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Theorem zarclsun 32116
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) ∣ 𝑖𝑗})
Assertion
Ref Expression
zarclsun ((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉𝑌 ∈ ran 𝑉) → (𝑋𝑌) ∈ 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.
StepHypRef Expression
1 simpllr 774 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) → 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗})
2 simpr 486 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) → 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})
31, 2uneq12d 4116 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) → (𝑋𝑌) = ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}))
4 unrab 4257 . . . . . . . . . 10 ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙𝑗𝑘𝑗)}
5 zarclsx.1 . . . . . . . . . . 11 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
6 eqid 2737 . . . . . . . . . . . . 13 (IDLsrg‘𝑅) = (IDLsrg‘𝑅)
7 eqid 2737 . . . . . . . . . . . . 13 (LIdeal‘𝑅) = (LIdeal‘𝑅)
8 eqid 2737 . . . . . . . . . . . . 13 (.r‘(IDLsrg‘𝑅)) = (.r‘(IDLsrg‘𝑅))
9 simpll 765 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
109crngringd 19891 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ Ring)
11 simplr 767 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑙 ∈ (LIdeal‘𝑅))
12 simpr 486 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑘 ∈ (LIdeal‘𝑅))
136, 7, 8, 10, 11, 12idlsrgmulrcl 31950 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ∈ (LIdeal‘𝑅))
14 sseq1 3961 . . . . . . . . . . . . . . 15 (𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘) → (𝑖𝑗 ↔ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗))
1514rabbidv 3412 . . . . . . . . . . . . . 14 (𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗})
1615eqeq2d 2748 . . . . . . . . . . . . 13 (𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙𝑗𝑘𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ↔ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙𝑗𝑘𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗}))
1716adantl 483 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘)) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙𝑗𝑘𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ↔ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙𝑗𝑘𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗}))
18 eqid 2737 . . . . . . . . . . . . . . . . 17 (.r𝑅) = (.r𝑅)
199ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙𝑗) → 𝑅 ∈ CRing)
2011ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙𝑗) → 𝑙 ∈ (LIdeal‘𝑅))
2112ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙𝑗) → 𝑘 ∈ (LIdeal‘𝑅))
226, 7, 8, 18, 19, 20, 21idlsrgmulrss1 31951 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑙)
23 simpr 486 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙𝑗) → 𝑙𝑗)
2422, 23sstrd 3946 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗)
2510ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘𝑗) → 𝑅 ∈ Ring)
2611ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘𝑗) → 𝑙 ∈ (LIdeal‘𝑅))
2712ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘𝑗) → 𝑘 ∈ (LIdeal‘𝑅))
286, 7, 8, 18, 25, 26, 27idlsrgmulrss2 31952 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑘)
29 simpr 486 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘𝑗) → 𝑘𝑗)
3028, 29sstrd 3946 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗)
3124, 30jaodan 956 . . . . . . . . . . . . . 14 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙𝑗𝑘𝑗)) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗)
32 eqid 2737 . . . . . . . . . . . . . . 15 (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
3310ad2antrr 724 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑅 ∈ Ring)
34 simplr 767 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑗 ∈ (PrmIdeal‘𝑅))
3511ad2antrr 724 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑙 ∈ (LIdeal‘𝑅))
3612ad2antrr 724 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑘 ∈ (LIdeal‘𝑅))
37 eqid 2737 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) = (Base‘𝑅)
38 eqid 2737 . . . . . . . . . . . . . . . . . . 19 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3937, 7lidlss 20587 . . . . . . . . . . . . . . . . . . . 20 (𝑙 ∈ (LIdeal‘𝑅) → 𝑙 ⊆ (Base‘𝑅))
4035, 39syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑙 ⊆ (Base‘𝑅))
4137, 7lidlss 20587 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (LIdeal‘𝑅) → 𝑘 ⊆ (Base‘𝑅))
4236, 41syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → 𝑘 ⊆ (Base‘𝑅))
4337, 38, 32, 33, 40, 42ringlsmss 31878 . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . . 19 (RSpan‘𝑅) = (RSpan‘𝑅)
4544, 37rspssid 20600 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Ring ∧ (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ (Base‘𝑅)) → (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ ((RSpan‘𝑅)‘(𝑙(LSSum‘(mulGrp‘𝑅))𝑘)))
4633, 43, 45syl2anc 585 . . . . . . . . . . . . . . . . 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)))
476, 7, 8, 38, 32, 33, 35, 36idlsrgmulrval 31949 . . . . . . . . . . . . . . . . 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)))
4846, 47sseqtrrd 3977 . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗)
5048, 49sstrd 3946 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙(LSSum‘(mulGrp‘𝑅))𝑘) ⊆ 𝑗)
5132, 33, 34, 35, 36, 50idlmulssprm 31912 . . . . . . . . . . . . . 14 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗) → (𝑙𝑗𝑘𝑗))
5231, 51impbida 799 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → ((𝑙𝑗𝑘𝑗) ↔ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗))
5352rabbidva 3411 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙𝑗𝑘𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗})
