zarclsiin - Mathbox for Thierry Arnoux

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Theorem zarclsiin 34270

Description: In a Zariski topology, the intersection of the closures of a family of ideals is the closure of the span of their union. (Contributed by Thierry Arnoux, 16-Jun-2024.)

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

**Assertion**
Ref	Expression
zarclsiin	⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙) = (𝑉‘(𝐾‘∪ 𝑇)))

Distinct variable groups: 𝑅,𝑖,𝑗,𝑙 𝑉,𝑙 𝑖,𝐾,𝑗,𝑙 𝑇,𝑖,𝑗,𝑙 Allowed substitution hints: 𝑉(𝑖, 𝑗)

Proof of Theorem zarclsiin Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3962	. . . . 5 ⊢ (𝑗 = 𝑝 → ((𝐾‘∪ 𝑇) ⊆ 𝑗 ↔ (𝐾‘∪ 𝑇) ⊆ 𝑝))
2		simpl3 1211	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) → 𝑇 ≠ ∅)
3		zarclsx.1	. . . . . . . . . . . 12 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
4	3	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑙 ∈ 𝑇) → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
5		sseq1 3961	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑙 → (𝑖 ⊆ 𝑗 ↔ 𝑙 ⊆ 𝑗))
6	5	rabbidv 3422	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑙 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
7	6	adantl 486	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑙 ∈ 𝑇) ∧ 𝑖 = 𝑙) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
8		simp2 1154	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → 𝑇 ⊆ (LIdeal‘𝑅))
9	8	sselda 3936	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑙 ∈ 𝑇) → 𝑙 ∈ (LIdeal‘𝑅))
10		fvex 6894	. . . . . . . . . . . . 13 ⊢ (PrmIdeal‘𝑅) ∈ V
11	10	rabex 5308	. . . . . . . . . . . 12 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∈ V
12	11	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑙 ∈ 𝑇) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∈ V)
13	4, 7, 9, 12	fvmptd 6997	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑙 ∈ 𝑇) → (𝑉‘𝑙) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
14		ssrab2 4033	. . . . . . . . . . 11 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ⊆ (PrmIdeal‘𝑅)
15	14	a1i 11	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑙 ∈ 𝑇) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ⊆ (PrmIdeal‘𝑅))
16	13, 15	eqsstrd 3970	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑙 ∈ 𝑇) → (𝑉‘𝑙) ⊆ (PrmIdeal‘𝑅))
17	16	sseld 3935	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑙 ∈ 𝑇) → (𝑝 ∈ (𝑉‘𝑙) → 𝑝 ∈ (PrmIdeal‘𝑅)))
18	17	ralimdva 3176	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → (∀𝑙 ∈ 𝑇 𝑝 ∈ (𝑉‘𝑙) → ∀𝑙 ∈ 𝑇 𝑝 ∈ (PrmIdeal‘𝑅)))
19		eliin 4960	. . . . . . . . 9 ⊢ (𝑝 ∈ V → (𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙) ↔ ∀𝑙 ∈ 𝑇 𝑝 ∈ (𝑉‘𝑙)))
20	19	elv 3459	. . . . . . . 8 ⊢ (𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙) ↔ ∀𝑙 ∈ 𝑇 𝑝 ∈ (𝑉‘𝑙))
21	20	biimpi 219	. . . . . . 7 ⊢ (𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙) → ∀𝑙 ∈ 𝑇 𝑝 ∈ (𝑉‘𝑙))
22	18, 21	impel 514	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) → ∀𝑙 ∈ 𝑇 𝑝 ∈ (PrmIdeal‘𝑅))
23		rspn0 4310	. . . . . . 7 ⊢ (𝑇 ≠ ∅ → (∀𝑙 ∈ 𝑇 𝑝 ∈ (PrmIdeal‘𝑅) → 𝑝 ∈ (PrmIdeal‘𝑅)))
