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Theorem zarclsun 32491
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 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
Assertion
Ref Expression
zarclsun ((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ∧ π‘Œ ∈ ran 𝑉) β†’ (𝑋 βˆͺ π‘Œ) ∈ ran 𝑉)
Distinct variable groups:   𝑅,𝑖,𝑗   𝑖,𝑋   𝑖,π‘Œ
Allowed substitution hints:   𝑉(𝑖,𝑗)   𝑋(𝑗)   π‘Œ(𝑗)

Proof of Theorem zarclsun
Dummy variables π‘˜ 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpllr 775 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ 𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗}) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) β†’ 𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗})
2 simpr 486 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ 𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗}) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) β†’ π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})
31, 2uneq12d 4129 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ 𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗}) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) β†’ (𝑋 βˆͺ π‘Œ) = ({𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗} βˆͺ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}))
4 unrab 4270 . . . . . . . . . 10 ({𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗} βˆͺ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙 βŠ† 𝑗 ∨ π‘˜ βŠ† 𝑗)}
5 zarclsx.1 . . . . . . . . . . 11 𝑉 = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
6 eqid 2737 . . . . . . . . . . . . 13 (IDLsrgβ€˜π‘…) = (IDLsrgβ€˜π‘…)
7 eqid 2737 . . . . . . . . . . . . 13 (LIdealβ€˜π‘…) = (LIdealβ€˜π‘…)
8 eqid 2737 . . . . . . . . . . . . 13 (.rβ€˜(IDLsrgβ€˜π‘…)) = (.rβ€˜(IDLsrgβ€˜π‘…))
9 simpll 766 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) β†’ 𝑅 ∈ CRing)
109crngringd 19984 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) β†’ 𝑅 ∈ Ring)
11 simplr 768 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) β†’ 𝑙 ∈ (LIdealβ€˜π‘…))
12 simpr 486 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) β†’ π‘˜ ∈ (LIdealβ€˜π‘…))
136, 7, 8, 10, 11, 12idlsrgmulrcl 32292 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) β†’ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) ∈ (LIdealβ€˜π‘…))
14 sseq1 3974 . . . . . . . . . . . . . . 15 (𝑖 = (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) β†’ (𝑖 βŠ† 𝑗 ↔ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗))
1514rabbidv 3418 . . . . . . . . . . . . . 14 (𝑖 = (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗})
1615eqeq2d 2748 . . . . . . . . . . . . 13 (𝑖 = (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) β†’ ({𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙 βŠ† 𝑗 ∨ π‘˜ βŠ† 𝑗)} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} ↔ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙 βŠ† 𝑗 ∨ π‘˜ βŠ† 𝑗)} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗}))
1716adantl 483 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑖 = (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜)) β†’ ({𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙 βŠ† 𝑗 ∨ π‘˜ βŠ† 𝑗)} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} ↔ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙 βŠ† 𝑗 ∨ π‘˜ βŠ† 𝑗)} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗}))
18 eqid 2737 . . . . . . . . . . . . . . . . 17 (.rβ€˜π‘…) = (.rβ€˜π‘…)
199ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑙 βŠ† 𝑗) β†’ 𝑅 ∈ CRing)
2011ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑙 βŠ† 𝑗) β†’ 𝑙 ∈ (LIdealβ€˜π‘…))
2112ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑙 βŠ† 𝑗) β†’ π‘˜ ∈ (LIdealβ€˜π‘…))
226, 7, 8, 18, 19, 20, 21idlsrgmulrss1 32293 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑙 βŠ† 𝑗) β†’ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑙)
23 simpr 486 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑙 βŠ† 𝑗) β†’ 𝑙 βŠ† 𝑗)
2422, 23sstrd 3959 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑙 βŠ† 𝑗) β†’ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗)
2510ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ π‘˜ βŠ† 𝑗) β†’ 𝑅 ∈ Ring)
2611ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ π‘˜ βŠ† 𝑗) β†’ 𝑙 ∈ (LIdealβ€˜π‘…))
2712ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ π‘˜ βŠ† 𝑗) β†’ π‘˜ ∈ (LIdealβ€˜π‘…))
286, 7, 8, 18, 25, 26, 27idlsrgmulrss2 32294 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ π‘˜ βŠ† 𝑗) β†’ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† π‘˜)
29 simpr 486 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ π‘˜ βŠ† 𝑗) β†’ π‘˜ βŠ† 𝑗)
3028, 29sstrd 3959 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ π‘˜ βŠ† 𝑗) β†’ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗)
3124, 30jaodan 957 . . . . . . . . . . . . . 14 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙 βŠ† 𝑗 ∨ π‘˜ βŠ† 𝑗)) β†’ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗)
32 eqid 2737 . . . . . . . . . . . . . . 15 (LSSumβ€˜(mulGrpβ€˜π‘…)) = (LSSumβ€˜(mulGrpβ€˜π‘…))
3310ad2antrr 725 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗) β†’ 𝑅 ∈ Ring)
34 simplr 768 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗) β†’ 𝑗 ∈ (PrmIdealβ€˜π‘…))
3511ad2antrr 725 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗) β†’ 𝑙 ∈ (LIdealβ€˜π‘…))
3612ad2antrr 725 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗) β†’ π‘˜ ∈ (LIdealβ€˜π‘…))
37 eqid 2737 . . . . . . . . . . . . . . . . . . 19 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
38 eqid 2737 . . . . . . . . . . . . . . . . . . 19 (mulGrpβ€˜π‘…) = (mulGrpβ€˜π‘…)
3937, 7lidlss 20696 . . . . . . . . . . . . . . . . . . . 20 (𝑙 ∈ (LIdealβ€˜π‘…) β†’ 𝑙 βŠ† (Baseβ€˜π‘…))
4035, 39syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗) β†’ 𝑙 βŠ† (Baseβ€˜π‘…))
4137, 7lidlss 20696 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ ∈ (LIdealβ€˜π‘…) β†’ π‘˜ βŠ† (Baseβ€˜π‘…))
4236, 41syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗) β†’ π‘˜ βŠ† (Baseβ€˜π‘…))
4337, 38, 32, 33, 40, 42ringlsmss 32216 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗) β†’ (𝑙(LSSumβ€˜(mulGrpβ€˜π‘…))π‘˜) βŠ† (Baseβ€˜π‘…))
44 eqid 2737 . . . . . . . . . . . . . . . . . . 19 (RSpanβ€˜π‘…) = (RSpanβ€˜π‘…)
4544, 37rspssid 20709 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Ring ∧ (𝑙(LSSumβ€˜(mulGrpβ€˜π‘…))π‘˜) βŠ† (Baseβ€˜π‘…)) β†’ (𝑙(LSSumβ€˜(mulGrpβ€˜π‘…))π‘˜) βŠ† ((RSpanβ€˜π‘…)β€˜(𝑙(LSSumβ€˜(mulGrpβ€˜π‘…))π‘˜)))
4633, 43, 45syl2anc 585 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗) β†’ (𝑙(LSSumβ€˜(mulGrpβ€˜π‘…))π‘˜) βŠ† ((RSpanβ€˜π‘…)β€˜(𝑙(LSSumβ€˜(mulGrpβ€˜π‘…))π‘˜)))
476, 7, 8, 38, 32, 33, 35, 36idlsrgmulrval 32291 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗) β†’ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) = ((RSpanβ€˜π‘…)β€˜(𝑙(LSSumβ€˜(mulGrpβ€˜π‘…))π‘˜)))
4846, 47sseqtrrd 3990 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗) β†’ (𝑙(LSSumβ€˜(mulGrpβ€˜π‘…))π‘˜) βŠ† (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜))
