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Theorem zarclsun 32451
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 485 . . . . . . . 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 4124 . . . . . . 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 4265 . . . . . . . . . 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 2736 . . . . . . . . . . . . 13 (IDLsrg‘𝑅) = (IDLsrg‘𝑅)
7 eqid 2736 . . . . . . . . . . . . 13 (LIdeal‘𝑅) = (LIdeal‘𝑅)
8 eqid 2736 . . . . . . . . . . . . 13 (.r‘(IDLsrg‘𝑅)) = (.r‘(IDLsrg‘𝑅))
9 simpll 765 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
109crngringd 19977 . . . . . . . . . . . . 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 485 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑘 ∈ (LIdeal‘𝑅))
136, 7, 8, 10, 11, 12idlsrgmulrcl 32252 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ∈ (LIdeal‘𝑅))
14 sseq1 3969 . . . . . . . . . . . . . . 15 (𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘) → (𝑖𝑗 ↔ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗))
1514rabbidv 3415 . . . . . . . . . . . . . 14 (𝑖 = (𝑙(.r‘(IDLsrg‘𝑅))𝑘) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑗})
1615eqeq2d 2747 . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . 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 32253 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑙)
23 simpr 485 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑙𝑗) → 𝑙𝑗)
2422, 23sstrd 3954 . . . . . . . . . . . . . . 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 32254 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘𝑗) → (𝑙(.r‘(IDLsrg‘𝑅))𝑘) ⊆ 𝑘)
29 simpr 485 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) ∧ 𝑘𝑗) → 𝑘𝑗)
3028, 29sstrd 3954 . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) = (Base‘𝑅)
38 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3937, 7lidlss 20680 . . . . . . . . . . . . . . . . . . . 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 20680 . . . . . . . . . . . . . . . . . . . 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 32176 . . . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . 19 (RSpan‘𝑅) = (RSpan‘𝑅)
4544, 37rspssid 20693 . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . 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 32251 . . . . . . . . . . . . . . . . 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 3985 . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . 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 3954 . . . . . . . . . . . . . . 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 32214 . . . . . . . . . . . . . 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 3414 . . . . . . . . . . . 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 3583 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → ∃𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ (PrmIdeal‘𝑅) ∣ (𝑙𝑗𝑘𝑗)} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
55 fvex 6855 . . . . . . . . . . . . 13 (PrmIdeal‘𝑅) ∈ V
5655rabex 5289 . . . . . . . . . . . 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 5916 . . . . . . . . . 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 481 . . . . . . 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 5289 . . . . . . . . 9 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ∈ V
655, 64elrnmpti 5915 . . . . . . . 8 (𝑌 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
66 sseq1 3969 . . . . . . . . . . 11 (𝑖 = 𝑘 → (𝑖𝑗𝑘𝑗))
6766rabbidv 3415 . . . . . . . . . 10 (𝑖 = 𝑘 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})
6867eqeq2d 2747 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ↔ 𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}))
6968cbvrexvw 3226 . . . . . . . 8 (∃𝑖 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})
70 biid 260 . . . . . . . 8 (∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗} ↔ ∃𝑘 ∈ (LIdeal‘𝑅)𝑌 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})
7165, 69, 703bitri 296 . . . . . . 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 3159 . . . 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 5915 . . . . . 6 (𝑋 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
77 sseq1 3969 . . . . . . . . 9 (𝑖 = 𝑙 → (𝑖𝑗𝑙𝑗))
7877rabbidv 3415 . . . . . . . 8 (𝑖 = 𝑙 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗})
7978eqeq2d 2747 . . . . . . 7 (𝑖 = 𝑙 → (𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ↔ 𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗}))
8079cbvrexvw 3226 . . . . . 6 (∃𝑖 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗})
81 biid 260 . . . . . 6 (∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗} ↔ ∃𝑙 ∈ (LIdeal‘𝑅)𝑋 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑙𝑗})
8276, 80, 813bitri 296 . . . . 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 3159 . 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 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  {crab 3407  Vcvv 3445  cun 3908  wss 3910  cmpt 5188  ran crn 5634  cfv 6496  (class class class)co 7357  Basecbs 17083  .rcmulr 17134  LSSumclsm 19416  mulGrpcmgp 19896  Ringcrg 19964  CRingccrg 19965  LIdealclidl 20631  RSpancrsp 20632  PrmIdealcprmidl 32207  IDLsrgcidlsrg 32242
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-lsm 19418  df-cmn 19564  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-lmod 20324  df-lss 20393  df-lsp 20433  df-sra 20633  df-rgmod 20634  df-lidl 20635  df-rsp 20636  df-prmidl 32208  df-idlsrg 32243
This theorem is referenced by:  zartopn  32456
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