Theorem intidl 39155

Description: Obsolete theorem, use intlidl 34040 instead. The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
intidl	⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∩ 𝐶 ∈ (Idl‘𝑅))

Proof of Theorem intidl

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

Step	Hyp	Ref	Expression
1		intssuni 4933	. . . 4 ⊢ (𝐶 ≠ ∅ → ∩ 𝐶 ⊆ ∪ 𝐶)
2	1	3ad2ant2 1152	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∩ 𝐶 ⊆ ∪ 𝐶)
3		ssel2 3908	. . . . . . . 8 ⊢ ((𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖 ∈ 𝐶) → 𝑖 ∈ (Idl‘𝑅))
4		eqid 2763	. . . . . . . . 9 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
5		eqid 2763	. . . . . . . . 9 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅)
6	4, 5	idlss 39142	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖 ⊆ ran (1st ‘𝑅))
7	3, 6	sylan2 605	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖 ∈ 𝐶)) → 𝑖 ⊆ ran (1st ‘𝑅))
8	7	anassrs 473	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖 ∈ 𝐶) → 𝑖 ⊆ ran (1st ‘𝑅))
9	8	ralrimiva 3158	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖 ∈ 𝐶 𝑖 ⊆ ran (1st ‘𝑅))
10	9	3adant2 1149	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖 ∈ 𝐶 𝑖 ⊆ ran (1st ‘𝑅))
11		unissb 4904	. . . 4 ⊢ (∪ 𝐶 ⊆ ran (1st ‘𝑅) ↔ ∀𝑖 ∈ 𝐶 𝑖 ⊆ ran (1st ‘𝑅))
12	10, 11	sylibr 237	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∪ 𝐶 ⊆ ran (1st ‘𝑅))
13	2, 12	sstrd 3923	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∩ 𝐶 ⊆ ran (1st ‘𝑅))
14		eqid 2763	. . . . . . . 8 ⊢ (GId‘(1st ‘𝑅)) = (GId‘(1st ‘𝑅))
15	4, 14	idl0cl 39144	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → (GId‘(1st ‘𝑅)) ∈ 𝑖)
16	3, 15	sylan2 605	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖 ∈ 𝐶)) → (GId‘(1st ‘𝑅)) ∈ 𝑖)
17	16	anassrs 473	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖 ∈ 𝐶) → (GId‘(1st ‘𝑅)) ∈ 𝑖)
18	17	ralrimiva 3158	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖 ∈ 𝐶 (GId‘(1st ‘𝑅)) ∈ 𝑖)
19		fvex 6921	. . . . 5 ⊢ (GId‘(1st ‘𝑅)) ∈ V
20	19	elint2 4917	. . . 4 ⊢ ((GId‘(1st ‘𝑅)) ∈ ∩ 𝐶 ↔ ∀𝑖 ∈ 𝐶 (GId‘(1st ‘𝑅)) ∈ 𝑖)
21	18, 20	sylibr 237	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (GId‘(1st ‘𝑅)) ∈ ∩ 𝐶)
22	21	3adant2 1149	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → (GId‘(1st ‘𝑅)) ∈ ∩ 𝐶)
23		vex 3458	. . . . . 6 ⊢ 𝑥 ∈ V
24	23	elint2 4917	. . . . 5 ⊢ (𝑥 ∈ ∩ 𝐶 ↔ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖)
25		vex 3458	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
26	25	elint2 4917	. . . . . . . . 9 ⊢ (𝑦 ∈ ∩ 𝐶 ↔ ∀𝑖 ∈ 𝐶 𝑦 ∈ 𝑖)
27		r19.26 3126	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝐶 (𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖) ↔ (∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖 ∧ ∀𝑖 ∈ 𝐶 𝑦 ∈ 𝑖))
28	4	idladdcl 39145	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖)) → (𝑥(1st ‘𝑅)𝑦) ∈ 𝑖)
29	28	ex 418	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖) → (𝑥(1st ‘𝑅)𝑦) ∈ 𝑖))
30	3, 29	sylan2 605	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖 ∈ 𝐶)) → ((𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖) → (𝑥(1st ‘𝑅)𝑦) ∈ 𝑖))
31	30	anassrs 473	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖 ∈ 𝐶) → ((𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖) → (𝑥(1st ‘𝑅)𝑦) ∈ 𝑖))
32	31	ralimdva 3178	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖 ∈ 𝐶 (𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖) → ∀𝑖 ∈ 𝐶 (𝑥(1st ‘𝑅)𝑦) ∈ 𝑖))
33		ovex 7476	. . . . . . . . . . . . 13 ⊢ (𝑥(1st ‘𝑅)𝑦) ∈ V
34	33	elint2 4917	. . . . . . . . . . . 12 ⊢ ((𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶 ↔ ∀𝑖 ∈ 𝐶 (𝑥(1st ‘𝑅)𝑦) ∈ 𝑖)
35	32, 34	imbitrrdi 255	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖 ∈ 𝐶 (𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖) → (𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶))
36	27, 35	biimtrrid 246	. . . . . . . . . 10 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ((∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖 ∧ ∀𝑖 ∈ 𝐶 𝑦 ∈ 𝑖) → (𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶))
37	36	expdimp 458	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → (∀𝑖 ∈ 𝐶 𝑦 ∈ 𝑖 → (𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶))
38	26, 37	biimtrid 245	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → (𝑦 ∈ ∩ 𝐶 → (𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶))
39	38	ralrimiv 3157	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → ∀𝑦 ∈ ∩ 𝐶(𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶)
40		eqid 2763	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
41	4, 40, 5	idllmulcl 39146	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥 ∈ 𝑖 ∧ 𝑧 ∈ ran (1st ‘𝑅))) → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖)
42	41	anass1rs 668	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ 𝑥 ∈ 𝑖) → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖)
43	42	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) → (𝑥 ∈ 𝑖 → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖))
44	43	an32s 665	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥 ∈ 𝑖 → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖))
45	3, 44	sylan2 605	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖 ∈ 𝐶)) → (𝑥 ∈ 𝑖 → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖))