24	23	imp 411	. . . . . 6 ⊢ ((𝑇 ≠ ∅ ∧ ∀𝑙 ∈ 𝑇 𝑝 ∈ (PrmIdeal‘𝑅)) → 𝑝 ∈ (PrmIdeal‘𝑅))
25	2, 22, 24	syl2anc 595	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) → 𝑝 ∈ (PrmIdeal‘𝑅))
26		simp1 1153	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → 𝑅 ∈ Ring)
27	26	adantr 485	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) → 𝑅 ∈ Ring)
28		prmidlidl 21480	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑝 ∈ (PrmIdeal‘𝑅)) → 𝑝 ∈ (LIdeal‘𝑅))
29	27, 25, 28	syl2anc 595	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) → 𝑝 ∈ (LIdeal‘𝑅))
30		nfv 1943	. . . . . . . . 9 ⊢ Ⅎ𝑙(𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅)
31		nfcv 2924	. . . . . . . . . 10 ⊢ Ⅎ𝑙𝑝
32		nfii1 4992	. . . . . . . . . 10 ⊢ Ⅎ𝑙∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)
33	31, 32	nfel 2938	. . . . . . . . 9 ⊢ Ⅎ𝑙 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)
34	30, 33	nfan 1928	. . . . . . . 8 ⊢ Ⅎ𝑙((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙))
35	20	bilani 509	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) → ∀𝑙 ∈ 𝑇 𝑝 ∈ (𝑉‘𝑙))
36	35	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) ∧ 𝑙 ∈ 𝑇) → ∀𝑙 ∈ 𝑇 𝑝 ∈ (𝑉‘𝑙))
37		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) ∧ 𝑙 ∈ 𝑇) → 𝑙 ∈ 𝑇)
38		rspa 3253	. . . . . . . . . . . . 13 ⊢ ((∀𝑙 ∈ 𝑇 𝑝 ∈ (𝑉‘𝑙) ∧ 𝑙 ∈ 𝑇) → 𝑝 ∈ (𝑉‘𝑙))
39	36, 37, 38	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) ∧ 𝑙 ∈ 𝑇) → 𝑝 ∈ (𝑉‘𝑙))
40	13	adantlr 727	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) ∧ 𝑙 ∈ 𝑇) → (𝑉‘𝑙) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
41	39, 40	eleqtrd 2864	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) ∧ 𝑙 ∈ 𝑇) → 𝑝 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
42		sseq2 3962	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑝 → (𝑙 ⊆ 𝑗 ↔ 𝑙 ⊆ 𝑝))
43	42	elrab 3649	. . . . . . . . . . 11 ⊢ (𝑝 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ↔ (𝑝 ∈ (PrmIdeal‘𝑅) ∧ 𝑙 ⊆ 𝑝))
44	41, 43	sylib 221	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) ∧ 𝑙 ∈ 𝑇) → (𝑝 ∈ (PrmIdeal‘𝑅) ∧ 𝑙 ⊆ 𝑝))
45	44	simprd 500	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) ∧ 𝑙 ∈ 𝑇) → 𝑙 ⊆ 𝑝)
46	45	ex 417	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) → (𝑙 ∈ 𝑇 → 𝑙 ⊆ 𝑝))
47	34, 46	ralrimi 3262	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) → ∀𝑙 ∈ 𝑇 𝑙 ⊆ 𝑝)
48		unissb 4905	. . . . . . 7 ⊢ (∪ 𝑇 ⊆ 𝑝 ↔ ∀𝑙 ∈ 𝑇 𝑙 ⊆ 𝑝)
49	47, 48	sylibr 237	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) → ∪ 𝑇 ⊆ 𝑝)
50		zarclsiin.1	. . . . . . 7 ⊢ 𝐾 = (RSpan‘𝑅)
51		eqid 2762	. . . . . . 7 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
52	50, 51	rspssp 21379	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑝 ∈ (LIdeal‘𝑅) ∧ ∪ 𝑇 ⊆ 𝑝) → (𝐾‘∪ 𝑇) ⊆ 𝑝)
53	27, 29, 49, 52	syl3anc 1397	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) → (𝐾‘∪ 𝑇) ⊆ 𝑝)
54	1, 25, 53	elrabd 3651	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) → 𝑝 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝐾‘∪ 𝑇) ⊆ 𝑗})
55	3	a1i 11	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
56		sseq1 3961	. . . . . . . . 9 ⊢ (𝑖 = (𝐾‘∪ 𝑇) → (𝑖 ⊆ 𝑗 ↔ (𝐾‘∪ 𝑇) ⊆ 𝑗))
