Theorem zarclsint 33906

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 20165	. . . . . . 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 4556	. . . . . . . . . . . 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 3930	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑖 ∈ 𝑟) → 𝑖 ∈ (LIdeal‘𝑅))
7		eqid 2733	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2733	. . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
9	7, 8	lidlss 21151	. . . . . . . . 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 3125	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∀𝑖 ∈ 𝑟 𝑖 ⊆ (Base‘𝑅))
12		unissb 4891	. . . . . . 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 2733	. . . . . . 7 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
15	14, 7, 8	rspcl 21174	. . . . . 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 3956	. . . . . . . 8 ⊢ (𝑖 = ((RSpan‘𝑅)‘∪ 𝑟) → (𝑖 ⊆ 𝑗 ↔ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗))
18	17	rabbidv 3403	. . . . . . 7 ⊢ (𝑖 = ((RSpan‘𝑅)‘∪ 𝑟) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗})
19	18	eqeq2d 2744	. . . . . 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 4902	. . . . . . 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 6525	. . . . . . . . 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 6841	. . . . . . . . . . 11 ⊢ (PrmIdeal‘𝑅) ∈ V
27	26	rabex 5279	. . . . . . . . . 10 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ∈ V
28	27, 23	dmmpti 6630	. . . . . . . . 9 ⊢ dom 𝑉 = (LIdeal‘𝑅)
29	5, 28	sseqtrrdi 3972	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑟 ⊆ dom 𝑉)
30		intimafv 32696	. . . . . . . 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 2768	. . . . . 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 6013	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → (𝑉 “ 𝑟) = (𝑉 “ ∅))
36		ima0 6030	. . . . . . . . . . 11 ⊢ (𝑉 “ ∅) = ∅
37	35, 36	eqtrdi 2784	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → (𝑉 “ 𝑟) = ∅)
38	33, 37	eqtrd 2768	. . . . . . . . 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 2934	. . . . . . . . 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 2936	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑟 ≠ ∅)
43	23, 14	zarclsiin 33905	. . . . . . 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 5279	. . . . . . . 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 6942	. . . . . 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 2772	. . . . 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 3575	. . . 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 5284	. . . . . . . 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 5902	. . . . . 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 6843	. . . . . 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 6629	. . . . . . . 8 ⊢ 𝑉 Fn (LIdeal‘𝑅)
63		fnima 6616	. . . . . . . 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 3972	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ (𝑉 “ (LIdeal‘𝑅)))
66		ssimaexg 6914	. . . . . 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 3441	. . . . . . . . . 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 4555	. . . . . . . 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 3058	. . . 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 3141	. 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 2929  ∀wral 3048  ∃wrex 3057  {crab 3396  Vcvv 3437   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4549  ∪ cuni 4858  ∩ cint 4897  ∩ ciin 4942   ↦ cmpt 5174  dom cdm 5619  ran crn 5620   “ cima 5622  Fun wfun 6480   Fn wfn 6481  ‘cfv 6486  Basecbs 17122  Ringcrg 20153  CRingccrg 20154  LIdealclidl 21145  RSpancrsp 21146  PrmIdealcprmidl 33407

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 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-mulr 17177  df-sca 17179  df-vsca 17180  df-ip 17181  df-0g 17347  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-grp 18851  df-minusg 18852  df-sbg 18853  df-subg 19038  df-mgp 20061  df-ur 20102  df-ring 20155  df-cring 20156  df-subrg 20487  df-lmod 20797  df-lss 20867  df-lsp 20907  df-sra 21109  df-rgmod 21110  df-lidl 21147  df-rsp 21148  df-prmidl 33408

This theorem is referenced by:  zartopn  33909