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Theorem ldsysgenld 30548
Description: The intersection of all lambda-systems containing a given collection of sets 𝐴, which is called the lambda-system generated by 𝐴, is itself also a lambda-system. (Contributed by Thierry Arnoux, 16-Jun-2020.)
Hypotheses
Ref Expression
isldsys.l 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}
ldsysgenld.1 (𝜑𝑂𝑉)
ldsysgenld.2 (𝜑𝐴 ⊆ 𝒫 𝑂)
Assertion
Ref Expression
ldsysgenld (𝜑 {𝑡𝐿𝐴𝑡} ∈ 𝐿)
Distinct variable groups:   𝑦,𝑠   𝑡,𝐿   𝑂,𝑠,𝑡,𝑥   𝑥,𝑉   𝑦,𝑡   𝐴,𝑠,𝑡,𝑥   𝐿,𝑠,𝑥   𝜑,𝑡,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑠)   𝐴(𝑦)   𝐿(𝑦)   𝑂(𝑦)   𝑉(𝑦,𝑡,𝑠)

Proof of Theorem ldsysgenld
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 ldsysgenld.1 . . . . 5 (𝜑𝑂𝑉)
2 pwsiga 30518 . . . . 5 (𝑂𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
31, 2syl 17 . . . 4 (𝜑 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
4 isldsys.l . . . . . . . 8 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}
54sigaldsys 30547 . . . . . . 7 (sigAlgebra‘𝑂) ⊆ 𝐿
65, 3sseldi 3796 . . . . . 6 (𝜑 → 𝒫 𝑂𝐿)
7 ldsysgenld.2 . . . . . 6 (𝜑𝐴 ⊆ 𝒫 𝑂)
8 sseq2 3824 . . . . . . 7 (𝑡 = 𝒫 𝑂 → (𝐴𝑡𝐴 ⊆ 𝒫 𝑂))
98elrab 3559 . . . . . 6 (𝒫 𝑂 ∈ {𝑡𝐿𝐴𝑡} ↔ (𝒫 𝑂𝐿𝐴 ⊆ 𝒫 𝑂))
106, 7, 9sylanbrc 574 . . . . 5 (𝜑 → 𝒫 𝑂 ∈ {𝑡𝐿𝐴𝑡})
11 intss1 4684 . . . . 5 (𝒫 𝑂 ∈ {𝑡𝐿𝐴𝑡} → {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑂)
1210, 11syl 17 . . . 4 (𝜑 {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑂)
133, 12sselpwd 5002 . . 3 (𝜑 {𝑡𝐿𝐴𝑡} ∈ 𝒫 𝒫 𝑂)
144isldsys 30544 . . . . . . . . . 10 (𝑡𝐿 ↔ (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))))
1514simprbi 486 . . . . . . . . 9 (𝑡𝐿 → (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡)))
1615simp1d 1165 . . . . . . . 8 (𝑡𝐿 → ∅ ∈ 𝑡)
1716adantl 469 . . . . . . 7 ((𝜑𝑡𝐿) → ∅ ∈ 𝑡)
1817a1d 25 . . . . . 6 ((𝜑𝑡𝐿) → (𝐴𝑡 → ∅ ∈ 𝑡))
1918ralrimiva 3154 . . . . 5 (𝜑 → ∀𝑡𝐿 (𝐴𝑡 → ∅ ∈ 𝑡))
20 0ex 4984 . . . . . 6 ∅ ∈ V
2120elintrab 4681 . . . . 5 (∅ ∈ {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 → ∅ ∈ 𝑡))
2219, 21sylibr 225 . . . 4 (𝜑 → ∅ ∈ {𝑡𝐿𝐴𝑡})
23 nfv 2005 . . . . . . . 8 𝑡𝜑
24 nfcv 2948 . . . . . . . . 9 𝑡𝑥
25 nfrab1 3311 . . . . . . . . . 10 𝑡{𝑡𝐿𝐴𝑡}
2625nfint 4679 . . . . . . . . 9 𝑡 {𝑡𝐿𝐴𝑡}
2724, 26nfel 2961 . . . . . . . 8 𝑡 𝑥 {𝑡𝐿𝐴𝑡}
2823, 27nfan 1990 . . . . . . 7 𝑡(𝜑𝑥 {𝑡𝐿𝐴𝑡})
29 simplr 776 . . . . . . . . . 10 ((((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑡𝐿)
30 vex 3394 . . . . . . . . . . . . . . 15 𝑥 ∈ V
3130elintrab 4681 . . . . . . . . . . . . . 14 (𝑥 {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡𝑥𝑡))
3231biimpi 207 . . . . . . . . . . . . 13 (𝑥 {𝑡𝐿𝐴𝑡} → ∀𝑡𝐿 (𝐴𝑡𝑥𝑡))
3332adantl 469 . . . . . . . . . . . 12 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → ∀𝑡𝐿 (𝐴𝑡𝑥𝑡))
3433r19.21bi 3120 . . . . . . . . . . 11 (((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) → (𝐴𝑡𝑥𝑡))
3534imp 395 . . . . . . . . . 10 ((((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥𝑡)
3615simp2d 1166 . . . . . . . . . . 11 (𝑡𝐿 → ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡)
3736r19.21bi 3120 . . . . . . . . . 10 ((𝑡𝐿𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
3829, 35, 37syl2anc 575 . . . . . . . . 9 ((((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → (𝑂𝑥) ∈ 𝑡)
3938ex 399 . . . . . . . 8 (((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) → (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡))
4039ex 399 . . . . . . 7 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → (𝑡𝐿 → (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡)))
4128, 40ralrimi 3145 . . . . . 6 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → ∀𝑡𝐿 (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡))
42 difexg 5003 . . . . . . . 8 (𝑂𝑉 → (𝑂𝑥) ∈ V)
43 elintrabg 4682 . . . . . . . 8 ((𝑂𝑥) ∈ V → ((𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡)))
441, 42, 433syl 18 . . . . . . 7 (𝜑 → ((𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡)))
4544adantr 468 . . . . . 6 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → ((𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡)))
