Theorem intidl 36945

Description: The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

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 4975	. . . 4 ⊢ (𝐶 ≠ ∅ → ∩ 𝐶 ⊆ ∪ 𝐶)
2	1	3ad2ant2 1135	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∩ 𝐶 ⊆ ∪ 𝐶)
3		ssel2 3978	. . . . . . . 8 ⊢ ((𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖 ∈ 𝐶) → 𝑖 ∈ (Idl‘𝑅))
4		eqid 2733	. . . . . . . . 9 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
5		eqid 2733	. . . . . . . . 9 ⊢ ran (1st ‘𝑅) = ran (1st ‘𝑅)
6	4, 5	idlss 36932	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖 ⊆ ran (1st ‘𝑅))
7	3, 6	sylan2 594	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖 ∈ 𝐶)) → 𝑖 ⊆ ran (1st ‘𝑅))
8	7	anassrs 469	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖 ∈ 𝐶) → 𝑖 ⊆ ran (1st ‘𝑅))
9	8	ralrimiva 3147	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖 ∈ 𝐶 𝑖 ⊆ ran (1st ‘𝑅))
10	9	3adant2 1132	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖 ∈ 𝐶 𝑖 ⊆ ran (1st ‘𝑅))
11		unissb 4944	. . . 4 ⊢ (∪ 𝐶 ⊆ ran (1st ‘𝑅) ↔ ∀𝑖 ∈ 𝐶 𝑖 ⊆ ran (1st ‘𝑅))
12	10, 11	sylibr 233	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∪ 𝐶 ⊆ ran (1st ‘𝑅))
13	2, 12	sstrd 3993	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∩ 𝐶 ⊆ ran (1st ‘𝑅))
14		eqid 2733	. . . . . . . 8 ⊢ (GId‘(1st ‘𝑅)) = (GId‘(1st ‘𝑅))
15	4, 14	idl0cl 36934	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → (GId‘(1st ‘𝑅)) ∈ 𝑖)
16	3, 15	sylan2 594	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖 ∈ 𝐶)) → (GId‘(1st ‘𝑅)) ∈ 𝑖)
17	16	anassrs 469	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖 ∈ 𝐶) → (GId‘(1st ‘𝑅)) ∈ 𝑖)
18	17	ralrimiva 3147	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖 ∈ 𝐶 (GId‘(1st ‘𝑅)) ∈ 𝑖)
19		fvex 6905	. . . . 5 ⊢ (GId‘(1st ‘𝑅)) ∈ V
20	19	elint2 4958	. . . 4 ⊢ ((GId‘(1st ‘𝑅)) ∈ ∩ 𝐶 ↔ ∀𝑖 ∈ 𝐶 (GId‘(1st ‘𝑅)) ∈ 𝑖)
21	18, 20	sylibr 233	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (GId‘(1st ‘𝑅)) ∈ ∩ 𝐶)
22	21	3adant2 1132	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → (GId‘(1st ‘𝑅)) ∈ ∩ 𝐶)
23		vex 3479	. . . . . 6 ⊢ 𝑥 ∈ V
24	23	elint2 4958	. . . . 5 ⊢ (𝑥 ∈ ∩ 𝐶 ↔ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖)
25		vex 3479	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
26	25	elint2 4958	. . . . . . . . 9 ⊢ (𝑦 ∈ ∩ 𝐶 ↔ ∀𝑖 ∈ 𝐶 𝑦 ∈ 𝑖)
27		r19.26 3112	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝐶 (𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖) ↔ (∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖 ∧ ∀𝑖 ∈ 𝐶 𝑦 ∈ 𝑖))
28	4	idladdcl 36935	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖)) → (𝑥(1st ‘𝑅)𝑦) ∈ 𝑖)
29	28	ex 414	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖) → (𝑥(1st ‘𝑅)𝑦) ∈ 𝑖))
30	3, 29	sylan2 594	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖 ∈ 𝐶)) → ((𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖) → (𝑥(1st ‘𝑅)𝑦) ∈ 𝑖))
31	30	anassrs 469	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖 ∈ 𝐶) → ((𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖) → (𝑥(1st ‘𝑅)𝑦) ∈ 𝑖))
32	31	ralimdva 3168	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖 ∈ 𝐶 (𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖) → ∀𝑖 ∈ 𝐶 (𝑥(1st ‘𝑅)𝑦) ∈ 𝑖))
33		ovex 7442	. . . . . . . . . . . . 13 ⊢ (𝑥(1st ‘𝑅)𝑦) ∈ V
34	33	elint2 4958	. . . . . . . . . . . 12 ⊢ ((𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶 ↔ ∀𝑖 ∈ 𝐶 (𝑥(1st ‘𝑅)𝑦) ∈ 𝑖)
35	32, 34	syl6ibr 252	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖 ∈ 𝐶 (𝑥 ∈ 𝑖 ∧ 𝑦 ∈ 𝑖) → (𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶))
36	27, 35	biimtrrid 242	. . . . . . . . . 10 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ((∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖 ∧ ∀𝑖 ∈ 𝐶 𝑦 ∈ 𝑖) → (𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶))
37	36	expdimp 454	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → (∀𝑖 ∈ 𝐶 𝑦 ∈ 𝑖 → (𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶))
38	26, 37	biimtrid 241	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → (𝑦 ∈ ∩ 𝐶 → (𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶))
39	38	ralrimiv 3146	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → ∀𝑦 ∈ ∩ 𝐶(𝑥(1st ‘𝑅)𝑦) ∈ ∩ 𝐶)
40		eqid 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
41	4, 40, 5	idllmulcl 36936	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥 ∈ 𝑖 ∧ 𝑧 ∈ ran (1st ‘𝑅))) → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖)
