Theorem zarclsun 31722

Description: The union of two closed sets of the Zariski topology is closed. (Contributed by Thierry Arnoux, 16-Jun-2024.)

Hypothesis

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

Assertion

Ref	Expression
zarclsun	⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ∧ 𝑌 ∈ ran 𝑉) → (𝑋 ∪ 𝑌) ∈ ran 𝑉)

Distinct variable groups:   𝑅,𝑖,𝑗   𝑖,𝑋   𝑖,𝑌

Allowed substitution hints:   𝑉(𝑖,𝑗)   𝑋(𝑗)   𝑌(𝑗)

Proof of Theorem zarclsun

Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpllr 772	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) → 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
2		simpr 484	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) → 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})
3	1, 2	uneq12d 4094	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) → (𝑋 ∪ 𝑌) = ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}))
4		unrab 4236	. . . . . . . . . 10 ⊢ ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)}
5		zarclsx.1	. . . . . . . . . . 11 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
6		eqid 2738	. . . . . . . . . . . . 13 ⊢ (IDLsrg‘𝑅) = (IDLsrg‘𝑅)
7		eqid 2738	. . . . . . . . . . . . 13 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
8		eqid 2738	. . . . . . . . . . . . 13 ⊢ (.r‘(IDLsrg‘𝑅)) = (.r‘(IDLsrg‘𝑅))
9		simpll 763	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
10	9	crngringd 19711	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ Ring)
11		simplr 765	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑙 ∈ (LIdeal‘𝑅))
12		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑘 ∈ (LIdeal‘𝑅))
13	6, 7, 8, 10, 11, 12	idlsrgmulrcl 31557	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ∈ (LIdeal‘𝑅))
14		sseq1 3942	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘) → (𝑖 ⊆ 𝑗 ↔ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗))
15	14	rabbidv 3404	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗})
16	15	eqeq2d 2749	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗}))
17	16	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘)) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗}))
18		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝑅) = (.r‘𝑅)
19	9	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙 ⊆ 𝑗) → 𝑅 ∈ CRing)
20	11	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙 ⊆ 𝑗) → 𝑙 ∈ (LIdeal‘𝑅))
21	12	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙 ⊆ 𝑗) → 𝑘 ∈ (LIdeal‘𝑅))
22	6, 7, 8, 18, 19, 20, 21	idlsrgmulrss1 31558	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙 ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑙)
23		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙 ⊆ 𝑗) → 𝑙 ⊆ 𝑗)
24	22, 23	sstrd 3927	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙 ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗)
25	10	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘 ⊆ 𝑗) → 𝑅 ∈ Ring)
26	11	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘 ⊆ 𝑗) → 𝑙 ∈ (LIdeal‘𝑅))
27	12	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘 ⊆ 𝑗) → 𝑘 ∈ (LIdeal‘𝑅))
28	6, 7, 8, 18, 25, 26, 27	idlsrgmulrss2 31559	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘 ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑘)
29		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘 ⊆ 𝑗) → 𝑘 ⊆ 𝑗)
30	28, 29	sstrd 3927	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘 ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗)
31	24, 30	jaodan 954	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗)
32		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
33	10	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑅 ∈ Ring)
34		simplr 765	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑗 ∈ (PrmIdeal‘𝑅))
35	11	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑙 ∈ (LIdeal‘𝑅))
36	12	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑘 ∈ (LIdeal‘𝑅))
37		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
38		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
39	37, 7	lidlss 20394	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ (LIdeal‘𝑅) → 𝑙 ⊆ (Base‘𝑅))
40	35, 39	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑙 ⊆ (Base‘𝑅))
41	37, 7	lidlss 20394	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (LIdeal‘𝑅) → 𝑘 ⊆ (Base‘𝑅))
42	36, 41	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑘 ⊆ (Base‘𝑅))
43	37, 38, 32, 33, 40, 42	ringlsmss 31485	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ (Base‘𝑅))
44		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
45	44, 37	rspssid 20407	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Ring ∧ (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ (Base‘𝑅)) → (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ ((RSpan‘𝑅)‘(𝑙(LSSum‘(mulGrp‘𝑅))𝑘)))
46	33, 43, 45	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ ((RSpan‘𝑅)‘(𝑙(LSSum‘(mulGrp‘𝑅))𝑘)))
47	6, 7, 8, 38, 32, 33, 35, 36	idlsrgmulrval 31556	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) = ((RSpan‘𝑅)‘(𝑙(LSSum‘(mulGrp‘𝑅))𝑘)))
