Theorem zarclsint 31724

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 19710	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	ad4antr 728	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑅 ∈ Ring)
3		elpwi 4539	. . . . . . . . . . . 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 3917	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑖 ∈ 𝑟) → 𝑖 ∈ (LIdeal‘𝑅))
7		eqid 2738	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2738	. . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
9	7, 8	lidlss 20394	. . . . . . . . 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 3107	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∀𝑖 ∈ 𝑟 𝑖 ⊆ (Base‘𝑅))
12		unissb 4870	. . . . . . 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 2738	. . . . . . 7 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
15	14, 7, 8	rspcl 20406	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ∪ 𝑟 ⊆ (Base‘𝑅)) → ((RSpan‘𝑅)‘∪ 𝑟) ∈ (LIdeal‘𝑅))
16	2, 13, 15	syl2anc 583	. . . . 5 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ((RSpan‘𝑅)‘∪ 𝑟) ∈ (LIdeal‘𝑅))
17		sseq1 3942	. . . . . . . 8 ⊢ (𝑖 = ((RSpan‘𝑅)‘∪ 𝑟) → (𝑖 ⊆ 𝑗 ↔ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗))
18	17	rabbidv 3404	. . . . . . 7 ⊢ (𝑖 = ((RSpan‘𝑅)‘∪ 𝑟) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘∪ 𝑟) ⊆ 𝑗})
19	18	eqeq2d 2749	. . . . . 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 4881	. . . . . . 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 6457	. . . . . . . . 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 6769	. . . . . . . . . . 11 ⊢ (PrmIdeal‘𝑅) ∈ V
27	26	rabex 5251	. . . . . . . . . 10 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ∈ V
28	27, 23	dmmpti 6561	. . . . . . . . 9 ⊢ dom 𝑉 = (LIdeal‘𝑅)
29	5, 28	sseqtrrdi 3968	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑟 ⊆ dom 𝑉)
30		intimafv 30945	. . . . . . . 8 ⊢ ((Fun 𝑉 ∧ 𝑟 ⊆ dom 𝑉) → ∩ (𝑉 “ 𝑟) = ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙))
31	25, 29, 30	syl2anc 583	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∩ (𝑉 “ 𝑟) = ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙))
32	22, 31	eqtrd 2778	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ∩ 𝑆 = ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙))
33		simplr 765	. . . . . . . . . 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 5958	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → (𝑉 “ 𝑟) = (𝑉 “ ∅))
36		ima0 5974	. . . . . . . . . . 11 ⊢ (𝑉 “ ∅) = ∅
37	35, 36	eqtrdi 2795	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → (𝑉 “ 𝑟) = ∅)
38	33, 37	eqtrd 2778	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → 𝑆 = ∅)
39		simp-4r 780	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → 𝑆 ≠ ∅)
40	39	neneqd 2947	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) ∧ 𝑟 = ∅) → ¬ 𝑆 = ∅)
41	38, 40	pm2.65da 813	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → ¬ 𝑟 = ∅)
42	41	neqned 2949	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉 “ 𝑟)) → 𝑟 ≠ ∅)
43	23, 14	zarclsiin 31723	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑟 ≠ ∅) → ∩ 𝑙 ∈ 𝑟 (𝑉‘𝑙) = (𝑉‘((RSpan‘𝑅)‘∪ 𝑟)))
44	2, 5, 42, 43	syl3anc 1369	. . . . . 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 5251	. . . . . . . 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 6864	. . . . . 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 2782	. . . . 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 3555	. . . 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 5256	. . . . . . . 8 ⊢ (𝑆 ≠ ∅ ↔ ∩ 𝑆 ∈ V)
53	52	biimpi 215	. . . . . . 7 ⊢ (𝑆 ≠ ∅ → ∩ 𝑆 ∈ V)
54	53	3ad2ant3 1133	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ V)
55	23	elrnmpt 5854	. . . . . 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 1362	. . . 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 6771	. . . . . 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 765	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ ran 𝑉)
62	27, 23	fnmpti 6560	. . . . . . . 8 ⊢ 𝑉 Fn (LIdeal‘𝑅)
63		fnima 6547	. . . . . . . 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 3968	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ (𝑉 “ (LIdeal‘𝑅)))
66		ssimaexg 6836	. . . . . 6 ⊢ (((LIdeal‘𝑅) ∈ V ∧ Fun 𝑉 ∧ 𝑆 ⊆ (𝑉 “ (LIdeal‘𝑅))) → ∃𝑟(𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)))
67	59, 60, 65, 66	syl3anc 1369	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∃𝑟(𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)))
68		vex 3426	. . . . . . . . . 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 4538	. . . . . . . 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 610	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ((𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟)) → (𝑟 ∈ 𝒫 (LIdeal‘𝑅) ∧ 𝑆 = (𝑉 “ 𝑟))))
74	73	eximdv 1921	. . . . 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 3069	. . . 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 3217	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ ran 𝑉)
79	78	3impa 1108	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ ran 𝑉)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836  ∩ cint 4876  ∩ ciin 4922   ↦ cmpt 5153  dom cdm 5580  ran crn 5581   “ cima 5583  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418  Basecbs 16840  Ringcrg 19698  CRingccrg 19699  LIdealclidl 20347  RSpancrsp 20348  PrmIdealcprmidl 31512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-ip 16906  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-mgp 19636  df-ur 19653  df-ring 19700  df-cring 19701  df-subrg 19937  df-lmod 20040  df-lss 20109  df-lsp 20149  df-sra 20349  df-rgmod 20350  df-lidl 20351  df-rsp 20352  df-prmidl 31513

This theorem is referenced by:  zartopn  31727