5413, 17, 53rspcedvd 3576 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → ∃𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙𝑗𝑘𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
55 fvex 6843 . . . . . . . . . . . . 13 (PrmIdeal‘𝑅) ∈ V
5655rabex 5281 . . . . . . . . . . . 12 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙𝑗𝑘𝑗)} ∈ V
5756a1i 11 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙𝑗𝑘𝑗)} ∈ V)
585, 54, 57elrnmptd 5907 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙𝑗𝑘𝑗)} ∈ ran 𝑉)
594, 58eqeltrid 2842 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) ∈ ran 𝑉)
6059adantlr 713 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) ∈ ran 𝑉)
6160adantr 482 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) → ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗} ∪ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) ∈ ran 𝑉)
623, 61eqeltrd 2838 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) → (𝑋𝑌) ∈ ran 𝑉)
6362adantl4r 753 . . . . 5 ((((((𝑅 ∈ CRing ∧ 𝑌 ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗}) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) → (𝑋𝑌) ∈ ran 𝑉)
6455rabex 5281 . . . . . . . . 9 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ∈ V
655, 64elrnmpti 5906 . . . . . . . 8 (𝑌 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
66 sseq1 3961 . . . . . . . . . . 11 (𝑖 = 𝑘 → (𝑖𝑗𝑘𝑗))
6766rabbidv 3412 . . . . . . . . . 10 (𝑖 = 𝑘 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})
6867eqeq2d 2748 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ↔ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}))
6968cbvrexvw 3223 . . . . . . . 8 (∃𝑖 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})
70 biid 261 . . . . . . . 8 (∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗} ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})
7165, 69, 703bitri 297 . . . . . . 7 (𝑌 ∈ ran 𝑉 ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})
7271biimpi 215 . . . . . 6 (𝑌 ∈ ran 𝑉 → ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})
7372ad3antlr 729 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝑌 ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗}) → ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})
7463, 73r19.29a 3156 . . . 4 ((((𝑅 ∈ CRing ∧ 𝑌 ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗}) → (𝑋𝑌) ∈ ran 𝑉)
7574adantl3r 748 . . 3 (((((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉) ∧ 𝑌 ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗}) → (𝑋𝑌) ∈ ran 𝑉)
765, 64elrnmpti 5906 . . . . . 6 (𝑋 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
77 sseq1 3961 . . . . . . . . 9 (𝑖 = 𝑙 → (𝑖𝑗𝑙𝑗))
7877rabbidv 3412 . . . . . . . 8 (𝑖 = 𝑙 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗})
7978eqeq2d 2748 . . . . . . 7 (𝑖 = 𝑙 → (𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ↔ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗}))
8079cbvrexvw 3223 . . . . . 6 (∃𝑖 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗})
81 biid 261 . . . . . 6 (∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗} ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗})
8276, 80, 813bitri 297 . . . . 5 (𝑋 ∈ ran 𝑉 ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗})
8382biimpi 215 . . . 4 (𝑋 ∈ ran 𝑉 → ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗})
8483ad2antlr 725 . . 3 (((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉) ∧ 𝑌 ∈ ran 𝑉) → ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗})
8575, 84r19.29a 3156 . 2 (((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉) ∧ 𝑌 ∈ ran 𝑉) → (𝑋𝑌) ∈ ran 𝑉)
86853impa 1110 1 ((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉𝑌 ∈ ran 𝑉) → (𝑋𝑌) ∈ ran 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 845  w3a 1087   = wceq 1541  wcel 2106  wrex 3071  {crab 3404  Vcvv 3442  cun 3900  wss 3902  cmpt 5180  ran crn 5626  cfv 6484  (class class class)co 7342  Basecbs 17010  .rcmulr 17061  LSSumclsm 19336  mulGrpcmgp 19815  Ringcrg 19878  CRingccrg 19879  LIdealclidl 20538  RSpancrsp 20539  PrmIdealcprmidl 31905  IDLsrgcidlsrg 31940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655  ax-cnex 11033  ax-resscn 11034  ax-1cn 11035  ax-icn 11036  ax-addcl 11037  ax-addrcl 11038  ax-mulcl 11039  ax-mulrcl 11040  ax-mulcom 11041  ax-addass 11042  ax-mulass 11043  ax-distr 11044  ax-i2m1 11045  ax-1ne0 11046  ax-1rid 11047  ax-rnegex 11048  ax-rrecex 11049  ax-cnre 11050  ax-pre-lttri 11051  ax-pre-lttrn 11052  ax-pre-ltadd 11053  ax-pre-mulgt0 11054
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4858  df-int 4900  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-om 7786  df-1st 7904  df-2nd 7905  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-1o 8372  df-er 8574  df-en 8810  df-dom 8811  df-sdom 8812  df-fin 8813  df-pnf 11117  df-mnf 11118  df-xr 11119  df-ltxr 11120  df-le 11121  df-sub 11313  df-neg 11314  df-nn 12080  df-2 12142  df-3 12143  df-4 12144  df-5 12145  df-6 12146  df-7 12147  df-8 12148  df-9 12149  df-n0 12340  df-z 12426  df-dec 12544  df-uz 12689  df-fz 13346  df-struct 16946  df-sets 16963  df-slot 16981  df-ndx 16993  df-base 17011  df-ress 17040  df-plusg 17073  df-mulr 17074  df-sca 17076  df-vsca 17077  df-ip 17078  df-tset 17079  df-ple 17080  df-0g 17250  df-mgm 18424  df-sgrp 18473  df-mnd 18484  df-grp 18677  df-minusg 18678  df-sbg 18679  df-subg 18849  df-lsm 19338  df-cmn 19484  df-mgp 19816  df-ur 19833  df-ring 19880  df-cring 19881  df-subrg 20127  df-lmod 20231  df-lss 20300  df-lsp 20340  df-sra 20540  df-rgmod 20541  df-lidl 20542  df-rsp 20543  df-prmidl 31906  df-idlsrg 31941
This theorem is referenced by:  zartopn  32121
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