Theorem qsidomlem1 33396

Description: If the quotient ring of a commutative ring relative to an ideal is an integral domain, that ideal must be prime. (Contributed by Thierry Arnoux, 16-Jan-2024.)

Hypothesis

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

Assertion

Ref	Expression
qsidomlem1	⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅))

Proof of Theorem qsidomlem1

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

Step	Hyp	Ref	Expression
1		crngring 20130	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	ad2antrr 726	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝑅 ∈ Ring)
3		simplr 768	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (LIdeal‘𝑅))
4		qsidom.1	. . . . . . . . 9 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
5		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝐼 = (Base‘𝑅))
6	5	oveq2d 7385	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (𝑅 ~QG 𝐼) = (𝑅 ~QG (Base‘𝑅)))
7	6	oveq2d 7385	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (𝑅 /s (𝑅 ~QG 𝐼)) = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
8	4, 7	eqtrid 2776	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑄 = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
9	8	fveq2d 6844	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘𝑄) = (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))))
10		ringgrp 20123	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
11	1, 10	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Grp)
12	11	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑅 ∈ Grp)
13		eqid 2729	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
14		eqid 2729	. . . . . . . . 9 ⊢ (𝑅 /s (𝑅 ~QG (Base‘𝑅))) = (𝑅 /s (𝑅 ~QG (Base‘𝑅)))
15	13, 14	qustriv 33308	. . . . . . . 8 ⊢ (𝑅 ∈ Grp → (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))) = {(Base‘𝑅)})
16	12, 15	syl 17	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))) = {(Base‘𝑅)})
17	9, 16	eqtrd 2764	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘𝑄) = {(Base‘𝑅)})
18	17	fveq2d 6844	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = (♯‘{(Base‘𝑅)}))
19		fvex 6853	. . . . . 6 ⊢ (Base‘𝑅) ∈ V
20		hashsng 14310	. . . . . 6 ⊢ ((Base‘𝑅) ∈ V → (♯‘{(Base‘𝑅)}) = 1)
21	19, 20	ax-mp 5	. . . . 5 ⊢ (♯‘{(Base‘𝑅)}) = 1
22	18, 21	eqtrdi 2780	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = 1)
23		1red 11151	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 1 ∈ ℝ)
24		isidom 20610	. . . . . . . . . 10 ⊢ (𝑄 ∈ IDomn ↔ (𝑄 ∈ CRing ∧ 𝑄 ∈ Domn))
25	24	simprbi 496	. . . . . . . . 9 ⊢ (𝑄 ∈ IDomn → 𝑄 ∈ Domn)
26		domnnzr 20591	. . . . . . . . 9 ⊢ (𝑄 ∈ Domn → 𝑄 ∈ NzRing)
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝑄 ∈ IDomn → 𝑄 ∈ NzRing)
28	27	ad2antlr 727	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑄 ∈ NzRing)
29		eqid 2729	. . . . . . . . 9 ⊢ (Base‘𝑄) = (Base‘𝑄)
30	29	isnzr2hash 20404	. . . . . . . 8 ⊢ (𝑄 ∈ NzRing ↔ (𝑄 ∈ Ring ∧ 1 < (♯‘(Base‘𝑄))))
31	30	simprbi 496	. . . . . . 7 ⊢ (𝑄 ∈ NzRing → 1 < (♯‘(Base‘𝑄)))
32	28, 31	syl 17	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 1 < (♯‘(Base‘𝑄)))
33	23, 32	gtned 11285	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) ≠ 1)
34	33	neneqd 2930	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → ¬ (♯‘(Base‘𝑄)) = 1)
35	22, 34	pm2.65da 816	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → ¬ 𝐼 = (Base‘𝑅))
36	35	neqned 2932	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ≠ (Base‘𝑅))
37	25	ad4antlr 733	. . . . . . 7 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → 𝑄 ∈ Domn)
38		ovex 7402	. . . . . . . . . 10 ⊢ (𝑅 ~QG 𝐼) ∈ V
39	38	ecelqsi 8720	. . . . . . . . 9 ⊢ (𝑥 ∈ (Base‘𝑅) → [𝑥](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
40	39	ad3antlr 731	. . . . . . . 8 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → [𝑥](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
41		simp-5l 784	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → 𝑅 ∈ CRing)
42	4	a1i 11	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)))
43		eqidd 2730	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
44		ovexd 7404	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ~QG 𝐼) ∈ V)
