Theorem smprngopr 38373

Description: A simple ring (one whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)

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 1137	. 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 38346	. . . 4 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
5	4	3ad2ant1 1134	. . 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 38348	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍}))
10		eqcom 2743	. . . . . . 7 ⊢ (𝑈 = 𝑍 ↔ 𝑍 = 𝑈)
11		eqcom 2743	. . . . . . 7 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍})
12	9, 10, 11	3bitr4g 314	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝑈 = 𝑍 ↔ {𝑍} = 𝑋))
13	12	necon3bid 2976	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑈 ≠ 𝑍 ↔ {𝑍} ≠ 𝑋))
14	13	biimpa 476	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → {𝑍} ≠ 𝑋)
15	14	3adant3 1133	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ≠ 𝑋)
16		df-pr 4570	. . . . . . . 8 ⊢ {{𝑍}, 𝑋} = ({{𝑍}} ∪ {𝑋})
17	16	eqeq2i 2749	. . . . . . 7 ⊢ ((Idl‘𝑅) = {{𝑍}, 𝑋} ↔ (Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}))
18		eleq2 2825	. . . . . . . . 9 ⊢ ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ ({{𝑍}} ∪ {𝑋})))
19		eleq2 2825	. . . . . . . . 9 ⊢ ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → (𝑗 ∈ (Idl‘𝑅) ↔ 𝑗 ∈ ({{𝑍}} ∪ {𝑋})))
20	18, 19	anbi12d 633	. . . . . . . 8 ⊢ ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} ∪ {𝑋}))))
21		elun 4093	. . . . . . . . . 10 ⊢ (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}))
22		velsn 4583	. . . . . . . . . . 11 ⊢ (𝑖 ∈ {{𝑍}} ↔ 𝑖 = {𝑍})
23		velsn 4583	. . . . . . . . . . 11 ⊢ (𝑖 ∈ {𝑋} ↔ 𝑖 = 𝑋)
24	22, 23	orbi12i 915	. . . . . . . . . 10 ⊢ ((𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
25	21, 24	bitri 275	. . . . . . . . 9 ⊢ (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
26		elun 4093	. . . . . . . . . 10 ⊢ (𝑗 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}))
27		velsn 4583	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {{𝑍}} ↔ 𝑗 = {𝑍})
28		velsn 4583	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑋} ↔ 𝑗 = 𝑋)
29	27, 28	orbi12i 915	. . . . . . . . . 10 ⊢ ((𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))
30	26, 29	bitri 275	. . . . . . . . 9 ⊢ (𝑗 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))
31	25, 30	anbi12i 629	. . . . . . . 8 ⊢ ((𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} ∪ {𝑋})) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)))
32	20, 31	bitrdi 287	. . . . . . 7 ⊢ ((Idl‘𝑅) = ({{𝑍}} ∪ {𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
33	17, 32	sylbi 217	. . . . . 6 ⊢ ((Idl‘𝑅) = {{𝑍}, 𝑋} → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))))
34	33	3ad2ant3 1136	. . . . 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 3980	. . . . . . . . . . 11 ⊢ (𝑖 = {𝑍} → 𝑖 ⊆ {𝑍})
36	35	orcd 874	. . . . . . . . . 10 ⊢ (𝑖 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
37	36	adantr 480	. . . . . . . . 9 ⊢ ((𝑖 = {𝑍} ∧ 𝑗 = {𝑍}) → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
38	37	a1d 25	. . . . . . . 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 3980	. . . . . . . . . . 11 ⊢ (𝑗 = {𝑍} → 𝑗 ⊆ {𝑍})
41	40	olcd 875	. . . . . . . . . 10 ⊢ (𝑗 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
42	41	adantl 481	. . . . . . . . 9 ⊢ ((𝑖 = 𝑋 ∧ 𝑗 = {𝑍}) → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
43	42	a1d 25	. . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝑖 = {𝑍} ∧ 𝑗 = 𝑋) → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))
46	45	a1d 25	. . . . . . . 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 5892	. . . . . . . . . . . . . 14 ⊢ ran 𝐺 = ran (1st ‘𝑅)
49	7, 48	eqtri 2759	. . . . . . . . . . . . 13 ⊢ 𝑋 = ran (1st ‘𝑅)
50	49, 6, 8	rngo1cl 38260	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
51	50	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → 𝑈 ∈ 𝑋)
52	6, 49, 8	rngolidm 38258	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑈𝐻𝑈) = 𝑈)
53	50, 52	mpdan 688	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ RingOps → (𝑈𝐻𝑈) = 𝑈)
54	53	eleq1d 2821	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 ∈ {𝑍}))
55	8	fvexi 6854	. . . . . . . . . . . . . . 15 ⊢ 𝑈 ∈ V
56	55	elsn 4582	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ {𝑍} ↔ 𝑈 = 𝑍)
57	54, 56	bitrdi 287	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 = 𝑍))
58	57	necon3bbid 2969	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → (¬ (𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 ≠ 𝑍))
59	58	biimpar 477	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ¬ (𝑈𝐻𝑈) ∈ {𝑍})
60		oveq1 7374	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑈 → (𝑥𝐻𝑦) = (𝑈𝐻𝑦))
61	60	eleq1d 2821	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑈 → ((𝑥𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑦) ∈ {𝑍}))
62	61	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑈 → (¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑦) ∈ {𝑍}))
63		oveq2 7375	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑈 → (𝑈𝐻𝑦) = (𝑈𝐻𝑈))
64	63	eleq1d 2821	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑈 → ((𝑈𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑈) ∈ {𝑍}))
65	64	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑈 → (¬ (𝑈𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑈) ∈ {𝑍}))
66	62, 65	rspc2ev 3577	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑈 ∈ 𝑋 ∧ ¬ (𝑈𝐻𝑈) ∈ {𝑍}) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍})
67	51, 51, 59, 66	syl3anc 1374	. . . . . . . . . 10 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍})
68		rexnal2 3119	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍})
69	67, 68	sylib 218	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ¬ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍})
70	69	pm2.21d 121	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))
71		raleq 3292	. . . . . . . . . 10 ⊢ (𝑖 = 𝑋 → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍}))
72		raleq 3292	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑋 → (∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
73	72	ralbidv 3160	. . . . . . . . . 10 ⊢ (𝑗 = 𝑋 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
74	71, 73	sylan9bb 509	. . . . . . . . 9 ⊢ ((𝑖 = 𝑋 ∧ 𝑗 = 𝑋) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍}))
75	74	imbi1d 341	. . . . . . . 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 247	. . . . . . 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 1039	. . . . . 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 1133	. . . . 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 240	. . . 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 3178	. . 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 38355	. . . 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 1134	. . 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 1344	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ∈ (PrIdl‘𝑅))
84	2, 3	isprrngo 38371	. 2 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
85	1, 83, 84	sylanbrc 584	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061   ∪ cun 3887   ⊆ wss 3889  {csn 4567  {cpr 4569  ran crn 5632  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  GIdcgi 30561  RingOpscrngo 38215  Idlcidl 38328  PrIdlcpridl 38329  PrRingcprrng 38367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  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-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-1st 7942  df-2nd 7943  df-grpo 30564  df-gid 30565  df-ginv 30566  df-ablo 30616  df-ass 38164  df-exid 38166  df-mgmOLD 38170  df-sgrOLD 38182  df-mndo 38188  df-rngo 38216  df-idl 38331  df-pridl 38332  df-prrngo 38369

This theorem is referenced by:  divrngpr  38374