49 simpr 486 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗) β†’ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗)
5048, 49sstrd 3959 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗) β†’ (𝑙(LSSumβ€˜(mulGrpβ€˜π‘…))π‘˜) βŠ† 𝑗)
5132, 33, 34, 35, 36, 50idlmulssprm 32254 . . . . . . . . . . . . . 14 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗) β†’ (𝑙 βŠ† 𝑗 ∨ π‘˜ βŠ† 𝑗))
5231, 51impbida 800 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) β†’ ((𝑙 βŠ† 𝑗 ∨ π‘˜ βŠ† 𝑗) ↔ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗))
5352rabbidva 3417 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙 βŠ† 𝑗 ∨ π‘˜ βŠ† 𝑗)} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙(.rβ€˜(IDLsrgβ€˜π‘…))π‘˜) βŠ† 𝑗})
5413, 17, 53rspcedvd 3586 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) β†’ βˆƒπ‘– ∈ (LIdealβ€˜π‘…){𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙 βŠ† 𝑗 ∨ π‘˜ βŠ† 𝑗)} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
55 fvex 6860 . . . . . . . . . . . . 13 (PrmIdealβ€˜π‘…) ∈ V
5655rabex 5294 . . . . . . . . . . . 12 {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙 βŠ† 𝑗 ∨ π‘˜ βŠ† 𝑗)} ∈ V
5756a1i 11 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙 βŠ† 𝑗 ∨ π‘˜ βŠ† 𝑗)} ∈ V)
585, 54, 57elrnmptd 5921 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ (𝑙 βŠ† 𝑗 ∨ π‘˜ βŠ† 𝑗)} ∈ ran 𝑉)
594, 58eqeltrid 2842 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) β†’ ({𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗} βˆͺ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) ∈ ran 𝑉)
6059adantlr 714 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ 𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗}) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) β†’ ({𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗} βˆͺ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) ∈ ran 𝑉)
6160adantr 482 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ 𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗}) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) β†’ ({𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗} βˆͺ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) ∈ ran 𝑉)
623, 61eqeltrd 2838 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ 𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗}) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) β†’ (𝑋 βˆͺ π‘Œ) ∈ ran 𝑉)
6362adantl4r 754 . . . . 5 ((((((𝑅 ∈ CRing ∧ π‘Œ ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ 𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗}) ∧ π‘˜ ∈ (LIdealβ€˜π‘…)) ∧ π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) β†’ (𝑋 βˆͺ π‘Œ) ∈ ran 𝑉)
6455rabex 5294 . . . . . . . . 9 {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} ∈ V
655, 64elrnmpti 5920 . . . . . . . 8 (π‘Œ ∈ ran 𝑉 ↔ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
66 sseq1 3974 . . . . . . . . . . 11 (𝑖 = π‘˜ β†’ (𝑖 βŠ† 𝑗 ↔ π‘˜ βŠ† 𝑗))
6766rabbidv 3418 . . . . . . . . . 10 (𝑖 = π‘˜ β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})
6867eqeq2d 2748 . . . . . . . . 9 (𝑖 = π‘˜ β†’ (π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} ↔ π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}))
6968cbvrexvw 3229 . . . . . . . 8 (βˆƒπ‘– ∈ (LIdealβ€˜π‘…)π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} ↔ βˆƒπ‘˜ ∈ (LIdealβ€˜π‘…)π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})