57	56	rabbidv 3422	. . . . . . . 8 ⊢ (𝑖 = (𝐾‘∪ 𝑇) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝐾‘∪ 𝑇) ⊆ 𝑗})
58	57	adantl 486	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑖 = (𝐾‘∪ 𝑇)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝐾‘∪ 𝑇) ⊆ 𝑗})
59	8	sselda 3936	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑖 ∈ 𝑇) → 𝑖 ∈ (LIdeal‘𝑅))
60		eqid 2762	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
61	60, 51	lidlss 21347	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (LIdeal‘𝑅) → 𝑖 ⊆ (Base‘𝑅))
62	59, 61	syl 18	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑖 ∈ 𝑇) → 𝑖 ⊆ (Base‘𝑅))
63	62	ralrimiva 3156	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → ∀𝑖 ∈ 𝑇 𝑖 ⊆ (Base‘𝑅))
64		unissb 4905	. . . . . . . . 9 ⊢ (∪ 𝑇 ⊆ (Base‘𝑅) ↔ ∀𝑖 ∈ 𝑇 𝑖 ⊆ (Base‘𝑅))
65	63, 64	sylibr 237	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → ∪ 𝑇 ⊆ (Base‘𝑅))
66	50, 60, 51	rspcl 21375	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ ∪ 𝑇 ⊆ (Base‘𝑅)) → (𝐾‘∪ 𝑇) ∈ (LIdeal‘𝑅))
67	26, 65, 66	syl2anc 595	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → (𝐾‘∪ 𝑇) ∈ (LIdeal‘𝑅))
68	10	rabex 5308	. . . . . . . 8 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝐾‘∪ 𝑇) ⊆ 𝑗} ∈ V
69	68	a1i 11	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝐾‘∪ 𝑇) ⊆ 𝑗} ∈ V)
70	55, 58, 67, 69	fvmptd 6997	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → (𝑉‘(𝐾‘∪ 𝑇)) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝐾‘∪ 𝑇) ⊆ 𝑗})
71	70	eleq2d 2848	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → (𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇)) ↔ 𝑝 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝐾‘∪ 𝑇) ⊆ 𝑗}))
72	71	adantr 485	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) → (𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇)) ↔ 𝑝 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝐾‘∪ 𝑇) ⊆ 𝑗}))
73	54, 72	mpbird 260	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙)) → 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇)))
74	71	biimpa 481	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) → 𝑝 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝐾‘∪ 𝑇) ⊆ 𝑗})
75	1	elrab 3649	. . . . . . . . . 10 ⊢ (𝑝 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝐾‘∪ 𝑇) ⊆ 𝑗} ↔ (𝑝 ∈ (PrmIdeal‘𝑅) ∧ (𝐾‘∪ 𝑇) ⊆ 𝑝))
76	74, 75	sylib 221	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) → (𝑝 ∈ (PrmIdeal‘𝑅) ∧ (𝐾‘∪ 𝑇) ⊆ 𝑝))
77	76	simpld 499	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) → 𝑝 ∈ (PrmIdeal‘𝑅))
78	77	adantr 485	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) ∧ 𝑙 ∈ 𝑇) → 𝑝 ∈ (PrmIdeal‘𝑅))
79		elssuni 4903	. . . . . . . . . 10 ⊢ (𝑙 ∈ 𝑇 → 𝑙 ⊆ ∪ 𝑇)
80	79	adantl 486	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) ∧ 𝑙 ∈ 𝑇) → 𝑙 ⊆ ∪ 𝑇)
81		simpll 778	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) ∧ 𝑙 ∈ 𝑇) → (𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅))
82	50, 60	rspssid 21376	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ∪ 𝑇 ⊆ (Base‘𝑅)) → ∪ 𝑇 ⊆ (𝐾‘∪ 𝑇))
83	26, 65, 82	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → ∪ 𝑇 ⊆ (𝐾‘∪ 𝑇))
84	81, 83	syl 18	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) ∧ 𝑙 ∈ 𝑇) → ∪ 𝑇 ⊆ (𝐾‘∪ 𝑇))