4641, 45mpbird 248 . . . . 5 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡})
4746ralrimiva 3154 . . . 4 (𝜑 → ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡})
4826nfpw 4365 . . . . . . . . . . 11 𝑡𝒫 {𝑡𝐿𝐴𝑡}
4924, 48nfel 2961 . . . . . . . . . 10 𝑡 𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}
5023, 49nfan 1990 . . . . . . . . 9 𝑡(𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡})
51 nfv 2005 . . . . . . . . 9 𝑡(𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)
5250, 51nfan 1990 . . . . . . . 8 𝑡((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦))
53 simplr 776 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑡𝐿)
54 simpr 473 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝑢 {𝑡𝐿𝐴𝑡})
55 simpllr 784 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝑡𝐿)
56 simplr 776 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝐴𝑡)
57 vex 3394 . . . . . . . . . . . . . . . . . . . 20 𝑢 ∈ V
5857elintrab 4681 . . . . . . . . . . . . . . . . . . 19 (𝑢 {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡𝑢𝑡))
5958biimpi 207 . . . . . . . . . . . . . . . . . 18 (𝑢 {𝑡𝐿𝐴𝑡} → ∀𝑡𝐿 (𝐴𝑡𝑢𝑡))
6059r19.21bi 3120 . . . . . . . . . . . . . . . . 17 ((𝑢 {𝑡𝐿𝐴𝑡} ∧ 𝑡𝐿) → (𝐴𝑡𝑢𝑡))
6160imp 395 . . . . . . . . . . . . . . . 16 (((𝑢 {𝑡𝐿𝐴𝑡} ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑢𝑡)
6254, 55, 56, 61syl21anc 857 . . . . . . . . . . . . . . 15 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝑢𝑡)
6362ex 399 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → (𝑢 {𝑡𝐿𝐴𝑡} → 𝑢𝑡))
6463ssrdv 3804 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → {𝑡𝐿𝐴𝑡} ⊆ 𝑡)
65 sspwb 5107 . . . . . . . . . . . . 13 ( {𝑡𝐿𝐴𝑡} ⊆ 𝑡 ↔ 𝒫 {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑡)
6664, 65sylib 209 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝒫 {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑡)
67 simp-4r 794 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡})
6866, 67sseldd 3799 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥 ∈ 𝒫 𝑡)
69 simpllr 784 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦))
7015simp3d 1167 . . . . . . . . . . . . 13 (𝑡𝐿 → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
7170r19.21bi 3120 . . . . . . . . . . . 12 ((𝑡𝐿𝑥 ∈ 𝒫 𝑡) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
7271imp 395 . . . . . . . . . . 11 (((𝑡𝐿𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥𝑡)
7353, 68, 69, 72syl21anc 857 . . . . . . . . . 10 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥𝑡)
7473ex 399 . . . . . . . . 9 ((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) → (𝐴𝑡 𝑥𝑡))
7574ex 399 . . . . . . . 8 (((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → (𝑡𝐿 → (𝐴𝑡 𝑥𝑡)))
7652, 75ralrimi 3145 . . . . . . 7 (((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → ∀𝑡𝐿 (𝐴𝑡 𝑥𝑡))
77 vuniex 7184 . . . . . . . 8 𝑥 ∈ V
7877elintrab 4681 . . . . . . 7 ( 𝑥 {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 𝑥𝑡))
7976, 78sylibr 225 . . . . . 6 (((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 {𝑡𝐿𝐴𝑡})
8079ex 399 . . . . 5 ((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))
8180ralrimiva 3154 . . . 4 (𝜑 → ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))
8222, 47, 813jca 1151 . . 3 (𝜑 → (∅ ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡})))
8313, 82jca 503 . 2 (𝜑 → ( {𝑡𝐿𝐴𝑡} ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))))
844isldsys 30544 . 2 ( {𝑡𝐿𝐴𝑡} ∈ 𝐿 ↔ ( {𝑡𝐿𝐴𝑡} ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))))
8583, 84sylibr 225 1 (𝜑 {𝑡𝐿𝐴𝑡} ∈ 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wral 3096  {crab 3100  Vcvv 3391  cdif 3766  wss 3769  c0 4116  𝒫 cpw 4351   cuni 4630   cint 4669  Disj wdisj 4812   class class class wbr 4844  cfv 6101  ωcom 7295  cdom 8190  sigAlgebracsiga 30495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785  ax-ac2 9570
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-er 7979  df-map 8094  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-oi 8654  df-card 9048  df-acn 9051  df-ac 9222  df-cda 9275  df-siga 30496
This theorem is referenced by:  ldgenpisys  30554  dynkin  30555
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