Theorem zarclsint 34587

Description: The intersection of a family of closed sets is closed in the Zariski topology. (Contributed by Thierry Arnoux, 16-Jun-2024.)

Hypothesis

Ref	Expression
zarclsx.1	⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})

Assertion

Ref	Expression
zarclsint	⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ ran 𝑉)

Distinct variable groups:   𝑅,𝑖,𝑗   𝑆,𝑖   𝑖,𝑉

Allowed substitution hints:   𝑆(𝑗)   𝑉(𝑗)

Proof of Theorem zarclsint

Dummy variables 𝑙 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngring 20505	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	ad4antr 745	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑅 ∈ Ring)
3		elpwi 4567	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ 𝒫 (LIdeal‘𝑅) → 𝑟 ⊆ (LIdeal‘𝑅))
4	3	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) → 𝑟 ⊆ (LIdeal‘𝑅))
5	4	adantr 486	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑟 ⊆ (LIdeal‘𝑅))
6	5	sselda 3913	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑖 ∈ 𝑟) → 𝑖 ∈ (LIdeal‘𝑅))
7		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2763	. . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
9	7, 8	lidlss 21523	. . . . . . . . 9 ⊢ (𝑖 ∈ (LIdeal‘𝑅) → 𝑖 ⊆ (Base‘𝑅))
10	6, 9	syl 18	. . . . . . . 8 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑖 ∈ 𝑟) → 𝑖 ⊆ (Base‘𝑅))
11	10	ralrimiva 3158	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∀𝑖 ∈ 𝑟 𝑖 ⊆ (Base‘𝑅))
12		unissb 4904	. . . . . . 7 ⊢ (∪ 𝑟 ⊆ (Base‘𝑅) ↔ ∀𝑖 ∈ 𝑟 𝑖 ⊆ (Base‘𝑅))
13	11, 12	sylibr 237	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∪ 𝑟 ⊆ (Base‘𝑅))
14		eqid 2763	. . . . . . 7 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
15	14, 7, 8	rspcl 21551	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ∪ 𝑟 ⊆ (Base‘𝑅)) → ((RSpan‘𝑅)‘∪ 𝑟) ∈ (LIdeal‘𝑅))
16	2, 13, 15	syl2anc 596	. . . . 5 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ((RSpan‘𝑅)‘∪ 𝑟) ∈ (LIdeal‘𝑅))
17		sseq1 3938	. . . . . . . 8 ⊢ (𝑖 = ((RSpan‘𝑅)‘∪ 𝑟) → (𝑖 ⊆ 𝑗 ↔ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗))
18	17	rabbidv 3423	. . . . . . 7 ⊢ (𝑖 = ((RSpan‘𝑅)‘∪ 𝑟) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗})
19	18	eqeq2d 2774	. . . . . 6 ⊢ (𝑖 = ((RSpan‘𝑅)‘∪ 𝑟) → (∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ ∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗}))
20	19	adantl 487	. . . . 5 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑖 = ((RSpan‘𝑅)‘∪ 𝑟)) → (∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ ∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗}))
21		simpr 490	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑆 = (𝑉 “ 𝑟))
22	21	inteqd 4915	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∩ 𝑆 = ∩ (𝑉 “ 𝑟))
23		zarclsx.1	. . . . . . . . . 10 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
24	23	funmpt2 6602	. . . . . . . . 9 ⊢ Fun 𝑉
25	24	a1i 11	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → Fun 𝑉)
26		fvex 6924	. . . . . . . . . . 11 ⊢ (PrmIdeal‘𝑅) ∈ V
27	26	rabex 5305	. . . . . . . . . 10 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ∈ V
28	27, 23	dmmpti 6708	. . . . . . . . 9 ⊢ dom 𝑉 = (LIdeal‘𝑅)
29	5, 28	sseqtrrdi 3954	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑟 ⊆ dom 𝑉)
30		intimafv 33378	. . . . . . . 8 ⊢ ((Fun 𝑉 ∧ 𝑟 ⊆ dom 𝑉) → ∩ (𝑉 “ 𝑟) = ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙))
31	25, 29, 30	syl2anc 596	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∩ (𝑉 “ 𝑟) = ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙))
32	22, 31	eqtrd 2798	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∩ 𝑆 = ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙))
33		simplr 781	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → 𝑆 = (𝑉 “ 𝑟))
34		simpr 490	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → 𝑟 = ∅)
35	34	imaeq2d 6065	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → (𝑉 “ 𝑟) = (𝑉 “ ∅))
36		ima0 6088	. . . . . . . . . . 11 ⊢ (𝑉 “ ∅) = ∅
37	35, 36	eqtrdi 2814	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → (𝑉 “ 𝑟) = ∅)
38	33, 37	eqtrd 2798	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → 𝑆 = ∅)
39		simp-4r 796	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → 𝑆 ≠ ∅)
40	39	neneqd 2964	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → ¬ 𝑆 = ∅)
41	38, 40	pm2.65da 829	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ¬ 𝑟 = ∅)
42	41	neqned 2966	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑟 ≠ ∅)
43	23, 14	zarclsiin 34586	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑟 ≠ ∅) → ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙) = (𝑉‘((RSpan‘𝑅)‘∪ 𝑟)))
44	2, 5, 42, 43	syl3anc 1399	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙) = (𝑉‘((RSpan‘𝑅)‘∪ 𝑟)))
45	23	a1i 11	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