85	80, 84	sstrd 3946	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) ∧ 𝑙 ∈ 𝑇) → 𝑙 ⊆ (𝐾‘∪ 𝑇))
86	76	simprd 500	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) → (𝐾‘∪ 𝑇) ⊆ 𝑝)
87	86	adantr 485	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) ∧ 𝑙 ∈ 𝑇) → (𝐾‘∪ 𝑇) ⊆ 𝑝)
88	85, 87	sstrd 3946	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) ∧ 𝑙 ∈ 𝑇) → 𝑙 ⊆ 𝑝)
89	42, 78, 88	elrabd 3651	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) ∧ 𝑙 ∈ 𝑇) → 𝑝 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
90	8	adantr 485	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) → 𝑇 ⊆ (LIdeal‘𝑅))
91	90	sselda 3936	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) ∧ 𝑙 ∈ 𝑇) → 𝑙 ∈ (LIdeal‘𝑅))
92	3	a1i 11	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑙 ∈ (LIdeal‘𝑅)) → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
93	6	adantl 486	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑖 = 𝑙) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
94		simpr 489	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑙 ∈ (LIdeal‘𝑅)) → 𝑙 ∈ (LIdeal‘𝑅))
95	11	a1i 11	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑙 ∈ (LIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∈ V)
96	92, 93, 94, 95	fvmptd 6997	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑙 ∈ (LIdeal‘𝑅)) → (𝑉‘𝑙) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
97	81, 91, 96	syl2anc 595	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) ∧ 𝑙 ∈ 𝑇) → (𝑉‘𝑙) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
98	89, 97	eleqtrrd 2865	. . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) ∧ 𝑙 ∈ 𝑇) → 𝑝 ∈ (𝑉‘𝑙))
99	98	ralrimiva 3156	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) → ∀𝑙 ∈ 𝑇 𝑝 ∈ (𝑉‘𝑙))
100	20	a1i 11	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) → (𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙) ↔ ∀𝑙 ∈ 𝑇 𝑝 ∈ (𝑉‘𝑙)))
101	99, 100	mpbird 260	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) ∧ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))) → 𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙))
102	73, 101	impbida 812	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → (𝑝 ∈ ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙) ↔ 𝑝 ∈ (𝑉‘(𝐾‘∪ 𝑇))))
103	102	eqrdv 2760	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙) = (𝑉‘(𝐾‘∪ 𝑇)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 ∀wral 3078 {crab 3415 Vcvv 3454 ⊆ wss 3904 ∅c0 4285 ∪ cuni 4871 ∩ ciin 4956 ↦ cmpt 5191 ‘cfv 6536 Basecbs 17275 Ringcrg 20321 LIdealclidl 21341 RSpancrsp 21342 PrmIdealcprmidl 21471

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-er 8692 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-sca 17332 df-vsca 17333 df-ip 17334 df-0g 17500 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-grp 19009 df-minusg 19010 df-sbg 19011 df-subg 19195 df-mgp 20223 df-ur 20270 df-ring 20323 df-subrg 20680 df-lmod 20994 df-lss 21064 df-lsp 21104 df-sra 21305 df-rgmod 21306 df-lidl 21343 df-rsp 21344 df-prmidl 21472

This theorem is used by: zarclsint 34271 zarcmplem 34280

W3C validator