42	41	anass1rs 654	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ 𝑥 ∈ 𝑖) → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖)
43	42	ex 414	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) → (𝑥 ∈ 𝑖 → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖))
44	43	an32s 651	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥 ∈ 𝑖 → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖))
45	3, 44	sylan2 594	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖 ∈ 𝐶)) → (𝑥 ∈ 𝑖 → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖))
46	45	an4s 659	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑧 ∈ ran (1st ‘𝑅) ∧ 𝑖 ∈ 𝐶)) → (𝑥 ∈ 𝑖 → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖))
47	46	anassrs 469	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ 𝑖 ∈ 𝐶) → (𝑥 ∈ 𝑖 → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖))
48	47	ralimdva 3168	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) → (∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖 → ∀𝑖 ∈ 𝐶 (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖))
49	48	imp 408	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → ∀𝑖 ∈ 𝐶 (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖)
50		ovex 7442	. . . . . . . . . . . 12 ⊢ (𝑧(2nd ‘𝑅)𝑥) ∈ V
51	50	elint2 4958	. . . . . . . . . . 11 ⊢ ((𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶 ↔ ∀𝑖 ∈ 𝐶 (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑖)
52	49, 51	sylibr 233	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → (𝑧(2nd ‘𝑅)𝑥) ∈ ∩ 𝐶)
53	4, 40, 5	idlrmulcl 36937	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥 ∈ 𝑖 ∧ 𝑧 ∈ ran (1st ‘𝑅))) → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖)
54	53	anass1rs 654	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ 𝑥 ∈ 𝑖) → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖)
55	54	ex 414	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) → (𝑥 ∈ 𝑖 → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖))
56	55	an32s 651	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥 ∈ 𝑖 → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖))
57	3, 56	sylan2 594	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖 ∈ 𝐶)) → (𝑥 ∈ 𝑖 → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖))
58	57	an4s 659	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑧 ∈ ran (1st ‘𝑅) ∧ 𝑖 ∈ 𝐶)) → (𝑥 ∈ 𝑖 → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖))
59	58	anassrs 469	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ 𝑖 ∈ 𝐶) → (𝑥 ∈ 𝑖 → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖))
60	59	ralimdva 3168	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) → (∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖 → ∀𝑖 ∈ 𝐶 (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖))
61	60	imp 408	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → ∀𝑖 ∈ 𝐶 (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖)
62		ovex 7442	. . . . . . . . . . . 12 ⊢ (𝑥(2nd ‘𝑅)𝑧) ∈ V
63	62	elint2 4958	. . . . . . . . . . 11 ⊢ ((𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶 ↔ ∀𝑖 ∈ 𝐶 (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑖)
64	61, 63	sylibr 233	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st ‘𝑅)) ∧ ∀𝑖 ∈ 𝐶 𝑥 ∈ 𝑖) → (𝑥(2nd ‘𝑅)𝑧) ∈ ∩ 𝐶)
65	52, 64	jca 513	. . . . . . . . 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 651	. . . . . . . 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 3147	. . . . . . 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 513	. . . . . 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 414	. . . . 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 241	. . . 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 3146	. . 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 1132	. 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 36930	. . 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 1134	. 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 1343	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∩ 𝐶 ∈ (Idl‘𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062   ⊆ wss 3949  ∅c0 4323  ∪ cuni 4909  ∩ cint 4951  ran crn 5678  ‘cfv 6544  (class class class)co 7409  1^st c1st 7973  2^nd c2nd 7974  GIdcgi 29774  RingOpscrngo 36810  Idlcidl 36923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-idl 36926

This theorem is referenced by:  inidl  36946  igenidl  36979