46	18	adantl 487	. . . . . . 7 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑖 = ((RSpan‘𝑅)‘∪ 𝑟)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗})
47	26	rabex 5305	. . . . . . . 8 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗} ∈ V
48	47	a1i 11	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗} ∈ V)
49	45, 46, 16, 48	fvmptd 7027	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → (𝑉‘((RSpan‘𝑅)‘∪ 𝑟)) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗})
50	32, 44, 49	3eqtrd 2802	. . . . 5 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗})
51	16, 20, 50	rspcedvd 3582	. . . 4 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∃𝑖 ∈ (LIdeal‘𝑅)∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
52		intex 5310	. . . . . . . 8 ⊢ (𝑆 ≠ ∅ ↔ ∩ 𝑆 ∈ V)
53	52	biimpi 219	. . . . . . 7 ⊢ (𝑆 ≠ ∅ → ∩ 𝑆 ∈ V)
54	53	3ad2ant3 1153	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ V)
55	23	elrnmpt 5948	. . . . . 6 ⊢ (∩ 𝑆 ∈ V → (∩ 𝑆 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
56	54, 55	syl 18	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → (∩ 𝑆 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
57	56	ad5ant123 1388	. . . 4 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → (∩ 𝑆 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
58	51, 57	mpbird 260	. . 3 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∩ 𝑆 ∈ ran 𝑉)
59		fvexd 6926	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → (LIdeal‘𝑅) ∈ V)
60	24	a1i 11	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → Fun 𝑉)
61		simplr 781	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ ran 𝑉)
62	27, 23	fnmpti 6707	. . . . . . . 8 ⊢ 𝑉 Fn (LIdeal‘𝑅)
63		fnima 6694	. . . . . . . 8 ⊢ (𝑉 Fn (LIdeal‘𝑅) → (𝑉 “ (LIdeal‘𝑅)) = ran 𝑉)
64	62, 63	ax-mp 5	. . . . . . 7 ⊢ (𝑉 “ (LIdeal‘𝑅)) = ran 𝑉
65	61, 64	sseqtrrdi 3954	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ (𝑉 “ (LIdeal‘𝑅)))
66		ssimaexg 6997	. . . . . 6 ⊢ (((LIdeal‘𝑅) ∈ V ∧ Fun 𝑉 ∧ 𝑆 ⊆ (𝑉 “ (LIdeal‘𝑅))) → ∃𝑟(𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)))
67	59, 60, 65, 66	syl3anc 1399	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∃𝑟(𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)))
68		vex 3458	. . . . . . . . . 10 ⊢ 𝑟 ∈ V
69	68	a1i 11	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ⊆ (LIdeal‘𝑅)) → 𝑟 ∈ V)
70		simpr 490	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ⊆ (LIdeal‘𝑅)) → 𝑟 ⊆ (LIdeal‘𝑅))
71	69, 70	elpwd 4566	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ⊆ (LIdeal‘𝑅)) → 𝑟 ∈ 𝒫 (LIdeal‘𝑅))
72	71	ex 418	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → (𝑟 ⊆ (LIdeal‘𝑅) → 𝑟 ∈ 𝒫 (LIdeal‘𝑅)))
73	72	anim1d 623	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ((𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)) → (𝑟 ∈ 𝒫 (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟))))
74	73	eximdv 1952	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → (∃𝑟(𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∃𝑟(𝑟 ∈ 𝒫 (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟))))
75	67, 74	mpd 16	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∃𝑟(𝑟 ∈ 𝒫 (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)))
76		df-rex 3091	. . . 4 ⊢ (∃𝑟 ∈ 𝒫 (LIdeal‘𝑅)𝑆 = (𝑉 “ 𝑟) ↔ ∃𝑟(𝑟 ∈ 𝒫 (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)))
77	75, 76	sylibr 237	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∃𝑟 ∈ 𝒫 (LIdeal‘𝑅)𝑆 = (𝑉 “ 𝑟))
78	58, 77	r19.29a 3174	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ ran 𝑉)
79	78	3impa 1127	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ ran 𝑉)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  ∪ cuni 4870  ∩ cint 4910  ∩ ciin 4955   ↦ cmpt 5190  dom cdm 5656  ran crn 5657   “ cima 5659  Fun wfun 6555   Fn wfn 6556  ‘cfv 6561  Basecbs 17419  Ringcrg 20492  CRingccrg 20493  LIdealclidl 21517  RSpancrsp 21518  PrmIdealcprmidl 21649

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-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  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-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  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-iun 4956  df-iin 4957  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-er 8748  df-en 9005  df-dom 9006  df-sdom 9007  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-2 12435  df-3 12436  df-4 12437  df-5 12438  df-6 12439  df-7 12440  df-8 12441  df-sets 17374  df-slot 17392  df-ndx 17404  df-base 17420  df-ress 17441  df-plusg 17473  df-mulr 17474  df-sca 17476  df-vsca 17477  df-ip 17478  df-0g 17644  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-grp 19179  df-minusg 19180  df-sbg 19181  df-subg 19365  df-mgp 20394  df-ur 20441  df-ring 20494  df-cring 20495  df-subrg 20855  df-lmod 21170  df-lss 21240  df-lsp 21280  df-sra 21481  df-rgmod 21482  df-lidl 21519  df-rsp 21520  df-prmidl 21650

This theorem is used by:  zartopn  34590