952  ax-cnre 10953  ax-pre-lttri 10954  ax-pre-lttrn 10955  ax-pre-ltadd 10956  ax-pre-mulgt0 10957

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-om 7722  df-1st 7840  df-2nd 7841  df-tpos 8051  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-er 8507  df-ec 8509  df-qs 8513  df-en 8743  df-dom 8744  df-sdom 8745  df-fin 8746  df-sup 9210  df-inf 9211  df-pnf 11020  df-mnf 11021  df-xr 11022  df-ltxr 11023  df-le 11024  df-sub 11216  df-neg 11217  df-nn 11983  df-2 12045  df-3 12046  df-4 12047  df-5 12048  df-6 12049  df-7 12050  df-8 12051  df-9 12052  df-n0 12243  df-z 12329  df-dec 12447  df-uz 12592  df-fz 13249  df-struct 16857  df-sets 16874  df-slot 16892  df-ndx 16904  df-base 16922  df-ress 16951  df-plusg 16984  df-mulr 16985  df-sca 16987  df-vsca 16988  df-ip 16989  df-tset 16990  df-ple 16991  df-ds 16993  df-0g 17161  df-imas 17228  df-qus 17229  df-mgm 18335  df-sgrp 18384  df-mnd 18395  df-grp 18589  df-minusg 18590  df-sbg 18591  df-subg 18761  df-nsg 18762  df-eqg 18763  df-cmn 19397  df-abl 19398  df-mgp 19730  df-ur 19747  df-ring 19794  df-cring 19795  df-oppr 19871  df-subrg 20031  df-lmod 20134  df-lss 20203  df-lsp 20243  df-sra 20443  df-rgmod 20444  df-lidl 20445  df-rsp 20446  df-2idl 20512  df-nzr 20538  df-domn 20564  df-idom 20565  df-prmidl 31620

This theorem is referenced by:  qsidom  31639