Theorem zarclsun 34326

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 788	. . . . . . . 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 490	. . . . . . . 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 4123	. . . . . . 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 4268	. . . . . . . . . 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 2765	. . . . . . . . . . . . 13 ⊢ (IDLsrg‘𝑅) = (IDLsrg‘𝑅)
7		eqid 2765	. . . . . . . . . . . . 13 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
8		eqid 2765	. . . . . . . . . . . . 13 ⊢ (.r‘(IDLsrg‘𝑅)) = (.r‘(IDLsrg‘𝑅))
9		simpll 779	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
10	9	crngringd 20374	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ Ring)
11		simplr 781	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑙 ∈ (LIdeal‘𝑅))
12		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑘 ∈ (LIdeal‘𝑅))
13	6, 7, 8, 10, 11, 12	idlsrgmulrcl 33866	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ∈ (LIdeal‘𝑅))
14		sseq1 3963	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘) → (𝑖 ⊆ 𝑗 ↔ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗))
15	14	rabbidv 3425	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗})
16	15	eqeq2d 2776	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝑅) = (.r‘𝑅)
19	9	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙 ⊆ 𝑗) → 𝑅 ∈ CRing)
20	11	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙 ⊆ 𝑗) → 𝑙 ∈ (LIdeal‘𝑅))
21	12	ad2antrr 739	. . . . . . . . . . . . . . . . 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 33867	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙 ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑙)
23		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙 ⊆ 𝑗) → 𝑙 ⊆ 𝑗)
24	22, 23	sstrd 3948	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙 ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗)
25	10	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘 ⊆ 𝑗) → 𝑅 ∈ Ring)
26	11	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘 ⊆ 𝑗) → 𝑙 ∈ (LIdeal‘𝑅))
27	12	ad2antrr 739	. . . . . . . . . . . . . . . . 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 33868	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘 ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑘)
29		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘 ⊆ 𝑗) → 𝑘 ⊆ 𝑗)
30	28, 29	sstrd 3948	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘 ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗)
31	24, 30	jaodan 972	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗)
32		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
33	10	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑅 ∈ Ring)
34		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑗 ∈ (PrmIdeal‘𝑅))
35	11	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑙 ∈ (LIdeal‘𝑅))
36	12	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑘 ∈ (LIdeal‘𝑅))
37		eqid 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
38		eqid 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
39	37, 7	lidlss 21388	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ (LIdeal‘𝑅) → 𝑙 ⊆ (Base‘𝑅))
40	35, 39	syl 18	. . . . . . . . . . . . . . . . . . 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 21388	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (LIdeal‘𝑅) → 𝑘 ⊆ (Base‘𝑅))
42	36, 41	syl 18	. . . . . . . . . . . . . . . . . . 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 33772	. . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
45	44, 37	rspssid 21417	. . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . 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 33865	. . . . . . . . . . . . . . . . 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 3975	. . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . 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 3948	. . . . . . . . . . . . . . 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 21519	. . . . . . . . . . . . . 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 813	. . . . . . . . . . . . 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 3424	. . . . . . . . . . . 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 3585	. . . . . . . . . . 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 6898	. . . . . . . . . . . . 13 ⊢ (PrmIdeal‘𝑅) ∈ V
56	55	rabex 5311	. . . . . . . . . . . 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 5955	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙 ⊆ 𝑗 ∨ 𝑘 ⊆ 𝑗)} ∈ ran 𝑉)
59	4, 58	eqeltrid 2869	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) ∈ ran 𝑉)
60	59	adantlr 728	. . . . . . . 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 486	. . . . . . 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 2865	. . . . . 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 768	. . . . 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 5311	. . . . . . . . 9 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ∈ V
65	5, 64	elrnmpti 5954	. . . . . . . 8 ⊢ (𝑌 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
66		sseq1 3963	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → (𝑖 ⊆ 𝑗 ↔ 𝑘 ⊆ 𝑗))
67	66	rabbidv 3425	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})
68	67	eqeq2d 2776	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}))
69	68	cbvrexvw 3246	. . . . . . . 8 ⊢ (∃𝑖 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})
70		biid 264	. . . . . . . 8 ⊢ (∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗} ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})
71	65, 69, 70	3bitri 300	. . . . . . 7 ⊢ (𝑌 ∈ ran 𝑉 ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})
72	71	biimpi 219	. . . . . 6 ⊢ (𝑌 ∈ ran 𝑉 → ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})
73	72	ad3antlr 744	. . . . 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 3175	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑌 ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}) → (𝑋 ∪ 𝑌) ∈ ran 𝑉)
75	74	adantl3r 763	. . 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 5954	. . . . . 6 ⊢ (𝑋 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
77		sseq1 3963	. . . . . . . . 9 ⊢ (𝑖 = 𝑙 → (𝑖 ⊆ 𝑗 ↔ 𝑙 ⊆ 𝑗))
78	77	rabbidv 3425	. . . . . . . 8 ⊢ (𝑖 = 𝑙 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
79	78	eqeq2d 2776	. . . . . . 7 ⊢ (𝑖 = 𝑙 → (𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗}))
80	79	cbvrexvw 3246	. . . . . 6 ⊢ (∃𝑖 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
81		biid 264	. . . . . 6 ⊢ (∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗} ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
82	76, 80, 81	3bitri 300	. . . . 5 ⊢ (𝑋 ∈ ran 𝑉 ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
83	82	biimpi 219	. . . 4 ⊢ (𝑋 ∈ ran 𝑉 → ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
84	83	ad2antlr 740	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉) ∧ 𝑌 ∈ ran 𝑉) → ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙 ⊆ 𝑗})
85	75, 84	r19.29a 3175	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉) ∧ 𝑌 ∈ ran 𝑉) → (𝑋 ∪ 𝑌) ∈ ran 𝑉)
86	85	3impa 1127	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ∧ 𝑌 ∈ ran 𝑉) → (𝑋 ∪ 𝑌) ∈ ran 𝑉)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3091  {crab 3418  Vcvv 3457   ∪ cun 3904   ⊆ wss 3906   ↦ cmpt 5194  ran crn 5664  ‘cfv 6540  (class class class)co 7419  Basecbs 17293  ._rcmulr 17335  LSSumclsm 19750  mulGrpcmgp 20262  Ringcrg 20361  CRingccrg 20362  LIdealclidl 21382  RSpancrsp 21383  PrmIdealcprmidl 21512  IDLsrgcidlsrg 33856

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-7 12325  df-8 12326  df-9 12327  df-n0 12522  df-z 12609  df-dec 12730  df-uz 12881  df-fz 13554  df-struct 17231  df-sets 17248  df-slot 17266  df-ndx 17278  df-base 17294  df-ress 17315  df-plusg 17347  df-mulr 17348  df-sca 17350  df-vsca 17351  df-ip 17352  df-tset 17353  df-ple 17354  df-0g 17518  df-mgm 18722  df-sgrp 18811  df-mnd 18827  df-grp 19049  df-minusg 19050  df-sbg 19051  df-subg 19235  df-lsm 19752  df-cmn 19898  df-abl 19899  df-mgp 20263  df-rng 20277  df-ur 20310  df-ring 20363  df-cring 20364  df-subrg 20721  df-lmod 21035  df-lss 21105  df-lsp 21145  df-sra 21346  df-rgmod 21347  df-lidl 21384  df-rsp 21385  df-prmidl 21513  df-idlsrg 33857

This theorem is used by:  zartopn  34331