Theorem zarclsint 34029

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 20180	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	ad4antr 732	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑅 ∈ Ring)
3		elpwi 4561	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ 𝒫 (LIdeal‘𝑅) → 𝑟 ⊆ (LIdeal‘𝑅))
4	3	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) → 𝑟 ⊆ (LIdeal‘𝑅))
5	4	adantr 480	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑟 ⊆ (LIdeal‘𝑅))
6	5	sselda 3933	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑖 ∈ 𝑟) → 𝑖 ∈ (LIdeal‘𝑅))
7		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2736	. . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
9	7, 8	lidlss 21167	. . . . . . . . 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 3128	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∀𝑖 ∈ 𝑟 𝑖 ⊆ (Base‘𝑅))
12		unissb 4896	. . . . . . 7 ⊢ (∪ 𝑟 ⊆ (Base‘𝑅) ↔ ∀𝑖 ∈ 𝑟 𝑖 ⊆ (Base‘𝑅))
13	11, 12	sylibr 234	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∪ 𝑟 ⊆ (Base‘𝑅))
14		eqid 2736	. . . . . . 7 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
15	14, 7, 8	rspcl 21190	. . . . . 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 3959	. . . . . . . 8 ⊢ (𝑖 = ((RSpan‘𝑅)‘∪ 𝑟) → (𝑖 ⊆ 𝑗 ↔ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗))
18	17	rabbidv 3406	. . . . . . 7 ⊢ (𝑖 = ((RSpan‘𝑅)‘∪ 𝑟) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗})
19	18	eqeq2d 2747	. . . . . 6 ⊢ (𝑖 = ((RSpan‘𝑅)‘∪ 𝑟) → (∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ ∩ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗}))
20	19	adantl 481	. . . . 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 484	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑆 = (𝑉 “ 𝑟))
22	21	inteqd 4907	. . . . . . 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 6531	. . . . . . . . 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 6847	. . . . . . . . . . 11 ⊢ (PrmIdeal‘𝑅) ∈ V
27	26	rabex 5284	. . . . . . . . . 10 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ∈ V
28	27, 23	dmmpti 6636	. . . . . . . . 9 ⊢ dom 𝑉 = (LIdeal‘𝑅)
29	5, 28	sseqtrrdi 3975	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑟 ⊆ dom 𝑉)
30		intimafv 32790	. . . . . . . 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 768	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → 𝑆 = (𝑉 “ 𝑟))
34		simpr 484	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → 𝑟 = ∅)
35	34	imaeq2d 6019	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → (𝑉 “ 𝑟) = (𝑉 “ ∅))
36		ima0 6036	. . . . . . . . . . 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 783	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → 𝑆 ≠ ∅)
40	39	neneqd 2937	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → ¬ 𝑆 = ∅)
41	38, 40	pm2.65da 816	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ¬ 𝑟 = ∅)
42	41	neqned 2939	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑟 ≠ ∅)
43	23, 14	zarclsiin 34028	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑟 ≠ ∅) → ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙) = (𝑉‘((RSpan‘𝑅)‘∪ 𝑟)))
44	2, 5, 42, 43	syl3anc 1373	. . . . . 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 481	. . . . . . 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 5284	. . . . . . . 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 6948	. . . . . 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 3578	. . . 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 5289	. . . . . . . 8 ⊢ (𝑆 ≠ ∅ ↔ ∩ 𝑆 ∈ V)
53	52	biimpi 216	. . . . . . 7 ⊢ (𝑆 ≠ ∅ → ∩ 𝑆 ∈ V)
54	53	3ad2ant3 1135	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ V)
55	23	elrnmpt 5907	. . . . . 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 1366	. . . 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 257	. . 3 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∩ 𝑆 ∈ ran 𝑉)
59		fvexd 6849	. . . . . 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 768	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ ran 𝑉)
62	27, 23	fnmpti 6635	. . . . . . . 8 ⊢ 𝑉 Fn (LIdeal‘𝑅)
63		fnima 6622	. . . . . . . 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 3975	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ (𝑉 “ (LIdeal‘𝑅)))
66		ssimaexg 6920	. . . . . 6 ⊢ (((LIdeal‘𝑅) ∈ V ∧ Fun 𝑉 ∧ 𝑆 ⊆ (𝑉 “ (LIdeal‘𝑅))) → ∃𝑟(𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)))
67	59, 60, 65, 66	syl3anc 1373	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∃𝑟(𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)))
68		vex 3444	. . . . . . . . . 10 ⊢ 𝑟 ∈ V
69	68	a1i 11	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ⊆ (LIdeal‘𝑅)) → 𝑟 ∈ V)
70		simpr 484	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ⊆ (LIdeal‘𝑅)) → 𝑟 ⊆ (LIdeal‘𝑅))
71	69, 70	elpwd 4560	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ⊆ (LIdeal‘𝑅)) → 𝑟 ∈ 𝒫 (LIdeal‘𝑅))
72	71	ex 412	. . . . . . 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 1918	. . . . 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 3061	. . . 4 ⊢ (∃𝑟 ∈ 𝒫 (LIdeal‘𝑅)𝑆 = (𝑉 “ 𝑟) ↔ ∃𝑟(𝑟 ∈ 𝒫 (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)))
77	75, 76	sylibr 234	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∃𝑟 ∈ 𝒫 (LIdeal‘𝑅)𝑆 = (𝑉 “ 𝑟))
78	58, 77	r19.29a 3144	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ ran 𝑉)
79	78	3impa 1109	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ ran 𝑉)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3399  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  ∪ cuni 4863  ∩ cint 4902  ∩ ciin 4947   ↦ cmpt 5179  dom cdm 5624  ran crn 5625   “ cima 5627  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492  Basecbs 17136  Ringcrg 20168  CRingccrg 20169  LIdealclidl 21161  RSpancrsp 21162  PrmIdealcprmidl 33516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-0g 17361  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19053  df-mgp 20076  df-ur 20117  df-ring 20170  df-cring 20171  df-subrg 20503  df-lmod 20813  df-lss 20883  df-lsp 20923  df-sra 21125  df-rgmod 21126  df-lidl 21163  df-rsp 21164  df-prmidl 33517

This theorem is referenced by:  zartopn  34032