Theorem qsidomlem1 33467

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 20205	. . 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 7421	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (𝑅 ~QG 𝐼) = (𝑅 ~QG (Base‘𝑅)))
7	6	oveq2d 7421	. . . . . . . . 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 2782	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑄 = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
9	8	fveq2d 6880	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘𝑄) = (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))))
10		ringgrp 20198	. . . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
14		eqid 2735	. . . . . . . . 9 ⊢ (𝑅 /s (𝑅 ~QG (Base‘𝑅))) = (𝑅 /s (𝑅 ~QG (Base‘𝑅)))
15	13, 14	qustriv 33379	. . . . . . . 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 2770	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘𝑄) = {(Base‘𝑅)})
18	17	fveq2d 6880	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = (♯‘{(Base‘𝑅)}))
19		fvex 6889	. . . . . 6 ⊢ (Base‘𝑅) ∈ V
20		hashsng 14387	. . . . . 6 ⊢ ((Base‘𝑅) ∈ V → (♯‘{(Base‘𝑅)}) = 1)
21	19, 20	ax-mp 5	. . . . 5 ⊢ (♯‘{(Base‘𝑅)}) = 1
22	18, 21	eqtrdi 2786	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = 1)
23		1red 11236	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 1 ∈ ℝ)
24		isidom 20685	. . . . . . . . . 10 ⊢ (𝑄 ∈ IDomn ↔ (𝑄 ∈ CRing ∧ 𝑄 ∈ Domn))
25	24	simprbi 496	. . . . . . . . 9 ⊢ (𝑄 ∈ IDomn → 𝑄 ∈ Domn)
26		domnnzr 20666	. . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (Base‘𝑄) = (Base‘𝑄)
30	29	isnzr2hash 20479	. . . . . . . 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 11370	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) ≠ 1)
34	33	neneqd 2937	. . . 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 2939	. 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 7438	. . . . . . . . . 10 ⊢ (𝑅 ~QG 𝐼) ∈ V
39	38	ecelqsi 8787	. . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
44		ovexd 7440	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝑅 ~QG 𝐼) ∈ V)
45		id 22	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CRing)
46	42, 43, 44, 45	qusbas 17559	. . . . . . . . 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 2836	. . . . . . 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 8787	. . . . . . . . 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 2836	. . . . . . 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 2735	. . . . . . . . . . . 12 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
54	53	lidlsubg 21184	. . . . . . . . . . 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 2735	. . . . . . . . . . 11 ⊢ (𝑅 ~QG 𝐼) = (𝑅 ~QG 𝐼)
59	58	eqg0el 19166	. . . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
64	13, 58	eqger 19161	. . . . . . . . . . 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 21242	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → (LIdeal‘𝑅) = (2Ideal‘𝑅))
68	67	eleq2d 2820	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (𝐼 ∈ (LIdeal‘𝑅) ↔ 𝐼 ∈ (2Ideal‘𝑅)))
69	68	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
70		eqid 2735	. . . . . . . . . . . 12 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
71		eqid 2735	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
72	13, 58, 70, 71	2idlcpbl 21233	. . . . . . . . . . 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 20210	. . . . . . . . . . 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 2735	. . . . . . . . . 10 ⊢ (.r‘𝑄) = (.r‘𝑄)
80	62, 63, 65, 66, 73, 78, 71, 79	qusmulval 17569	. . . . . . . . 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 21209	. . . . . . . . . . . 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 2735	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
85	4, 84	qus0 19172	. . . . . . . . . . 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 19163	. . . . . . . . . . 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 2772	. . . . . . . . 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 2780	. . . . . . 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 2735	. . . . . . . . 9 ⊢ (0g‘𝑄) = (0g‘𝑄)
93	29, 79, 92	domneq0 20668	. . . . . . . 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 2746	. . . . . . . . 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 19166	. . . . . . . . . 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 2746	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = (0g‘𝑄) ↔ [𝑦](𝑅 ~QG 𝐼) = 𝐼))
102	58	eqg0el 19166	. . . . . . . . . 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 3187	. 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 33456	. 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 2108   ≠ wne 2932  ∀wral 3051  Vcvv 3459  {csn 4601   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405   Er wer 8716  [cec 8717   / cqs 8718  1c1 11130   < clt 11269  ♯chash 14348  Basecbs 17228  ._rcmulr 17272  0_gc0g 17453   /_s cqus 17519  Grpcgrp 18916  SubGrpcsubg 19103  NrmSGrpcnsg 19104   ~_QG cqg 19105  Ringcrg 20193  CRingccrg 20194  NzRingcnzr 20472  Domncdomn 20652  IDomncidom 20653  LIdealclidl 21167  2Idealc2idl 21210  PrmIdealcprmidl 33450

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8719  df-ec 8721  df-qs 8725  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-sup 9454  df-inf 9455  df-dju 9915  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-xnn0 12575  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-hash 14349  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-ds 17293  df-0g 17455  df-imas 17522  df-qus 17523  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-grp 18919  df-minusg 18920  df-sbg 18921  df-subg 19106  df-nsg 19107  df-eqg 19108  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-cring 20196  df-oppr 20297  df-nzr 20473  df-subrg 20530  df-domn 20655  df-idom 20656  df-lmod 20819  df-lss 20889  df-lsp 20929  df-sra 21131  df-rgmod 21132  df-lidl 21169  df-rsp 21170  df-2idl 21211  df-prmidl 33451

This theorem is referenced by:  qsidom  33469