Theorem zarclsint 32542

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 19990	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	ad4antr 730	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑅 ∈ Ring)
3		elpwi 4572	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ 𝒫 (LIdeal‘𝑅) → 𝑟 ⊆ (LIdeal‘𝑅))
4	3	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) → 𝑟 ⊆ (LIdeal‘𝑅))
5	4	adantr 481	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑟 ⊆ (LIdeal‘𝑅))
6	5	sselda 3947	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑖 ∈ 𝑟) → 𝑖 ∈ (LIdeal‘𝑅))
7		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2731	. . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
9	7, 8	lidlss 20739	. . . . . . . . 9 ⊢ (𝑖 ∈ (LIdeal‘𝑅) → 𝑖 ⊆ (Base‘𝑅))
10	6, 9	syl 17	. . . . . . . 8 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑖 ∈ 𝑟) → 𝑖 ⊆ (Base‘𝑅))
11	10	ralrimiva 3139	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∀𝑖 ∈ 𝑟 𝑖 ⊆ (Base‘𝑅))
12		unissb 4905	. . . . . . 7 ⊢ (∪ 𝑟 ⊆ (Base‘𝑅) ↔ ∀𝑖 ∈ 𝑟 𝑖 ⊆ (Base‘𝑅))
13	11, 12	sylibr 233	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∪ 𝑟 ⊆ (Base‘𝑅))
14		eqid 2731	. . . . . . 7 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
15	14, 7, 8	rspcl 20751	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ∪ 𝑟 ⊆ (Base‘𝑅)) → ((RSpan‘𝑅)‘∪ 𝑟) ∈ (LIdeal‘𝑅))
16	2, 13, 15	syl2anc 584	. . . . 5 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ((RSpan‘𝑅)‘∪ 𝑟) ∈ (LIdeal‘𝑅))
17		sseq1 3972	. . . . . . . 8 ⊢ (𝑖 = ((RSpan‘𝑅)‘∪ 𝑟) → (𝑖 ⊆ 𝑗 ↔ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗))
18	17	rabbidv 3413	. . . . . . 7 ⊢ (𝑖 = ((RSpan‘𝑅)‘∪ 𝑟) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗})
19	18	eqeq2d 2742	. . . . . 6 ⊢ (𝑖 = ((RSpan‘𝑅)‘∪ 𝑟) → (∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ ∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗}))
20	19	adantl 482	. . . . 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 485	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑆 = (𝑉 “ 𝑟))
22	21	inteqd 4917	. . . . . . 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 6545	. . . . . . . . 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 6860	. . . . . . . . . . 11 ⊢ (PrmIdeal‘𝑅) ∈ V
27	26	rabex 5294	. . . . . . . . . 10 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ∈ V
28	27, 23	dmmpti 6650	. . . . . . . . 9 ⊢ dom 𝑉 = (LIdeal‘𝑅)
29	5, 28	sseqtrrdi 3998	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑟 ⊆ dom 𝑉)
30		intimafv 31692	. . . . . . . 8 ⊢ ((Fun 𝑉 ∧ 𝑟 ⊆ dom 𝑉) → ∩ (𝑉 “ 𝑟) = ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙))
31	25, 29, 30	syl2anc 584	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∩ (𝑉 “ 𝑟) = ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙))
32	22, 31	eqtrd 2771	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∩ 𝑆 = ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙))
33		simplr 767	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → 𝑆 = (𝑉 “ 𝑟))
34		simpr 485	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → 𝑟 = ∅)
35	34	imaeq2d 6018	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → (𝑉 “ 𝑟) = (𝑉 “ ∅))
36		ima0 6034	. . . . . . . . . . 11 ⊢ (𝑉 “ ∅) = ∅
37	35, 36	eqtrdi 2787	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → (𝑉 “ 𝑟) = ∅)
38	33, 37	eqtrd 2771	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → 𝑆 = ∅)
39		simp-4r 782	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → 𝑆 ≠ ∅)
40	39	neneqd 2944	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → ¬ 𝑆 = ∅)
41	38, 40	pm2.65da 815	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ¬ 𝑟 = ∅)
42	41	neqned 2946	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑟 ≠ ∅)
43	23, 14	zarclsiin 32541	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑟 ≠ ∅) → ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙) = (𝑉‘((RSpan‘𝑅)‘∪ 𝑟)))