45		id 22	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing)
46	42, 43, 44, 45	qusbas 17484	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → ((Base‘𝑅) / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
47	41, 46	syl 17	. . . . . . . 8 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → ((Base‘𝑅) / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
48	40, 47	eleqtrd 2830	. . . . . . 7 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → [𝑥](𝑅 ~QG 𝐼) ∈ (Base‘𝑄))
49	38	ecelqsi 8720	. . . . . . . . 9 ⊢ (𝑦 ∈ (Base‘𝑅) → [𝑦](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
50	49	ad2antlr 727	. . . . . . . 8 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → [𝑦](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
51	50, 47	eleqtrd 2830	. . . . . . 7 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → [𝑦](𝑅 ~QG 𝐼) ∈ (Base‘𝑄))
52	41, 1, 10	3syl 18	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → 𝑅 ∈ Grp)
53		eqid 2729	. . . . . . . . . . . 12 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
54	53	lidlsubg 21109	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
55	1, 54	sylan 580	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
56	55	ad4antr 732	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → 𝐼 ∈ (SubGrp‘𝑅))
57		simpr 484	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐼)
58		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑅 ~QG 𝐼) = (𝑅 ~QG 𝐼)
59	58	eqg0el 19091	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([(𝑥(.r‘𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼 ↔ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼))
60	59	biimpar 477	. . . . . . . . 9 ⊢ (((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → [(𝑥(.r‘𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼)
61	52, 56, 57, 60	syl21anc 837	. . . . . . . 8 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → [(𝑥(.r‘𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼)
62	4	a1i 11	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)))
63		eqidd 2730	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
64	13, 58	eqger 19086	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → (𝑅 ~QG 𝐼) Er (Base‘𝑅))
65	55, 64	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑅 ~QG 𝐼) Er (Base‘𝑅))
66		simpl 482	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
67	53	crng2idl 21167	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → (LIdeal‘𝑅) = (2Ideal‘𝑅))
68	67	eleq2d 2814	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (𝐼 ∈ (LIdeal‘𝑅) ↔ 𝐼 ∈ (2Ideal‘𝑅)))
69	68	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
70		eqid 2729	. . . . . . . . . . . 12 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
71		eqid 2729	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
72	13, 58, 70, 71	2idlcpbl 21158	. . . . . . . . . . 11 ⊢ ((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)))
73	1, 69, 72	syl2an2r 685	. . . . . . . . . 10 ⊢ ((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)))
74	1	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
75		simprl 770	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑒 ∈ (Base‘𝑅))
76		simprr 772	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑓 ∈ (Base‘𝑅))
77	13, 71	ringcl 20135	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅)) → (𝑒(.r‘𝑅)𝑓) ∈ (Base‘𝑅))
78	74, 75, 76, 77	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → (𝑒(.r‘𝑅)𝑓) ∈ (Base‘𝑅))
79		eqid 2729	. . . . . . . . . 10 ⊢ (.r‘𝑄) = (.r‘𝑄)
80	62, 63, 65, 66, 73, 78, 71, 79	qusmulval 17494	. . . . . . . . 9 ⊢ (((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))
81	80	ad5ant134 1369	. . . . . . . 8 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → ([𝑥](𝑅 ~QG 𝐼)(.r‘𝑄)[𝑦](𝑅 ~QG 𝐼)) = [(𝑥(.r‘𝑅)𝑦)](𝑅 ~QG 𝐼))
82		lidlnsg 21134	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
83	1, 82	sylan 580	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
84		eqid 2729	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
85	4, 84	qus0 19097	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → [(0g‘𝑅)](𝑅 ~QG 𝐼) = (0g‘𝑄))
86	83, 85	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → [(0g‘𝑅)](𝑅 ~QG 𝐼) = (0g‘𝑄))