70 biid 261 . . . . . . . 8 (βˆƒπ‘˜ ∈ (LIdealβ€˜π‘…)π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗} ↔ βˆƒπ‘˜ ∈ (LIdealβ€˜π‘…)π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})
7165, 69, 703bitri 297 . . . . . . 7 (π‘Œ ∈ ran 𝑉 ↔ βˆƒπ‘˜ ∈ (LIdealβ€˜π‘…)π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})
7271biimpi 215 . . . . . 6 (π‘Œ ∈ ran 𝑉 β†’ βˆƒπ‘˜ ∈ (LIdealβ€˜π‘…)π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})
7372ad3antlr 730 . . . . 5 ((((𝑅 ∈ CRing ∧ π‘Œ ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ 𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗}) β†’ βˆƒπ‘˜ ∈ (LIdealβ€˜π‘…)π‘Œ = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})
7463, 73r19.29a 3160 . . . 4 ((((𝑅 ∈ CRing ∧ π‘Œ ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ 𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗}) β†’ (𝑋 βˆͺ π‘Œ) ∈ ran 𝑉)
7574adantl3r 749 . . 3 (((((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉) ∧ π‘Œ ∈ ran 𝑉) ∧ 𝑙 ∈ (LIdealβ€˜π‘…)) ∧ 𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗}) β†’ (𝑋 βˆͺ π‘Œ) ∈ ran 𝑉)
765, 64elrnmpti 5920 . . . . . 6 (𝑋 ∈ ran 𝑉 ↔ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
77 sseq1 3974 . . . . . . . . 9 (𝑖 = 𝑙 β†’ (𝑖 βŠ† 𝑗 ↔ 𝑙 βŠ† 𝑗))
7877rabbidv 3418 . . . . . . . 8 (𝑖 = 𝑙 β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗})
7978eqeq2d 2748 . . . . . . 7 (𝑖 = 𝑙 β†’ (𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} ↔ 𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗}))
8079cbvrexvw 3229 . . . . . 6 (βˆƒπ‘– ∈ (LIdealβ€˜π‘…)𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} ↔ βˆƒπ‘™ ∈ (LIdealβ€˜π‘…)𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗})
81 biid 261 . . . . . 6 (βˆƒπ‘™ ∈ (LIdealβ€˜π‘…)𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗} ↔ βˆƒπ‘™ ∈ (LIdealβ€˜π‘…)𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗})
8276, 80, 813bitri 297 . . . . 5 (𝑋 ∈ ran 𝑉 ↔ βˆƒπ‘™ ∈ (LIdealβ€˜π‘…)𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗})
8382biimpi 215 . . . 4 (𝑋 ∈ ran 𝑉 β†’ βˆƒπ‘™ ∈ (LIdealβ€˜π‘…)𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗})
8483ad2antlr 726 . . 3 (((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉) ∧ π‘Œ ∈ ran 𝑉) β†’ βˆƒπ‘™ ∈ (LIdealβ€˜π‘…)𝑋 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑙 βŠ† 𝑗})
8575, 84r19.29a 3160 . 2 (((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉) ∧ π‘Œ ∈ ran 𝑉) β†’ (𝑋 βˆͺ π‘Œ) ∈ ran 𝑉)
86853impa 1111 1 ((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ∧ π‘Œ ∈ ran 𝑉) β†’ (𝑋 βˆͺ π‘Œ) ∈ ran 𝑉)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074  {crab 3410  Vcvv 3448   βˆͺ cun 3913   βŠ† wss 3915   ↦ cmpt 5193  ran crn 5639  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  .rcmulr 17141  LSSumclsm 19423  mulGrpcmgp 19903  Ringcrg 19971  CRingccrg 19972  LIdealclidl 20647  RSpancrsp 20648  PrmIdealcprmidl 32247  IDLsrgcidlsrg 32282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-fz 13432  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-sca 17156  df-vsca 17157  df-ip 17158  df-tset 17159  df-ple 17160  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-minusg 18759  df-sbg 18760  df-subg 18932  df-lsm 19425  df-cmn 19571  df-mgp 19904  df-ur 19921  df-ring 19973  df-cring 19974  df-subrg 20236  df-lmod 20340  df-lss 20409  df-lsp 20449  df-sra 20649  df-rgmod 20650  df-lidl 20651  df-rsp 20652  df-prmidl 32248  df-idlsrg 32283
This theorem is referenced by:  zartopn  32496
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