46	45	an4s 673	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑧 ∈ ran (1st ‘𝑅) ∧ 𝑖 ∈ 𝐶)) → (𝑥 ∈ 𝑖 → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖))
47	46	anassrs 473	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ 𝑖 ∈ 𝐶) → (𝑥 ∈ 𝑖 → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖))
48	47	ralimdva 3178	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) → (∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖 → ∀𝑖 ∈ 𝐶 (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖))
49	48	imp 412	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → ∀𝑖 ∈ 𝐶 (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖)
50		ovex 7476	. . . . . . . . . . . 12 ⊢ (𝑧(2nd ‘𝑅)𝑥) ∈ V
51	50	elint2 4917	. . . . . . . . . . 11 ⊢ ((𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶 ↔ ∀𝑖 ∈ 𝐶 (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖)
52	49, 51	sylibr 237	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → (𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶)
53	4, 40, 5	idlrmulcl 39147	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥 ∈ 𝑖 ∧ 𝑧 ∈ ran (1st ‘𝑅))) → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖)
54	53	anass1rs 668	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ 𝑥 ∈ 𝑖) → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖)
55	54	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) → (𝑥 ∈ 𝑖 → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖))
56	55	an32s 665	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥 ∈ 𝑖 → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖))
57	3, 56	sylan2 605	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖 ∈ 𝐶)) → (𝑥 ∈ 𝑖 → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖))
58	57	an4s 673	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑧 ∈ ran (1st ‘𝑅) ∧ 𝑖 ∈ 𝐶)) → (𝑥 ∈ 𝑖 → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖))
59	58	anassrs 473	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ 𝑖 ∈ 𝐶) → (𝑥 ∈ 𝑖 → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖))
60	59	ralimdva 3178	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) → (∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖 → ∀𝑖 ∈ 𝐶 (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖))
61	60	imp 412	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → ∀𝑖 ∈ 𝐶 (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖)
62		ovex 7476	. . . . . . . . . . . 12 ⊢ (𝑥(2nd ‘𝑅)𝑧) ∈ V
63	62	elint2 4917	. . . . . . . . . . 11 ⊢ ((𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶 ↔ ∀𝑖 ∈ 𝐶 (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖)
64	61, 63	sylibr 237	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → (𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶)
65	52, 64	jca 521	. . . . . . . . 9 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → ((𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶))
66	65	an32s 665	. . . . . . . 8 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) ∧ 𝑧 ∈ ran (1st ‘𝑅)) → ((𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶))
67	66	ralrimiva 3158	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → ∀𝑧 ∈ ran (1st ‘𝑅)((𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶))
68	39, 67	jca 521	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → (∀𝑦 ∈ ∩ 𝐶(𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶 ∧ ∀𝑧 ∈ ran (1st ‘𝑅)((𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶)))
69	68	ex 418	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖 → (∀𝑦 ∈ ∩ 𝐶(𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶 ∧ ∀𝑧 ∈ ran (1st ‘𝑅)((𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶))))
70	24, 69	biimtrid 245	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (𝑥 ∈ ∩ 𝐶 → (∀𝑦 ∈ ∩ 𝐶(𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶 ∧ ∀𝑧 ∈ ran (1st ‘𝑅)((𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶))))
71	70	ralrimiv 3157	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑥 ∈ ∩ 𝐶(∀𝑦 ∈ ∩ 𝐶(𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶 ∧ ∀𝑧 ∈ ran (1st ‘𝑅)((𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶)))
72	71	3adant2 1149	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑥 ∈ ∩ 𝐶(∀𝑦 ∈ ∩ 𝐶(𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶 ∧ ∀𝑧 ∈ ran (1st ‘𝑅)((𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶)))
73	4, 40, 5, 14	isidl 39140	. . 3 ⊢ (𝑅 ∈ RingOps → (∩ 𝐶 ∈ (Idl‘𝑅) ↔ (∩ 𝐶 ⊆ ran (1st ‘𝑅) ∧ (GId‘(1st ‘𝑅)) ∈ ∩ 𝐶 ∧ ∀𝑥 ∈ ∩ 𝐶(∀𝑦 ∈ ∩ 𝐶(𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶 ∧ ∀𝑧 ∈ ran (1st ‘𝑅)((𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶)))))
74	73	3ad2ant1 1151	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∩ 𝐶 ∈ (Idl‘𝑅) ↔ (∩ 𝐶 ⊆ ran (1st ‘𝑅) ∧ (GId‘(1st ‘𝑅)) ∈ ∩ 𝐶 ∧ ∀𝑥 ∈ ∩ 𝐶(∀𝑦 ∈ ∩ 𝐶(𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶 ∧ ∀𝑧 ∈ ran (1st ‘𝑅)((𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶)))))
75	13, 22, 72, 74	mpbir3and 1361	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∩ 𝐶 ∈ (Idl‘𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080   ⊆ wss 3900  ∅c0 4281  ∪ cuni 4870  ∩ cint 4910  ran crn 5656  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  2^nd c2nd 8024  GIdcgi 31122  RingOpscrngo 39020  Idlcidl 39133

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-idl 39136

This theorem is used by:  inidl  39156  igenidl  39189