Theorem qsidomlem1 33445

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 20272	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	ad2antrr 725	. 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 7464	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (𝑅 ~QG 𝐼) = (𝑅 ~QG (Base‘𝑅)))
7	6	oveq2d 7464	. . . . . . . . 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 2792	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑄 = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
9	8	fveq2d 6924	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘𝑄) = (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))))
10		ringgrp 20265	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
11	1, 10	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Grp)
12	11	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑅 ∈ Grp)
13		eqid 2740	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
14		eqid 2740	. . . . . . . . 9 ⊢ (𝑅 /s (𝑅 ~QG (Base‘𝑅))) = (𝑅 /s (𝑅 ~QG (Base‘𝑅)))
15	13, 14	qustriv 33357	. . . . . . . 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 2780	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘𝑄) = {(Base‘𝑅)})
18	17	fveq2d 6924	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = (♯‘{(Base‘𝑅)}))
19		fvex 6933	. . . . . 6 ⊢ (Base‘𝑅) ∈ V
20		hashsng 14418	. . . . . 6 ⊢ ((Base‘𝑅) ∈ V → (♯‘{(Base‘𝑅)}) = 1)
21	19, 20	ax-mp 5	. . . . 5 ⊢ (♯‘{(Base‘𝑅)}) = 1
22	18, 21	eqtrdi 2796	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = 1)
23		1red 11291	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 1 ∈ ℝ)
24		isidom 20747	. . . . . . . . . 10 ⊢ (𝑄 ∈ IDomn ↔ (𝑄 ∈ CRing ∧ 𝑄 ∈ Domn))
25	24	simprbi 496	. . . . . . . . 9 ⊢ (𝑄 ∈ IDomn → 𝑄 ∈ Domn)
26		domnnzr 20728	. . . . . . . . 9 ⊢ (𝑄 ∈ Domn → 𝑄 ∈ NzRing)
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝑄 ∈ IDomn → 𝑄 ∈ NzRing)
28	27	ad2antlr 726	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑄 ∈ NzRing)
29		eqid 2740	. . . . . . . . 9 ⊢ (Base‘𝑄) = (Base‘𝑄)
30	29	isnzr2hash 20545	. . . . . . . 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 11425	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) ≠ 1)
34	33	neneqd 2951	. . . 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 2953	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ≠ (Base‘𝑅))
37	25	ad4antlr 732	. . . . . . 7 ⊢ ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝐼) → 𝑄 ∈ Domn)
38		ovex 7481	. . . . . . . . . 10 ⊢ (𝑅 ~QG 𝐼) ∈ V
39	38	ecelqsi 8831	. . . . . . . . 9 ⊢ (𝑥 ∈ (Base‘𝑅) → [𝑥](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
40	39	ad3antlr 730	. . . . . . . 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 2741	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
44		ovexd 7483	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ~QG 𝐼) ∈ V)
45		id 22	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing)
46	42, 43, 44, 45	qusbas 17605	. . . . . . . . 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 2846	. . . . . . 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 8831	. . . . . . . . 9 ⊢ (𝑦 ∈ (Base‘𝑅) → [𝑦](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
50	49	ad2antlr 726	. . . . . . . 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 2846	. . . . . . 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 2740	. . . . . . . . . . . 12 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
54	53	lidlsubg 21256	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
55	1, 54	sylan 579	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
56	55	ad4antr 731	. . . . . . . . 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 2740	. . . . . . . . . . 11 ⊢ (𝑅 ~QG 𝐼) = (𝑅 ~QG 𝐼)
59	58	eqg0el 19223	. . . . . . . . . 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 2741	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
64	13, 58	eqger 19218	. . . . . . . . . . 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 21314	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → (LIdeal‘𝑅) = (2Ideal‘𝑅))
68	67	eleq2d 2830	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (𝐼 ∈ (LIdeal‘𝑅) ↔ 𝐼 ∈ (2Ideal‘𝑅)))
69	68	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
70		eqid 2740	. . . . . . . . . . . 12 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
71		eqid 2740	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
72	13, 58, 70, 71	2idlcpbl 21305	. . . . . . . . . . 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 684	. . . . . . . . . 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 725	. . . . . . . . . . 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 20277	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅)) → (𝑒(.r‘𝑅)𝑓) ∈ (Base‘𝑅))
78	74, 75, 76, 77	syl3anc 1371	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → (𝑒(.r‘𝑅)𝑓) ∈ (Base‘𝑅))
79		eqid 2740	. . . . . . . . . 10 ⊢ (.r‘𝑄) = (.r‘𝑄)
80	62, 63, 65, 66, 73, 78, 71, 79	qusmulval 17615	. . . . . . . . 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 1367	. . . . . . . 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 21281	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
83	1, 82	sylan 579	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
84		eqid 2740	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
85	4, 84	qus0 19229	. . . . . . . . . . 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 19220	. . . . . . . . . . 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 2782	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (0g‘𝑄) = 𝐼)
90	89	ad4antr 731	. . . . . . . 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 2790	. . . . . . 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 2740	. . . . . . . . 9 ⊢ (0g‘𝑄) = (0g‘𝑄)
93	29, 79, 92	domneq0 20730	. . . . . . . 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 1373	. . . . . 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 2751	. . . . . . . . 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 19223	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑥 ∈ 𝐼))
99	97, 55, 98	syl2anc 583	. . . . . . . . 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 2751	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = (0g‘𝑄) ↔ [𝑦](𝑅 ~QG 𝐼) = 𝐼))
102	58	eqg0el 19223	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼 ↔ 𝑦 ∈ 𝐼))
103	97, 55, 102	syl2anc 583	. . . . . . . . 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 917	. . . . . . 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 731	. . . . . 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 3208	. 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 33434	. 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 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  Vcvv 3488  {csn 4648   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448   Er wer 8760  [cec 8761   / cqs 8762  1c1 11185   < clt 11324  ♯chash 14379  Basecbs 17258  ._rcmulr 17312  0_gc0g 17499   /_s cqus 17565  Grpcgrp 18973  SubGrpcsubg 19160  NrmSGrpcnsg 19161   ~_QG cqg 19162  Ringcrg 20260  CRingccrg 20261  NzRingcnzr 20538  Domncdomn 20714  IDomncidom 20715  LIdealclidl 21239  2Idealc2idl 21282  PrmIdealcprmidl 33428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-er 8763  df-ec 8765  df-qs 8769  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-xnn0 12626  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-hash 14380  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-0g 17501  df-imas 17568  df-qus 17569  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-sbg 18978  df-subg 19163  df-nsg 19164  df-eqg 19165  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-cring 20263  df-oppr 20360  df-nzr 20539  df-subrg 20597  df-domn 20717  df-idom 20718  df-lmod 20882  df-lss 20953  df-lsp 20993  df-sra 21195  df-rgmod 21196  df-lidl 21241  df-rsp 21242  df-2idl 21283  df-prmidl 33429

This theorem is referenced by:  qsidom  33447