48	46, 47	sseqtrrd 3958	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ (𝑙(.r‘(IDLsrg‘𝑅))𝑘))
49		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗)
50	48, 49	sstrd 3927	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ 𝑗)
51	32, 33, 34, 35, 36, 50	idlmulssprm 31519	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗))
52	31, 51	impbida 797	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → ((𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗) ↔ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗))
53	52	rabbidva 3402	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗})
54	13, 17, 53	rspcedvd 3555	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → ∃𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
55		fvex 6769	. . . . . . . . . . . . 13 ⊢ (PrmIdeal‘𝑅) ∈ V
56	55	rabex 5251	. . . . . . . . . . . 12 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} ∈ V
57	56	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} ∈ V)
58	5, 54, 57	elrnmptd 5859	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} ∈ ran 𝑉)
59	4, 58	eqeltrid 2843	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) ∈ ran 𝑉)
60	59	adantlr 711	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) ∈ ran 𝑉)
61	60	adantr 480	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) ∈ ran 𝑉)
62	3, 61	eqeltrd 2839	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) → (𝑋 ∪ 𝑌) ∈ ran 𝑉)
63	62	adantl4r 751	. . . . 5 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑌 ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) → (𝑋 ∪ 𝑌) ∈ ran 𝑉)
64	55	rabex 5251	. . . . . . . . 9 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ∈ V
65	5, 64	elrnmpti 5858	. . . . . . . 8 ⊢ (𝑌 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
66		sseq1 3942	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → (𝑖 ⊆ 𝑗 ↔ 𝑘 ⊆ 𝑗))
67	66	rabbidv 3404	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})
68	67	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}))
69	68	cbvrexvw 3373	. . . . . . . 8 ⊢ (∃𝑖 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})
70		biid 260	. . . . . . . 8 ⊢ (∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗} ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})
71	65, 69, 70	3bitri 296	. . . . . . 7 ⊢ (𝑌 ∈ ran 𝑉 ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})
72	71	biimpi 215	. . . . . 6 ⊢ (𝑌 ∈ ran 𝑉 → ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})
73	72	ad3antlr 727	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑌 ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) → ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})
74	63, 73	r19.29a 3217	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑌 ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) → (𝑋 ∪ 𝑌) ∈ ran 𝑉)
75	74	adantl3r 746	. . 3 ⊢ (((((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉) ∧ 𝑌 ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) → (𝑋 ∪ 𝑌) ∈ ran 𝑉)
76	5, 64	elrnmpti 5858	. . . . . 6 ⊢ (𝑋 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
77		sseq1 3942	. . . . . . . . 9 ⊢ (𝑖 = 𝑙 → (𝑖 ⊆ 𝑗 ↔ 𝑙 ⊆ 𝑗))
78	77	rabbidv 3404	. . . . . . . 8 ⊢ (𝑖 = 𝑙 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
79	78	eqeq2d 2749	. . . . . . 7 ⊢ (𝑖 = 𝑙 → (𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}))
80	79	cbvrexvw 3373	. . . . . 6 ⊢ (∃𝑖 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
81		biid 260	. . . . . 6 ⊢ (∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
82	76, 80, 81	3bitri 296	. . . . 5 ⊢ (𝑋 ∈ ran 𝑉 ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
83	82	biimpi 215	. . . 4 ⊢ (𝑋 ∈ ran 𝑉 → ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
84	83	ad2antlr 723	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉) ∧ 𝑌 ∈ ran 𝑉) → ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
85	75, 84	r19.29a 3217	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉) ∧ 𝑌 ∈ ran 𝑉) → (𝑋 ∪ 𝑌) ∈ ran 𝑉)
86	85	3impa 1108	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ∧ 𝑌 ∈ ran 𝑉) → (𝑋 ∪ 𝑌) ∈ ran 𝑉)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  {crab 3067  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883   ↦ cmpt 5153  ran crn 5581  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  ._rcmulr 16889  LSSumclsm 19154  mulGrpcmgp 19635  Ringcrg 19698  CRingccrg 19699  LIdealclidl 20347  RSpancrsp 20348  PrmIdealcprmidl 31512  IDLsrgcidlsrg 31547

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-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-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  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-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-struct 16776  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-tset 16907  df-ple 16908  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-lsm 19156  df-cmn 19303  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  df-idlsrg 31548

This theorem is referenced by:  zartopn  31727