87	13, 58, 84	eqgid 19088	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → [(0g‘𝑅)](𝑅 ~QG 𝐼) = 𝐼)
88	55, 87	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → [(0g‘𝑅)](𝑅 ~QG 𝐼) = 𝐼)
89	86, 88	eqtr3d 2766	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (0g‘𝑄) = 𝐼)
90	89	ad4antr 732	. . . . . . . 8 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → (0g‘𝑄) = 𝐼)
91	61, 81, 90	3eqtr4d 2774	. . . . . . 7 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → ([𝑥](𝑅 ~QG 𝐼)(.r‘𝑄)[𝑦](𝑅 ~QG 𝐼)) = (0g‘𝑄))
92		eqid 2729	. . . . . . . . 9 ⊢ (0g‘𝑄) = (0g‘𝑄)
93	29, 79, 92	domneq0 20593	. . . . . . . 8 ⊢ ((𝑄 ∈ Domn ∧ [𝑥](𝑅 ~QG 𝐼) ∈ (Base‘𝑄) ∧ [𝑦](𝑅 ~QG 𝐼) ∈ (Base‘𝑄)) → (([𝑥](𝑅 ~QG 𝐼)(.r‘𝑄)[𝑦](𝑅 ~QG 𝐼)) = (0g‘𝑄) ↔ ([𝑥](𝑅 ~QG 𝐼) = (0g‘𝑄) ∨ [𝑦](𝑅 ~QG 𝐼) = (0g‘𝑄))))
94	93	biimpa 476	. . . . . . 7 ⊢ (((𝑄 ∈ Domn ∧ [𝑥](𝑅 ~QG 𝐼) ∈ (Base‘𝑄) ∧ [𝑦](𝑅 ~QG 𝐼) ∈ (Base‘𝑄)) ∧ ([𝑥](𝑅 ~QG 𝐼)(.r‘𝑄)[𝑦](𝑅 ~QG 𝐼)) = (0g‘𝑄)) → ([𝑥](𝑅 ~QG 𝐼) = (0g‘𝑄) ∨ [𝑦](𝑅 ~QG 𝐼) = (0g‘𝑄)))
95	37, 48, 51, 91, 94	syl31anc 1375	. . . . . 6 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → ([𝑥](𝑅 ~QG 𝐼) = (0g‘𝑄) ∨ [𝑦](𝑅 ~QG 𝐼) = (0g‘𝑄)))
96	89	eqeq2d 2740	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = (0g‘𝑄) ↔ [𝑥](𝑅 ~QG 𝐼) = 𝐼))
97	66, 1, 10	3syl 18	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ Grp)
98	58	eqg0el 19091	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑥 ∈ 𝐼))
99	97, 55, 98	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑥 ∈ 𝐼))
100	96, 99	bitrd 279	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = (0g‘𝑄) ↔ 𝑥 ∈ 𝐼))
101	89	eqeq2d 2740	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = (0g‘𝑄) ↔ [𝑦](𝑅 ~QG 𝐼) = 𝐼))
102	58	eqg0el 19091	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑦 ∈ 𝐼))
103	97, 55, 102	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑦 ∈ 𝐼))
104	101, 103	bitrd 279	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = (0g‘𝑄) ↔ 𝑦 ∈ 𝐼))
105	100, 104	orbi12d 918	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (([𝑥](𝑅 ~QG 𝐼) = (0g‘𝑄) ∨ [𝑦](𝑅 ~QG 𝐼) = (0g‘𝑄)) ↔ (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)))
106	105	ad4antr 732	. . . . . 6 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → (([𝑥](𝑅 ~QG 𝐼) = (0g‘𝑄) ∨ [𝑦](𝑅 ~QG 𝐼) = (0g‘𝑄)) ↔ (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)))
107	95, 106	mpbid 232	. . . . 5 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼))
108	107	ex 412	. . . 4 ⊢ (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)))
109	108	anasss 466	. . 3 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)))
110	109	ralrimivva 3178	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)))
111	13, 71	prmidl2 33385	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝐼 ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) ∈ 𝐼 → (𝑥 ∈ 𝐼 ∨ 𝑦 ∈ 𝐼)))) → 𝐼 ∈ (PrmIdeal‘𝑅))
112	2, 3, 36, 110, 111	syl22anc 838	1 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3444  {csn 4585   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369   Er wer 8645  [cec 8646   / cqs 8647  1c1 11045   < clt 11184  ♯chash 14271  Basecbs 17155  ._rcmulr 17197  0_gc0g 17378   /_s cqus 17444  Grpcgrp 18841  SubGrpcsubg 19028  NrmSGrpcnsg 19029   ~_QG cqg 19030  Ringcrg 20118  CRingccrg 20119  NzRingcnzr 20397  Domncdomn 20577  IDomncidom 20578  LIdealclidl 21092  2Idealc2idl 21135  PrmIdealcprmidl 33379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-oadd 8415  df-er 8648  df-ec 8650  df-qs 8654  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-inf 9370  df-dju 9830  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-xnn0 12492  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-hash 14272  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-0g 17380  df-imas 17447  df-qus 17448  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-grp 18844  df-minusg 18845  df-sbg 18846  df-subg 19031  df-nsg 19032  df-eqg 19033  df-cmn 19688  df-abl 19689  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-cring 20121  df-oppr 20222  df-nzr 20398  df-subrg 20455  df-domn 20580  df-idom 20581  df-lmod 20744  df-lss 20814  df-lsp 20854  df-sra 21056  df-rgmod 21057  df-lidl 21094  df-rsp 21095  df-2idl 21136  df-prmidl 33380

This theorem is referenced by:  qsidom  33398