Theorem smprngopr 38586

Description: Obsolete theorem, use smprngprmrng 48986 instead. A simple ring (one whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
smprngpr.1	⊢ 𝐺 = (1st ‘𝑅)
smprngpr.2	⊢ 𝐻 = (2nd ‘𝑅)
smprngpr.3	⊢ 𝑋 = ran 𝐺
smprngpr.4	⊢ 𝑍 = (GId‘𝐺)
smprngpr.5	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
smprngopr	⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing)

Proof of Theorem smprngopr

Dummy variables 𝑖 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1152	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ RingOps)
2		smprngpr.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑅)
3		smprngpr.4	. . . . 5 ⊢ 𝑍 = (GId‘𝐺)
4	2, 3	0idl 38559	. . . 4 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
5	4	3ad2ant1 1149	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ∈ (Idl‘𝑅))
6		smprngpr.2	. . . . . . . 8 ⊢ 𝐻 = (2nd ‘𝑅)
7		smprngpr.3	. . . . . . . 8 ⊢ 𝑋 = ran 𝐺
8		smprngpr.5	. . . . . . . 8 ⊢ 𝑈 = (GId‘𝐻)
9	2, 6, 7, 3, 8	0rngo 38561	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍}))
10		eqcom 2776	. . . . . . 7 ⊢ (𝑈 = 𝑍 ↔ 𝑍 = 𝑈)
11		eqcom 2776	. . . . . . 7 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍})
12	9, 10, 11	3bitr4g 317	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑈 = 𝑍 ↔ {𝑍} = 𝑋))
13	12	necon3bid 3008	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑈 ≠ 𝑍 ↔ {𝑍} ≠ 𝑋))
14	13	biimpa 481	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → {𝑍} ≠ 𝑋)
15	14	3adant3 1148	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ≠ 𝑋)
16		df-pr 4594	. . . . . . . 8 ⊢ {{𝑍}, 𝑋} = ({{𝑍}} ∪ {𝑋})
17	16	eqeq2i 2782	. . . . . . 7 ⊢ ((Idl‘𝑅) = {{𝑍}, 𝑋} ↔ (Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}))
18		eleq2 2858	. . . . . . . . 9 ⊢ ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ ({{𝑍}} ∪ {𝑋})))
19		eleq2 2858	. . . . . . . . 9 ⊢ ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → (𝑗 ∈ (Idl‘𝑅) ↔ 𝑗 ∈ ({{𝑍}} ∪ {𝑋})))
20	18, 19	anbi12d 643	. . . . . . . 8 ⊢ ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} ∪ {𝑋}))))
21		elun 4115	. . . . . . . . . 10 ⊢ (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}))
22		velsn 4607	. . . . . . . . . . 11 ⊢ (𝑖 ∈ {{𝑍}} ↔ 𝑖 = {𝑍})
23		velsn 4607	. . . . . . . . . . 11 ⊢ (𝑖 ∈ {𝑋} ↔ 𝑖 = 𝑋)
24	22, 23	orbi12i 927	. . . . . . . . . 10 ⊢ ((𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
25	21, 24	bitri 278	. . . . . . . . 9 ⊢ (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
26		elun 4115	. . . . . . . . . 10 ⊢ (𝑗 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}))
27		velsn 4607	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {{𝑍}} ↔ 𝑗 = {𝑍})
28		velsn 4607	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑋} ↔ 𝑗 = 𝑋)
29	27, 28	orbi12i 927	. . . . . . . . . 10 ⊢ ((𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))
30	26, 29	bitri 278	. . . . . . . . 9 ⊢ (𝑗 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))
31	25, 30	anbi12i 639	. . . . . . . 8 ⊢ ((𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} ∪ {𝑋})) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)))
32	20, 31	bitrdi 290	. . . . . . 7 ⊢ ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
33	17, 32	sylbi 220	. . . . . 6 ⊢ ((Idl‘𝑅) = {{𝑍}, 𝑋} → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
34	33	3ad2ant3 1151	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
35		eqimss 4003	. . . . . . . . . . 11 ⊢ (𝑖 = {𝑍} → 𝑖 ⊆ {𝑍})
36	35	orcd 886	. . . . . . . . . 10 ⊢ (𝑖 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
37	36	adantr 485	. . . . . . . . 9 ⊢ ((𝑖 = {𝑍} ∧ 𝑗 = {𝑍}) → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
38	37	a1d 26	. . . . . . . 8 ⊢ ((𝑖 = {𝑍} ∧ 𝑗 = {𝑍}) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
39	38	a1i 11	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ((𝑖 = {𝑍} ∧ 𝑗 = {𝑍}) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
40		eqimss 4003	. . . . . . . . . . 11 ⊢ (𝑗 = {𝑍} → 𝑗 ⊆ {𝑍})
41	40	olcd 887	. . . . . . . . . 10 ⊢ (𝑗 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
42	41	adantl 486	. . . . . . . . 9 ⊢ ((𝑖 = 𝑋 ∧ 𝑗 = {𝑍}) → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
43	42	a1d 26	. . . . . . . 8 ⊢ ((𝑖 = 𝑋 ∧ 𝑗 = {𝑍}) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
44	43	a1i 11	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ((𝑖 = 𝑋 ∧ 𝑗 = {𝑍}) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
45	36	adantr 485	. . . . . . . . 9 ⊢ ((𝑖 = {𝑍} ∧ 𝑗 = 𝑋) → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
46	45	a1d 26	. . . . . . . 8 ⊢ ((𝑖 = {𝑍} ∧ 𝑗 = 𝑋) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
47	46	a1i 11	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ((𝑖 = {𝑍} ∧ 𝑗 = 𝑋) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