44	2, 5, 42, 43	syl3anc 1371	. . . . . 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 482	. . . . . . 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 5294	. . . . . . . 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 6960	. . . . . 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 2775	. . . . 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 3584	. . . 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 5299	. . . . . . . 8 ⊢ (𝑆 ≠ ∅ ↔ ∩ 𝑆 ∈ V)
53	52	biimpi 215	. . . . . . 7 ⊢ (𝑆 ≠ ∅ → ∩ 𝑆 ∈ V)
54	53	3ad2ant3 1135	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ V)
55	23	elrnmpt 5916	. . . . . 6 ⊢ (∩ 𝑆 ∈ V → (∩ 𝑆 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
56	54, 55	syl 17	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → (∩ 𝑆 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
57	56	ad5ant123 1364	. . . 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 256	. . 3 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∩ 𝑆 ∈ ran 𝑉)
59		fvexd 6862	. . . . . 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 767	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ ran 𝑉)
62	27, 23	fnmpti 6649	. . . . . . . 8 ⊢ 𝑉 Fn (LIdeal‘𝑅)
63		fnima 6636	. . . . . . . 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 3998	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ (𝑉 “ (LIdeal‘𝑅)))
66		ssimaexg 6932	. . . . . 6 ⊢ (((LIdeal‘𝑅) ∈ V ∧ Fun 𝑉 ∧ 𝑆 ⊆ (𝑉 “ (LIdeal‘𝑅))) → ∃𝑟(𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)))
67	59, 60, 65, 66	syl3anc 1371	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∃𝑟(𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)))
68		vex 3450	. . . . . . . . . 10 ⊢ 𝑟 ∈ V
69	68	a1i 11	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ⊆ (LIdeal‘𝑅)) → 𝑟 ∈ V)
70		simpr 485	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ⊆ (LIdeal‘𝑅)) → 𝑟 ⊆ (LIdeal‘𝑅))
71	69, 70	elpwd 4571	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ⊆ (LIdeal‘𝑅)) → 𝑟 ∈ 𝒫 (LIdeal‘𝑅))
72	71	ex 413	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → (𝑟 ⊆ (LIdeal‘𝑅) → 𝑟 ∈ 𝒫 (LIdeal‘𝑅)))
73	72	anim1d 611	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ((𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)) → (𝑟 ∈ 𝒫 (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟))))
74	73	eximdv 1920	. . . . 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 15	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∃𝑟(𝑟 ∈ 𝒫 (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)))
76		df-rex 3070	. . . 4 ⊢ (∃𝑟 ∈ 𝒫 (LIdeal‘𝑅)𝑆 = (𝑉 “ 𝑟) ↔ ∃𝑟(𝑟 ∈ 𝒫 (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)))
77	75, 76	sylibr 233	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∃𝑟 ∈ 𝒫 (LIdeal‘𝑅)𝑆 = (𝑉 “ 𝑟))
78	58, 77	r19.29a 3155	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ ran 𝑉)
79	78	3impa 1110	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ ran 𝑉)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3405  Vcvv 3446   ⊆ wss 3913  ∅c0 4287  𝒫 cpw 4565  ∪ cuni 4870  ∩ cint 4912  ∩ ciin 4960   ↦ cmpt 5193  dom cdm 5638  ran crn 5639   “ cima 5641  Fun wfun 6495   Fn wfn 6496  ‘cfv 6501  Basecbs 17094  Ringcrg 19978  CRingccrg 19979  LIdealclidl 20690  RSpancrsp 20691  PrmIdealcprmidl 32283

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-sets 17047  df-slot 17065  df-ndx 17077  df-base 17095  df-ress 17124  df-plusg 17160  df-mulr 17161  df-sca 17163  df-vsca 17164  df-ip 17165  df-0g 17337  df-mgm 18511  df-sgrp 18560  df-mnd 18571  df-grp 18765  df-minusg 18766  df-sbg 18767  df-subg 18939  df-mgp 19911  df-ur 19928  df-ring 19980  df-cring 19981  df-subrg 20268  df-lmod 20380  df-lss 20450  df-lsp 20490  df-sra 20692  df-rgmod 20693  df-lidl 20694  df-rsp 20695  df-prmidl 32284

This theorem is referenced by:  zartopn  32545