48	2	rneqi 5925	. . . . . . . . . . . . . 14 ⊢ ran 𝐺 = ran (1st ‘𝑅)
49	7, 48	eqtri 2792	. . . . . . . . . . . . 13 ⊢ 𝑋 = ran (1st ‘𝑅)
50	49, 6, 8	rngo1cl 38473	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
51	50	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → 𝑈 ∈ 𝑋)
52	6, 49, 8	rngolidm 38471	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑈𝐻𝑈) = 𝑈)
53	50, 52	mpdan 699	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ RingOps → (𝑈𝐻𝑈) = 𝑈)
54	53	eleq1d 2854	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
55	8	fvexi 6893	. . . . . . . . . . . . . . 15 ⊢ 𝑈 ∈ V
56	55	elsn 4606	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ {𝑍} ↔ 𝑈 = 𝑍)
57	54, 56	bitrdi 290	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 = 𝑍))
58	57	necon3bbid 3001	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → (¬ (𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 ≠ 𝑍))
59	58	biimpar 482	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ¬ (𝑈𝐻𝑈) ∈ {𝑍})
60		oveq1 7415	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑈 → (𝑥𝐻𝑦) = (𝑈𝐻𝑦))
61	60	eleq1d 2854	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑈 → ((𝑥𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑦) ∈ {𝑍}))
62	61	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑈 → (¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑦) ∈ {𝑍}))
63		oveq2 7416	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑈 → (𝑈𝐻𝑦) = (𝑈𝐻𝑈))
64	63	eleq1d 2854	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑈 → ((𝑈𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑈) ∈ {𝑍}))
65	64	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑈 → (¬ (𝑈𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑈) ∈ {𝑍}))
66	62, 65	rspc2ev 3603	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑈 ∈ 𝑋 ∧ ¬ (𝑈𝐻𝑈) ∈ {𝑍}) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍})
67	51, 51, 59, 66	syl3anc 1396	. . . . . . . . . 10 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍})
68		rexnal2 3153	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍})
69	67, 68	sylib 221	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ¬ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍})
70	69	pm2.21d 122	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
71		raleq 3326	. . . . . . . . . 10 ⊢ (𝑖 = 𝑋 → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍}))
72		raleq 3326	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑋 → (∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
73	72	ralbidv 3194	. . . . . . . . . 10 ⊢ (𝑗 = 𝑋 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
74	71, 73	sylan9bb 518	. . . . . . . . 9 ⊢ ((𝑖 = 𝑋 ∧ 𝑗 = 𝑋) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
75	74	imbi1d 344	. . . . . . . 8 ⊢ ((𝑖 = 𝑋 ∧ 𝑗 = 𝑋) → ((∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})) ↔ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
76	70, 75	syl5ibrcom 250	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ((𝑖 = 𝑋 ∧ 𝑗 = 𝑋) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
77	39, 44, 47, 76	ccased 1052	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → (((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
78	77	3adant3 1148	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → (((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
79	34, 78	sylbid 243	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))
80	79	ralrimivv 3212	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ∀𝑖 ∈ (Idl‘𝑅)∀𝑗 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
81	2, 6, 7	ispridl 38568	. . . 4 ⊢ (𝑅 ∈ RingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑖 ∈ (Idl‘𝑅)∀𝑗 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))))
82	81	3ad2ant1 1149	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑖 ∈ (Idl‘𝑅)∀𝑗 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))))
83	5, 15, 80, 82	mpbir3and 1359	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ∈ (PrIdl‘𝑅))
84	2, 3	isprrngo 38584	. 2 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
85	1, 83, 84	sylanbrc 594	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∪ cun 3911   ⊆ wss 3913  {csn 4591  {cpr 4593  ran crn 5660  ‘cfv 6533  (class class class)co 7408  1^st c1st 7980  2^nd c2nd 7981  GIdcgi 30779  RingOpscrngo 38428  Idlcidl 38541  PrIdlcpridl 38542  PrRingcprrng 38580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-1st 7982  df-2nd 7983  df-grpo 30782  df-gid 30783  df-ginv 30784  df-ablo 30834  df-ass 38377  df-exid 38379  df-mgmOLD 38383  df-sgrOLD 38395  df-mndo 38401  df-rngo 38429  df-idl 38544  df-pridl 38545  df-prrngo 38582

This theorem is referenced by:  divrngpr  38587